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Real-Time Estimation of Delay at Signalized Intersections

Permanent Link: http://ufdc.ufl.edu/UFE0021660/00001

Material Information

Title: Real-Time Estimation of Delay at Signalized Intersections
Physical Description: 1 online resource (381 p.)
Language: english
Creator: Buckholz, Jeffrey W
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: delay, intersection, signal, signalized, traffic
Civil and Coastal Engineering -- Dissertations, Academic -- UF
Genre: Civil Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: To evaluate improvements at signalized intersections it is important to know the resulting change in vehicular delay. However, it is difficult to collect delay data during over-saturated conditions even though this is when knowledge of delay levels is critical. Extensive peak hour queuing thwarts our ability to collect key data, such as arrivals at the back of queue. This incomplete information makes it impossible to calculate the resulting delay. The research presents a real-time procedure for estimating delay during over-saturated conditions with limited information. The procedure utilizes a series of adjustments to the measured arrival rate entering the field of view to estimate the true arrival rate at the back of the queue. An advantage of the procedure is that estimated queues and associated delay are calculated on a second-by-second basis in real time. A disadvantage is that no theoretical relationship exists between the measured arrival rate and the real arrival rate. Fortunately, it is possible to calculate a set of theoretical upper and lower bounds on the solution space by using historical minimum peak hour factors. The theoretical bounds take the form of cumulative arrival curves. Delay is obtained through consideration of the area between these arrival curves and the associated departure curve. Trajectory analysis during over-saturated conditions is used to reconcile the difference between stopped delay and the area between the curves. This research also demonstrates that the Highway Capacity Manual (HCM) definition of an initial (residual) queue is incorrect. To identify the true residual queue, the situation must be evaluated at the end of the red interval and thruput during the subsequent green interval must be deducted. Failure to do so leads to overestimation of both the initial queue and the corresponding delay. Another finding is that the random component of the HCM?s incremental delay term incorrectly contributes to delay during over-saturated periods preceded by an initial queue. A remedial modification to the d2 term is proposed. Finally, it is demonstrated that the HCM?s period-based queue accumulation procedure has drawbacks that can produce substantial errors in delay during over-saturated conditions. A remedial cycle-based counting technique is proposed.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Jeffrey W Buckholz.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Courage, Ken G.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021660:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021660/00001

Material Information

Title: Real-Time Estimation of Delay at Signalized Intersections
Physical Description: 1 online resource (381 p.)
Language: english
Creator: Buckholz, Jeffrey W
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: delay, intersection, signal, signalized, traffic
Civil and Coastal Engineering -- Dissertations, Academic -- UF
Genre: Civil Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: To evaluate improvements at signalized intersections it is important to know the resulting change in vehicular delay. However, it is difficult to collect delay data during over-saturated conditions even though this is when knowledge of delay levels is critical. Extensive peak hour queuing thwarts our ability to collect key data, such as arrivals at the back of queue. This incomplete information makes it impossible to calculate the resulting delay. The research presents a real-time procedure for estimating delay during over-saturated conditions with limited information. The procedure utilizes a series of adjustments to the measured arrival rate entering the field of view to estimate the true arrival rate at the back of the queue. An advantage of the procedure is that estimated queues and associated delay are calculated on a second-by-second basis in real time. A disadvantage is that no theoretical relationship exists between the measured arrival rate and the real arrival rate. Fortunately, it is possible to calculate a set of theoretical upper and lower bounds on the solution space by using historical minimum peak hour factors. The theoretical bounds take the form of cumulative arrival curves. Delay is obtained through consideration of the area between these arrival curves and the associated departure curve. Trajectory analysis during over-saturated conditions is used to reconcile the difference between stopped delay and the area between the curves. This research also demonstrates that the Highway Capacity Manual (HCM) definition of an initial (residual) queue is incorrect. To identify the true residual queue, the situation must be evaluated at the end of the red interval and thruput during the subsequent green interval must be deducted. Failure to do so leads to overestimation of both the initial queue and the corresponding delay. Another finding is that the random component of the HCM?s incremental delay term incorrectly contributes to delay during over-saturated periods preceded by an initial queue. A remedial modification to the d2 term is proposed. Finally, it is demonstrated that the HCM?s period-based queue accumulation procedure has drawbacks that can produce substantial errors in delay during over-saturated conditions. A remedial cycle-based counting technique is proposed.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Jeffrey W Buckholz.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Courage, Ken G.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021660:00001


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REAL-TIME ESTIMATION OF DELAY AT SIGNALIZED INTERSECTIONS


By

JEFFREY W. BUCKHOLZ
















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007



























2007 Jeffrey W. Buckholz


































To my dogs Zack and Sweet Pea,
who always provided me with free "fuzz therapy."
I wish that service was still available.









ACKNOWLEDGMENTS

Special thanks go to Mr. Seokjoo Lee for his programming assistance and to Mr. Petra

Vintu for checking the mathematical derivations









TABLE OF CONTENTS
page

A CK N O W LED G M EN T S ................................................................. ........... ............. .....

LIST OF TABLES ......... .............................................................. 8

LIST OF FIGURES ................................... .. .... .... ................. 10

A B S T R A C T ............ ................... ............................................................ 14

CHAPTER

1 INTRODUCTION AND PROBLEM STATEMENT...................................16

B background D discussion ........................... .... ...................... .. .... ........ .... ..... 16
Problem Statem ent ............................................................... .... ..... ........ 19

2 OBJECTIVES AND RESEARCH APPROACH .............. ...........................................22

3 CURREN T STATE OF THE ART ............................................... ............................. 30

Real-Time Measurement of Intersection Delay ....................................... ...............30
Vehicle Reidentification via Inductance Loops................................... ....................... 36
Perform ance of Video D election System s.................................... ........................... ......... 45
Signalized Intersection Queuing and D elay ........................................ ........ ............... 52
Probe M monitoring ................................ ............................... .... ..... ......... 65
Extending the Body of K now ledge......................................... ............... ............... 67

4 ESTIMATING NON-VISIBLE DELAY .................................... ................... ............... 68

Data Analysis Programs .............................. ....... .. .. ............. ........ 68
Prediction Algorithm for N on-Visible Delay ........................................ ...... ............... 80
Non-Visible Queue Estimation Technique........... ............................... ...............80
Non-Visible Queue Adjustment Technique: ................................ ...............82
Non-Visible Queue Re-Adjustment Technique: ............................. ...............83
E xam ples.......... ........................ ....................................... ........ ...... 84
Q ueue P reduction .............................................................................87
Stopped D elay Prediction ............................................ .. .. .... ........ ......... 88
Control D elay Prediction ................................................. ...... .............. .. 90
V ariability Considerations ................ ............................. .. .. .. ............ 91
Limitations to the Delay Prediction Procedure....................... ..... ...............92

5 THEORETICAL BOUNDS FOR DELAY ESTIMATION ...........................................129

D eriv ation of th e B ou n d s ........................... .............................................. .................... 13 1
Derivation of the Upper Bound ............................................ ............................134
D erivation of the Low er B ound......... ................... ................................ ............. 138









Analysis of Bounds Summary ......... .... ......... ....................... 146
Derivation of Delay for Upper and Lower Bounds ......................................... ..........148
Derivation of the Bounds with Visible Period 1 Queue ............................. ... ............... 166
Derivation of Upper Bound with Visible Period 1 Queue .......................................166
Derivation of Lower Bound with Visible Period 1 Queue........................................172
Analysis of Bounds Summary with Visible Period 1 Queue .....................................173
Derivation of Delay with Visible Period 1 Queue......................................... .................... 174
Derivation of the Bounds When Queue is Visible During Three Periods...........................176
Derivation of the Bounds When Analysis Time Frame is Greater Than One Hour............. 176
Derivation of the Five Period Upper Bound...... ...................... ...........177
Derivation of the Five Period Lower Bound..... ...................... ...........183
Five Period Analysis of Bounds Summary ....................................... ............... 197
Generalized Analysis of Bounds Summary ...............................................200
H historical Peak H our Factors......... ................. ......................................... ................. 203
Limitations to the Theoretical Bracketing Procedure.............................................205

6 COMPARISONS WITH VEHICLE TRAJECTORY ANALYSIS...................................228

T rajectory E x am ple ............ .. .. ........................................................ ....................2 30
Cumulative Arrival/Departure Curve Example.............................................................. 232
Reconciling the Difference Between Cumulative Curves and Trajectories.........................233
Calculating Trajectory-Based Delay Components for the BuckQ Examples.......................236
Calculating Cumulative Curve Delay for the BuckQ Examples .......................................237
Bracketing the Stopped Delay Prediction Results ............................. ... .............240

7 PERIOD ISSUES DURING OVER-SATURATED FLOW.................. ...... .............276

Simplified Example of Cycle-Period Issues in Calculating d3 ..........................................276
R esidual Q ueue D iscrepancy ............... ........ ................ .............................................281
Detailed Example of Cycle-Period Issues in Calculating d3 ..........................................283

8 CONCLUSIONS, APPLICATIONS, AND RECOMMENDATIONS.............................299

R e search F in d in g s.................................................................................................... .. 2 9 9
A application of the R esearch......... ................. ......................... ....... ... ............... 301
Example 1: Signal System Retiming Evaluation....................................... ............... 302
Example 2: Real-Time Traffic Signal Control ........................................ ............... 303
Example 3: Signalized Intersection Capacity Analysis..................................................... 304
Potential Areas of Extended Research ......... ...... ......... ........................ 304

APPENDIX

A DATA SETS FOR BUCKQ TESTING ........................................ .......................... 308

B TYPICAL PEAK HOUR FACTORS...................... .................................. ............... 331

C GENERALIZED CYCLE-PERIOD DELAY EXAMPLE: ...........................................353









R E F E R E N C E S ..................................................................................................................3 7 6

BIOGRAPHICAL SKETCH ...............................................................381









LIST OF TABLES


Table page

4-1 Example summary volume and capacity ......................... ...... ...............122

4-2 Example summary queue discharge, delay check and goodness-of-fit......................123

4-3 Queue prediction........ .......... ............ .. ... .... ...... ......... 124

4-4 Stopped delay prediction .......................................................... ..................................125

4-5 C control delay prediction ......................................................................... ................... 126

4-6 Comparison of variation in actual and predicted stopped delay ........... .....................127

4-7 P-value determination for difference in median values.............................128

6-1 Calculation of cumulative curve delay conversion factors, volume pattern
625_700_650_350vph.............................................................................. ...............249

6-2 Calculation of cumulative curve delay conversion factors, volume pattern
700_725_625_350vph ............. .............................. ...... ............................... 251

6-3 Calculation of cumulative curve delay conversion factors, volume pattern
700_700_700_350vph .............. .............................. ..... .... ..................... 253

6-4 Calculation of cumulative curve delay conversion factors, volume pattern
725_700_700_350vph ............. .............................. ...... ............................... 255

6-5 Cumulative curve delay for standard 4-period case............................................. 257

6-6 Cumulative curve delay with multiple visible periods ......................................... 258

6-7 Stopped delay prediction results for 700_725_625_350vph volume pattern ..................259

6-8 Average stopped delay prediction results for 700_725_625_350vph volume pattern ....262

6-9 Stopped delay prediction results for 700_700_700_350vph volume pattern ..................263

6-10 Average stopped delay prediction results for 700_700_700_350vph volume pattern ....266

6-11 Stopped delay prediction results for 725_700_700_350vph volume pattern ..................267

6-12 Average stopped delay prediction results for 725_700_700_350vph volume pattern ....270

6-13 Stopped delay prediction results for 625_700_650_350vph volume pattern ..................271

6-14 Average stopped delay prediction results for 625_700_650_350vph volume pattern ....274









6-15 P reduction com prison .......................................................................... .....................275

7-1 Generalized example of cycle-period delay discrepancies data ....................................292

7-2 Generalized example of cycle-period delay discrepancies summary .........................293

7-3 Detailed example of cycle-period delay discrepancies, residual queue determination ...296

7-4 Detailed example of cycle-period delay discrepancies, delay comparison....................297

7-5 Detailed example of cycle-period delay discrepancies, delay comparison with
m modified d2 term ............. ......... .. .... ............ ......................... 297

7-6 Detailed example of cycle-period delay discrepancies, delay comparison with d3
ad ju stm en t ...................................... ................................................... 2 9 7

Bl US 1 machine counts (Southern St. Johns County)...................................................334

B-2 US1 M machine counts (northern St. Johns County).........................................................339

B-3 Atlantic Boulevard m machine counts...........................................................................342

B-4 University Boulevard machine counts (Jacksonville)............................345

B-5 SR A1A machine counts (Crescent Beach) ........................................ ............... 348

B-6 SR A1A machine counts (Ponte Vedra) PDF 17 KB ............................................... 351

B -7 A ppendix B data sum m ary...................................................................... ...................352

C-1 Generalized example of cycle-period delay discrepancies data. ................................354









LIST OF FIGURES

Figure page

4 -1 Q u eu e relation ship s........ ........................................................................ ...... .............94

4-2 Signalized intersection delay components .................................................. .............. 95

4-3 M measured versus estim ated delay............................................................ .....................96

4-4 V isible and non-visible variables............................................................ .....................97

4-5 Relationship between v/c ratio and ratio of control delay to stopped delay ....................98

4-6 Re-queuing that results in simultaneous queues ..................................... .................99

4-7 Re-queuing that does not result in simultaneous queues ..............................................100

4-8 E xam ple of a blind period.. ..................................................... ........................................ 101

4-9 Exam ple of adjacent blind periods........................................................... ............... 102

4-10 C ounters and queue statu s........................................................................ .................. 103

4-11 Base case for P, C and X; stopped delay comparison......................................................104

4-12 Effect of increasing the power constant on stopped delay comparison........................... 105

4-13 Q ueue propagation exam ple ................................................. ............................... 106

4-14 A actual vehicle queues ............................................................................ ....................107

4-15 A average queue length com parison....................................................................... ...... 108

4-16 M axim um queue length com parison.......................................... .......................... 109

4-17 98th percentile back of queue comparison................................................110

4-18 V vehicle re-queuing ......... .. .................................. .. .. .. ............................ ... 111

4-19 Stopped delay com prison ......... ................................ ...................................... 112

4-20 Stopped delay prediction, 12 FOV ...... ............................................................ 113

4-21 Comparison of actual and predicted stopped delay ............. ..................................... 114

4-22 Adjacent blind period counter v. stopped delay..........................................................115

4-23 Control delay com prison .............................................................. ............. 116



10









4-24 Ratio of control delay to stopped delay .................................. .......................... ......... 117

4-25 Graphical control delay comparison, ......................................................... ...... ......... 118

4-26 C control delay estim ates.......................................................................... ......... ........... 119

4-27 C control delay com position ...................................................................... ...................120

4-28 Ratio of control delay to stopped plus move-up delay ...............................................121

5-1 Cumulative arrival-departure curves and overflow delay.................................... 207

5-2 Critical time and volume points for period 4..........................................................208

5-3 O overflow delay in period 4 ........................................... .................. ............... 209

5-4 M aximum reasonable cumulative arrival curve................................... ..................210

5-5 M minimum reasonable cum ulative arrival curve ..............................................................211

5-6 Minimum overall reasonable cumulative arrival curve ....................................... 212

5-7 Minimum reasonable cumulative arrival curve (minimum V4 for minimum Vi and
V 2) ........................................................................... ..................... 2 13

5-8 Minimum reasonable cumulative arrival curve (minimum V4 for minimum Vi) ..........214

5-9 Period 1 delay for the upper bound............................. ......................... ............... 215

5-10 Period 2 delay for the upper bound............................. ......................... ............... 216

5-11 Period 3 and period 4 delay for the upper bound..................... .................................217

5-12 Reasonable overflow delay region for 600 vph capacity and 0.75 minimum PHF .........218

5-13 Reasonable overflow delay region for 600 vph capacity and 0.80 minimum PHF .........219

5-14 Reasonable overflow delay region for 600 vph capacity and 0.85 minimum PHF .........220

5-15 Maximum delay estimation error for 0.75 minimum PHF ..............................................221

5-16 Maximum delay estimation error for 0.80 minimum PHF ...........................................222

5-17 Maximum delay estimation error for 0.85 minimum PHF ...........................................223

5-18 Maximum reasonable cumulative arrival curve with period 1 visible...........................224

5-19 Minimum reasonable cumulative arrival curve with period 1 visible ...........................225

5-20 Maximum reasonable cumulative arrival curve with 5 periods ............... .................226









5-21 Minimum reasonable cumulative arrival curve with 5 periods ......................................227

6-1 Trajectory example A) Complete chart B) Detailed view of circled area in upper
rig h t c o rn e r ...................................... ................................................... 2 4 4

6-2 Cumulative arrival-departure curve example........................................ ............... 246

6-3 Trajectory conversion of cumulative curve example........... .............................. 247

6-4 D elay and travel tim e com ponents........................................................ ............... 248

7-1 Cycle v. period initial queue delay analysis................................ ....................... 294

7-2 Cycle v. period "control delay" analysis..................................... ......................... 295

7-3 Upward bias in HCM residual queue calculation ................................. ..................... 298

A-i Queue discharge headw ay histogram ........................................ ........................... 309

A -2 Start-up lost tim e histogram ....... ......... ......... ............................... ............... 310

A-3 Comparison of control delay and stopped delay by cycle length (g/C =0.30)...............311

A-4 Comparison of control delay and stopped delay (g/C =0.30) ................. ............... 312

A-5 Comparison of control delay and stopped plus queue move-up delay by cycle length
(g /C = 0 .3 0 ) ....................................................................... .. 3 1 3

A-6 Comparison of control delay and stopped delay plus queue move-up delay (g/C
= 0.3 0) ................... ................ ...........................................3 14

A-7 Relationship between v/c ratio and stopped delay..................... ........ ............. ......... 315

A-8 Relationship between v/c ratio and stopped delay by cycle length ..............................316

A-9 Relationship between v/c ratio and stopped plus queue move-up delay .......................317

A-10 Relationship between v/c ratio and stopped plus queue move-up delay by cycle
len g th ................ ......... .................................. ............................... 18

A-11 Relationship between v/c ratio and control delay .............................. ...............319

A-12 Relationship between v/c ratio and control delay by cycle length................................320

A-13 Relationship between vehicle re-queues and control delay ..........................................321

A-14 Relationship between v/c ratio and vehicle re-queues...............................................3.22

A-15 Relationship between v/c ratio and vehicle re-queues by cycle length .........................323









A-16 Relationship between v/c ratio and cycles with phase failure ......................................324

A-17 Relationship between v/c ratio and cycles with phase failure by cycle length..............325

A-18 Percentage of cycles in 1 hour with phase failure by cycle length...............................326

A-19 Percentage of cycles in 1 hour with phase failure................................. ............... 327

A-20 Linear relationship between ABPC and stopped delay.................. ............................ 328

A-21 Exponential relationship between ABPC and stopped delay........................................329

A-22 Relationship between ABPC and control delay................................... ............... 330

B-l US 1 S. PM peak hour factor, southbound (outbound) flow .............. .............. 332

B-2 US 1 S. PM peak period factor, southbound (outbound) flow........................................ 333

B-3 US 1 N. PM peak hour factor, northbound (outbound) flow..................................... .....337

B-4 US 1 N. PM peak period factor, northbound (outbound) flow........................................338

B-5 Atlantic Boulevard PM peak hour factor, eastbound (outbound) flow............................ 340

B-6 Atlantic Boulevard PM peak period factor, eastbound (outbound) flow......................... 341

B-7 University Blvd. PM peak hour factor, northbound (outbound) flow ............................. 343

B-8 University Blvd. PM peak period factor, northbound (outbound) flow .......................... 344

B-9 SR A1A S. PM peak hour factor, southbound (outbound) flow..............................346

B-10 SR A1A S. PM peak period factor, southbound (outbound) flow..............................347

B-11 SR A1A N. PM peak hour factor, southbound (outbound) flow ..............................349

B-12 SR A1A N. PM peak period factor, southbound (outbound) flow .............................350









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

REAL-TIME ESTIMATION OF DELAY AT SIGNALIZED INTERSECTIONS

By

Jeffrey W. Buckholz

December 2007

Chair: Ken Courage
Major: Civil and Coastal Engineering

To evaluate improvements at signalized intersections it is important to know the resulting

change in vehicular delay. However, it is difficult to collect delay data during over-saturated

conditions even though this is when knowledge of delay levels is critical. Extensive peak hour

queuing thwarts our ability to collect key data, such as arrivals at the back of queue. This

incomplete information makes it impossible to calculate the resulting delay.

The research presents a real-time procedure for estimating delay during over-saturated

conditions with limited information. The procedure utilizes a series of adjustments to the

measured arrival rate entering the field of view to estimate the true arrival rate at the back of the

queue. An advantage of the procedure is that estimated queues and associated delay are

calculated on a second-by-second basis in real time. A disadvantage is that no theoretical

relationship exists between the measured arrival rate and the real arrival rate.

Fortunately, it is possible to calculate a set of theoretical upper and lower bounds on the

solution space by using historical minimum peak hour factors. The theoretical bounds take the

form of cumulative arrival curves. Delay is obtained through consideration of the area between

these arrival curves and the associated departure curve. Trajectory analysis during over-









saturated conditions is used to reconcile the difference between stopped delay and the area

between the curves.

This research also demonstrates that the Highway Capacity Manual (HCM) definition of

an initial (residual) queue is incorrect. To identify the true residual queue, the situation must be

evaluated at the end of the red interval and thruput during the subsequent green interval must be

deducted. Failure to do so leads to overestimation of both the initial queue and the

corresponding delay.

Another finding is that the random component of the HCM's incremental delay term

incorrectly contributes to delay during over-saturated periods preceded by an initial queue. A

remedial modification to the d2 term is proposed.

Finally, it is demonstrated that the HCM's period-based queue accumulation procedure

has drawbacks that can produce substantial errors in delay during over-saturated conditions. A

remedial cycle-based counting technique is proposed.









CHAPTER 1
INTRODUCTION AND PROBLEM STATEMENT

Since the efficient operation of signalized intersections is a pertinent topic throughout the

world, providing a real-time evaluation system that allows such intersections to be operated at

maximum efficiency has the potential for tremendous benefit. Reductions in travel time would

be the primary benefit, along with associated reductions in fuel usage and vehicle emissions.

The benefits would accrue "24/7" in that signalized intersections function around the clock. In

the United States alone there are approximately 265,000 signalized intersections and the delays

at these signalized intersections contribute an estimated 25% to total highway system delay [1].

Background Discussion

To properly evaluate improvements made at a signalized intersection it is important to

know the resulting change in various Measures of Effectiveness (MOEs), including what may be

the most important MOE, vehicular delay. Delay is a particularly attractive measure of

effectiveness because, as discussed by Hurdle [2], it can: "be measured; it has obvious economic

worth; and it is easily understood by both technical and non-technical people." As recognized by

Dowling [3], many MOEs (such as queue length, speed, stops, and density) are relatively

invariant during highly over-saturated conditions where little vehicle movement occurs. Delay,

on the other hand, continues to increase under such conditions, which is a highly desirable trait.

The benefit of corridor re-timing programs, signal phasing changes, and intersection

geometric improvements can be properly evaluated only if a realistic assessment of the change in

overall vehicular delay is determined. Collecting delay data by hand, as described in Chapter 16,

Appendix A of the 2000 Highway Capacity Manual [4] is a labor-intensive task that must, by

practical necessity, be limited to brief data collection periods. As Saito, et al. [5] put it:

Manual field observations require large number of personnel and large amounts of other
resources if delay estimates must be done frequently, such is the case if delay estimates are









needed for Advanced Traffic Management Systems (ATMS's). The method is meant for
occasional checks of delays at signalized intersections; it is not meant for continuous
monitoring of the LOS (level of service) of signalized intersections. A more advantageous
method would be to create automated methods of estimating delay from direct observation
of queued vehicles. This significantly reduces the amount of data that needs to be
collected and (eliminates) unnecessary assumptions. When such methods work, they allow
traffic engineers to continuously monitor the LOS at intersections and estimate the arterial
LOS ...

In addition, it is particularly difficult to collect delay data during over-saturated conditions

even though this is exactly when knowledge of delay levels is most critical. Consequently, under

congested conditions, delay calculations that are based on manual information can be considered

both piecemeal and of dubious accuracy. As Engelbrecht, et al. [6] explain

From a practical point of view it is very difficult to accurately measure over-saturation
delay in the field. Long queues and restricted sight distance may make the actual counting
of queued vehicles impossible. Also, counting a large number of vehicles in a short 10-
second interval may be very difficult. Furthermore, not all vehicles in the queue may be
stationary at a single point in time, as internal shock waves due to the stopping and starting
of traffic at the stop line may travel through the queue continuously. Because of the
presence of non-stationary vehicles in the queue, transformation of the measured stopped
delay into the overall delay predicted by most of the delay equations may be the most
difficult task of all.

A properly automated method for collecting delay data, either on a cycle-by-cycle basis or

on a periodic basis, could provide the needed evaluation data for all pertinent periods. Such a

system would also provide reasonable estimations of delay, even during over-saturated

conditions. Resulting delay data could then be used for project evaluation or for real-time

modification of controller settings.

Using real-time delay obtained from intersection-based field measurements for project

evaluation purposes (such as signal retiming evaluation) provides an important supplement to

traditional before and after travel time runs, which completely ignore the delay experienced by

side street motorists or main street left turn motorists. A rather large leap forward in project









evaluation could be taken if we are able to develop a widely applicable, robust procedure for

calculating vehicular delay on the fly.

Video detection systems, vehicle re-identification systems using inductance loops, and

probe monitoring all offer the potential of being able to calculate (or reasonably estimate)

vehicular delay in real time.

Unfortunately, direct measurement of stopped delay via video detection or inductance

loops falls prey to a number of practical limitations, ranging from detection inaccuracies to field

of view limitations. The accuracy of any intersection-based delay measurement system is

essentially limited by the detection technology available at the approaches under study. For

example, if an intersection approach has video detection oriented to "see" from the stop bar to a

point far upstream (the best case scenario) then the resulting estimation of delay can be expected

to be relatively good whereas, if the approach only has a stop bar loop (other than no detection,

the worst case scenario), then the delay estimation will be relatively poor.

In addition, the accurate estimation of approach delay is of most interest during peak

periods when traffic demand is at its greatest. It is during these critical periods that extensive

queues typically form; queues that can extend well beyond the field of view of any intersection-

based detection system. Consequently, when we most need an accurate estimation of approach

delay is exactly when we are least likely to obtain it from conventional detection systems.

Theoretical delay models for signalized intersection approaches, such as those described in

the Highway Capacity Manual (HCM), offer another means of determining delay. One would

expect that these models could be used in a real-time manner to obtain real-time delay results.

However, to produce reasonable results the models must be based on reasonably accurate input

data. If this needed data cannot be accurately obtained, then the models are of little value. This









brings us right back to the problems associated with obtaining accurate data under peak hour

conditions. Extensive peak hour queuing essentially thwarts our ability to collect key approach

data, such as the rate of vehicle arrivals at the back of the queue.

The use of probe vehicles provides a fresh alternative for collecting delay data. However,

a host of challenging technical and privacy issues still need to be worked-out before probe

vehicles can provide the needed detail to accurately estimate approach delay. On the technical

side, a team of researchers in Florida recently discovered that cell phone technology, a promising

probe alternative, is not accurate in congested traffic conditions and that the level of accuracy

decreases rapidly as congestion increases.

Problem Statement

The latest edition of the Highway Capacity Manual provides a well-recognized analytical

procedure for calculating control delay at signalized intersections, with control delay being

defined as the sum of deceleration delay, stopped delay, queue move-up delay, and acceleration

delay. This procedure has been automated in the form of the signalized intersection module of

the HCS+ software suite. The HCS+ software offers a direct, user-friendly procedure for

calculating lane group, approach, and intersection control delay and their associated levels of

service. However, the HCM methodology assumes that, on a given approach, certain average

conditions apply over the entire analysis period (saturation flow rate, start-up lost time, g/C ratio,

arrival type) and that the vehicle arrival rate on the approach remains constant within each of the

four 15-minute periods. In reality, conditions change on a cycle-by-cycle basis depending on

random fluctuations in approach volumes and driver composition. For example, the considerable

variation in cycle-by-cycle saturation flow rates at signalized intersections was documented in

two recent papers, one citing data from the United States [7] and one citing data from Taiwan

[8].









In addition to this cycle-by-cycle variation in conditions on a given approach, variations

also occur between different approaches due to unique characteristics of the approach. For this

reason, the HCM recommends collecting field data to establish such items as ideal saturation

flow rate. The HCM recognizes that true site-specific delay can only be evaluated accurately by

field measurement. Unfortunately, the field measurement of delay requires knowledge of the

entire extent of the queue, and survey techniques required to capture the entire extent of the

queue must utilize costly resources such as aerial surveillance or multiple coordinated ground

observers. Less expensive observation techniques, such as a video camera located at a single

point, can estimate delay only if the back of the queue is always in sight, which is typically not

the case when peak hour congestion occurs.

Recognizing these limitations, a new procedure is needed that can reasonably estimate

delay over a wide variety of conditions, including grossly over-saturated conditions. In order to

properly measure delay during over-saturated conditions, multi-period analysis becomes a must

in order to ensure that that no initial queues exist either at the start or at the end of the analysis.

Keeping track of the various components of control delay (stopped delay, move-up delay,

acceleration delay prior to the stop line, acceleration delay beyond the stop line, and deceleration

delay) becomes more difficult as volume exceeds capacity for any significant length of time.

Predicting control delay in real-time with limited information, and being able to do so even with

over-saturated conditions, is the challenge addressed in the research at hand.

Key to this problem statement is the idea of limited information. Obviously, if we have

perfect knowledge of each and every vehicle trajectory then we can rather easily compute a

complete set of arrival rates, departure rates, queue lengths, and the resulting control delay.

However, detailed vehicle trajectory information can be very difficult to obtain and trying to









secure it for more than a few locations quickly becomes cost-prohibitive given current

technology. The crux of the problem is to find a method that uses more easily obtainable data to

approximate the same delay information that a complete set of accurate vehicle trajectories

would produce. The most easily obtainable data are usually data that occurs in proximity to the

stop line. Current vehicle detection systems, including most video and inductance loop systems,

are best suited to obtaining data at this location. The quest is to develop a practical, real-time

delay estimation system that is supported by theoretical considerations and which also makes use

of readily obtainable data.









CHAPTER 2
OBJECTIVES AND RESEARCH APPROACH

The following objectives were established for the research.

OBJECTIVE 1: Develop a methodology and associated real-time procedure that can reasonably
estimate delay associated with vehicles that are beyond the reach of the detection system. The
procedure should function during both under-saturated and over-saturated, obtaining reasonable
estimates of vehicular delay even when queues are long and multiple phase failures occur.

OBJECTIVE 2: Identify variables to be used in the procedure that are important in the
prediction of delay beyond the detection area (non-visible delay).

OBJECTIVE 3: Establish and clearly define any new terminology needed to document the
methodology.

OBJECTIVE 4: If the proposed procedure is empirical in nature, develop theoretical limits on
the solution space that can be established using readily available information.

OBJECTIVE 5: Ensure that all delay estimates are consistent with trajectory analysis and
reflect the true nature of control delay.

OBJECTIVE 6: Ensure that all delay estimates are reconciled to the procedures contained in the
2000 Highway Capacity Manual and the current version of the HCS+ software. Document any
needed modifications to the manual or the software based on the research.

OBJECTIVE 7: Provide examples of how the procedure could be used to address real-world
traffic analysis or traffic control issues.

OBJECTIVE 8: Indicate areas of future research.

Objectives of the research would best be achieved using actual field data. However,

detailed field data are not only expensive and time consuming to collect; one cannot safely or

expeditiously manipulate field data in order to experiment at controlled volume levels or cycle

lengths. Analyzing substantially over-saturated systems is also very difficult using actual field

data as queue lengths can become quite extensive; spilling over into adjacent signalized

intersections.

Therefore, theoretical research work was conducted in the laboratory using the CORSIM

micro-simulation model. CORSIM allows us to quickly simulate a variety of real-world

conditions in a relatively realistic manner and to accumulate important measures of









effectiveness, including delay. CORSIM was used because it is a well-accepted and well-

understood model that has the capability to accommodate a wide range of input variables,

including variable combinations that produce grossly over-saturated conditions with multiple

phase failures. CORSIM also allows the user to vary the set of random number seeds to order to

investigate changes in the results that occur due to random fluctuations. This ability is important

since the stochastic nature of micro-simulation models can result in a level of variation that

masks cause-and-effect relationships.

CORSIM was specifically used to examine how measured delay differs from actual delay

when queues exceed the limits of the detection system. In order to investigate such differences,

it was necessary to assume a certain "field of view" for the simulation runs. The field of view is

defined as the number of vehicles on an intersection approach lane that can be accurately

measured by the detection system when the vehicles are queued at the stop bar. A field of view

of 12 vehicles was used in most of the examples associated with this theoretical work. This

would be a reasonable field of view for a modem video detection system.

Using various fields of view and cycle lengths, a reasonably accurate method for

estimating actual stopped delay was developed. For example, the back-of-queue on a single lane

approach might extend to 20 vehicles whereas a video detection system may only be able to

accurately "see" a queue extent of 12 vehicles. If this happens, the delay associated with the

remaining 8 vehicles (the vehicles queued in the "blind" area) cannot be measured and must

instead be estimated in some reasonably accurate manner. Knowing the time during which a

queue existed in the "blind" area, which may extend over multiple cycles, and knowing the

number of vehicles that "come into sight" after such a period of blind queuing, the procedures

developed in this endeavor allow us to obtain a workable estimate of the "non-visible delay" that









occurred. The procedure developed is capable of handling both under-saturated conditions

(having little or no "blindness") and over-saturated conditions (with blind periods occurring over

multiple cycles; referred to in this document as adjacent blind periods). The development of this

procedure is one of the primary contributions to the literature dealing with signalized intersection

delay.

A limited field of view produces a situation where arrivals at the back of the queue cannot

be observed. This incomplete information makes it impossible to calculate the resulting delay.

However, using the methodology contained in this dissertation, the delay can be reasonably

estimated under a rather wide variety of conditions. The procedure that was developed in

response to the challenge of estimating non-visible delay begins by calculating an "estimated

arrival rate" (which is actually the departure rate). If the back end of the queue is not visible, the

procedure modifies the estimated arrival rate upward using a power function in an attempt to

predict the real arrival rate. This power function adjusts the rate in a manner that varies with the

amount of time during which the back end of the queue is not visible. A major advantage of this

approach is that the resulting estimated queues and associated delay are immediately calculated

on a second-by-second basis, in real time. A major disadvantage of the approach is that there is

no theoretical relationship between the departure rate and the real arrival rate. Hence, two

different arrival patterns that result in the same number of vehicles crossing the stop line during

the analysis period can produce similar delay results. This problem is most evident when the

length of time that the end of the queue is not visible covers most of the analysis period.

Fortunately, it is possible to calculate a set of theoretical upper and lower bounds on the

solution space by using information obtained at the end of the analysis period, when all queues

are visible and the arrival rate equals the departure rate. In order to make any type of reasonable









delay estimation, all queues must dissipate prior to the end of the analysis period. Once queues

become fully visible, an accurate calculation of the arrival rate can be made. Knowing this

arrival/departure rate and knowing the total number of vehicles that have crossed the stop line

during the entire hour we can, by assuming a reasonable minimum peak hour factor, work

backwards through the period to identify minimum and maximum cumulative arrival curves.

From these curves we can then calculate both lower and upper bounds on the overflow delay.

These theoretical bounds can be used, in an ex post facto manner, to bracket the previously

discussed real-time delay estimation procedure. They can also be used to identify an

independent "most probable" arrival pattern by selecting an intermediate curve between the

upper and lower bounds that minimizes the maximum percent error between the estimate and the

actual delay. The development of these theoretical bounds is another important contribution to

the literature dealing with signalized intersection delay.

The theoretical upper and lower bounds on the delay solution are calculated using

cumulative arrival and departure curves. Vehicular delay is obtained through consideration of

the area between these curves. Within this document it is demonstrated that, contrary to popular

belief, the area between the arrival and departure curves is not the delay incurred by approaching

vehicles. An evaluation of trajectory analysis during over-saturated conditions is used to

reconcile the difference between the true delay and the area between the cumulative arrival and

cumulative departure curves so that a consistent set of upper and lower bounds are provided.

This reconciliation is another contribution to the literature dealing with signalized intersection

delay.

The multi-period signalized intersection analysis procedure that is currently contained in

the 2000 Highway Capacity Manual is codified as part of the HCS+ version 5.21 software suite.









The period-based procedure for queue accumulation that is described in this manual has certain

drawbacks that can produce substantial errors when calculating control delay during over-

saturated conditions. A description of these errors and the presentation of a cycle-based

technique for eliminating them is yet another contribution to the literature dealing with

signalized intersection delay.

The following detailed work tasks were developed in order to carry out this research

approach:

TASK 1: Select a micro-simulation model for conducting the research and develop tools to
extract needed information from the model.

TASK 2: Develop a comprehensive software tool that will facility the evaluation of real-time
second-by-second delay estimation procedures for a one-hour analysis timeframe.

TASK 3: Develop data test sets for use in identifying the preferred delay estimation procedure.
Various v/c ratios, cycle lengths, and fields of view should be reflected in this test set.

TASK 4: Using the test sets, identify the preferred delay estimation procedure.

TASK 5: Use the delay estimation procedure to analyze multiple replicates of four examples
and document the results

TASK 6: Examine statistical variability issues by using a large number of replicates of a single
example.

The first 6 tasks are documented in Chapter 4.

TASK 7: If the delay estimation procedure is empirical in nature, develop a theoretical
technique for constraining the solution space.

Task 7 is documented in Chapter 5.

TASK 8: Develop a software tool for extracting trajectory information from the selected micro-
simulation model.

TASK 9: Develop a software tool that will analyze all components of control delay associated
with vehicle trajectories. The tool should summarize the resulting delay by 15-minute period for
a one-hour analysis timeframe.

TASK 10: If necessary, modify the delay estimation procedure or the theoretical constraints to
reflect true control delay concepts.









Tasks 8 through 10 are documented in Chapter 6.

TASK 11: Compare the results obtained with results produced by the 2000 Highway Capacity
Manual and reconcile all differences.

Task 11 is documented in Chapter 7

TASK 12: Summarize the results and identify potential areas for further research.

Task 12 is documented in Chapter 8.

The end result of this research is the development of a theoretically constrained delay

estimation procedure that is based on limited information. The delay estimation procedure

makes use of available data to predict arrivals at the back of the non-visible queue as well as

departures from the front of the non-visible queue at each point in time, information that would

otherwise be unknown. Knowing the arrivals and departures we can predict the length of the

non-visible queue at each point in time. This predicted non-visible queue length is then added to

the measured visible queue length to obtain the total queue length with stopped delay being

obtained directly from the queue length. Theoretical bounds based on historical minimum peak

hour factors are then imposed on the delay estimate to ensure a reasonable result.

Use of the procedure to estimate control delay on an over-saturated intersection approach for

a one-hour analysis time frame would proceed as follows:

1. Using the vehicle detection equipment for the approach of interest, real-time second-
by-second data are collected on the number of vehicles crossing the stop bar, the
number of vehicles entering the field of view, the length of the visible queue, and the
presence or absence of a stationary vehicle in the last queue position of the field of
view.

2. This data set is entered into the delay estimation software, which measures the length
of the visible queue and estimates the length of the non-visible queue at every second
of the one-hour analysis time frame. Second-by-second cumulative stopped delay is
then calculated using this queue information.

3. The stopped delay prediction is converted to control delay using a series of
conversion ratios that vary by cycle length and v/c ratio. The conversion ratio varies









between 1.2 and 1.4 with 1.3 being a typical value. The predicted control delay is
considered the final control delay for use in real-time traffic control.

4. The time during the last 15-minute period at which the end of the queue becomes
visible is recorded, as is the cumulative number of vehicles that have crossed the stop
bar at that time. At the end of the one-hour analysis time frame, the cumulative
number of vehicles that have crossed the stop bar is also recorded. This information
is used to calculate the arrival rate during the last 15-minute period.

5. The minimum reasonable Peak Hour Factor (PHF) for the approach and time period
in question is obtained from historical traffic counts. The analysis software
constructs a theoretical set of minimum and maximum cumulative arrival curves
using this minimum PHF and the calculated arrival rate during the last 15-minute
period.

6. The analysis software then calculates the cumulative curve delay (overflow delay)
associated with the minimum and maximum cumulative arrival curves.

7. The cumulative curve delay is then converted to stopped delay by the application of a
correction factor (approximately 0.77) derived from trajectory analysis.

8. The corrected maximum theoretical stopped delay is used as an upper bound for the
predicted stopped delay and the corrected minimum theoretical stopped delay is used
as a lower bound. If the predicted stopped delay falls outside of the theoretical
bounds during any of the four 15-minute periods, then the predicted delay is
appropriately adjusted to remain within the bounds. The resulting "hybrid" stopped
delay is considered the final stopped delay prediction. Note that the theoretical
bracketing of the predicted stopped delay is carried-out in an ex post facto manner,
after the analysis time frame has expired.

9. The hybrid stopped delay results are converted to control delay using a series of
conversion ratios that vary by cycle length and v/c ratio. The conversion ratio varies
between 1.2 and 1.4 with 1.3 being a typical value. The hybrid control delay is
considered the final control delay prediction for project evaluation purposes.

By using the maximum amount of information available and by recognizing the true

characteristics of overflow delay, this procedure produces, for over-saturated conditions, a delay

estimate that is generally superior to that found in the Highway Capacity Manual and does so

in real time.. The proposed delay estimation technique should prove useful for both real-time

traffic control and project evaluation. It is envisioned that the eventual end product of this

theoretical research will be a self-contained delay estimation module that could be attached to









either a closed-loop or centralized signal control system, or could be inserted within the software

of a local traffic signal controller.









CHAPTER 3
CURRENT STATE OF THE ART

A literature review was conducted to identify both past and ongoing research efforts

affecting the area of interest. The studies obtained from this search can be segregated into the

following general areas: Real Time Measurement of Intersection Delay, Vehicle Re-

identification via Inductance Loops, Performance of Video Detection Systems, Signalized

Intersection Queuing and Delay, and Probe Vehicle Monitoring. Quite a bit is known about

intersection control delay, especially for under-saturated conditions and for situations where all

of the information needed to calculate delay is known. The current state of knowledge with

respect to over-saturated conditions is more primitive and the results less tested.

Real-Time Measurement of Intersection Delay

In 1994, Maddula [9] studied signalized intersection delay using an AUTOSCOPE 2003

video detection system. This system is based on a tripwire approach and has count, presence and

speed detectors. The system can provide interval data (from 10 seconds to 1 hour) and event

data. The computational model developed makes use of a mandatory detection pattern that has 4

detectors in each lane. The first upstream detector (position 1) is located "as far upstream as

possible such that section length includes all delay associated with the signal" and identifies the

beginning of the Approach Delay Section (defined as the section where most, or all, of the

approach delay is incurred) and reports arrival events. Position 2 is an additional upstream

detector located between position 1 and the stop line. This detector accounts for vehicles

changing lanes. It is used to estimate any missing data at other positions. Position 3 is at the

stop line and defines the end of the approach delay section and reports departure events. Position

4 is beyond the stop bar and is used to determine the signal indication. Position 4 houses a

directional detector.









The first step is the identification of each event in their chronological order. This step

includes the removal of all events that lead to unrealistic headways (FILTER I).

The second step in the process is the use of the data from detector positions 3 and 4 to

determine the signal status associated with every recorded event. The following user input is

required to conduct the search: 1) beginning of red indication for first cycle, 2) limits of travel

time between positions 3 and 4, and 3) limits of red indication for the phase. Each event is

associated with a signal indication (red or green) and a cycle number. This step includes the

removal of all events that lead to departures when there is no right-of-way (FILTER II).

The third and final step is the computation of the MOEs (throughput, stops, saturation

headways, and saturation flow rate). Volume is computed from throughput and the estimated

green time is treated as effective green time. Delay is then calculated using the 1985 HCM

formula and LOS is identified via the HCM signalized intersection LOS table. The calculations

are done using a computer program written in C called ADELAY. The inputs to ADELAY are

an ASCII detection file from the video system with extension TXT (the events) and a text file

with extension VXT (other required information) from the VIADET user interface program.

The report defines the Approach Free Flow Time as the time used by an unimpeded

vehicle to traverse the approach delay section and defines the Approach Time as the time used

by an impeded vehicle to traverse the approach delay section. Approach Delay (defined as the

Approach Time minus the Approach Free Flow Time) is converted to Stopped Delay (defined as

the time that the vehicle is stopped with stationary wheels) for comparison to field observations

by dividing by a factor of 1.3 The raw data are converted to usable data using three filters:


* FILTER I. False detections (glare, reflections, turn signals) resulting in unrealistic headways
(1 second is used as a minimum realistic headway)









* FILTER II. Detections at position 3 that lead to departures when there is no right of way
(detections during red produced by pedestrians, crossing vehicles, etc.)

* FILTER III. Unrealistically high throughput (continuous detection due to shadows, turn
signals) Maximum Throughput = Green Time / Minimum Headway

All vehicles that arrive on the approach delay section and depart before the end of the

green of the current cycle are reported as throughput for the cycle. If a vehicle could not clear

the intersection before the end of the green, it is reported as throughput for the next cycle. When

the throughput reported at various positions in the lane is different (due to lane changing or

detection errors), the maximum number of vehicles reported at any position is taken as the

throughput for the cycle.

Every vehicle that arrives before the beginning of the green indication, minus the free flow

travel time within a current cycle, is automatically treated as a stop. The free flow travel time for

the vehicles that arrive after the stated time is calculated at 5 miles per hour (mph). If the travel

time of the vehicle is more than this time, it is treated as a stop for that vehicle. (i.e. a vehicle is

defined to have stopped if the actual travel time is more then the free flow travel time calculated

at a speed of 5 mph.)

Reported departure times are used for determining saturation headways and calculating the

saturation flow rate. Headways associated with the first 3 vehicles in the queue, and headways

of more than 3 seconds, are not used. If the number of vehicles in the queue never exceeds three

throughout the study, then default saturation flow rates are used that vary by lane type (1756 for

a thru lane, 1946 for a single left turn lane, and 1651 for a dual left turn lane).

A preliminary study for a limited number of observations indicated that, for queues of

passenger cars, average distance headway (front bumper to front bumper) is 25.1 feet and

average spacing between cars is 9.0 feet. This yields an average car length of 16.1 feet.









The report defines the Time-in-Queue Delay (a.k.a. Time-in-Queue) as the time from the

vehicle's first stop to the vehicle's exit across the stop line. The report also defines Percentage

of Vehicles Stopping as the number of vehicles incurring Stopped Delay divided by the number

of vehicles crossing the stop line.

Of the traffic parameters investigated, vehicle count, delay and level of service were

obtained accurately from the data reported by VIDS (Video Image Detection System). However,

throughput and stops were not. Minor changes in detector size, placement and orientation caused

noticeable variation in the results. Data missing at a particular detector location was often

available at another detector location, which argues for the use of multiple detection systems for

evaluation.

The basic limitation of this work with respect to the research at hand is that it relied on a

relatively optimum detection configuration and was not used for estimating delay during over-

saturated conditions (a time when delay estimation is most critical).

In 1998, Lall, et al. [10], developed a speed-based procedure for calculating delay on a

signalized intersection approach. For a 15 minute study period, traffic volumes and average

speeds were recorded every 10 seconds using AUTOSCOPE at 5 distances from the stop bar (20

ft, 65 ft, 88 ft, 267 ft & 500 ft). Free-flow speeds (for vehicles not stopping) and "prevailing

speeds" (for vehicles stopping) were calculated and associated travel times compared to estimate

delay. The comparison checked well with "control delay" calculated for the approach using the

HCM. If posted speed is used instead of prevailing speed the delay calculated is substantially

higher and probably corresponds to "total delay", wherein total delay is defined as the difference









between the travel time actually experienced and the reference travel time that would result

during ideal conditions. 1

The authors noted that the longer the lens' focal length (view more zoomed in), the easier

and more robust is vehicle tracking and detection. The shorter the focal length of the lens, the

smaller the objects are on the image, but the larger the field of view. If the vehicle image is

smaller than 5 pixels of the image that is analyzed by their video system, the tracking of vehicles

becomes rather unreliable.

Two types of shadow problems were revealed. The first problem occurs when a tree, tall

building or some other tall object is close to the section of roadway being monitored. On sunny

days, the object's shadow will cover the monitored roadway at certain times of the day. If a

vehicle enters the shadow, it may become barely visible, especially if the vehicle is dark. If a

detection zone is located in the area covered by the shadow, the detection performance from this

zone may be seriously impaired.

A second type of shadow problem occurs due to vehicle shadows. A shadow of a moving

vehicle in one lane may sweep over the detection zone in another lane. This sweeping shadow

may be taken for a vehicle. The authors "solved" the problem of thru lane vehicles activating

left turn lane detection through the use of a 1.2 second detector delay setting (for a 6 foot

detector length).

However, experience with this site indicates that the accuracy of video detection is

adequate (the average maximum error is only about 5%). It is better than the accuracy of loop

detectors at this location, which gave a maximum error rate of 10%.



1 The important delay calculations contained in Tables 2 and 3 of this report cannot be followed
given the information contained in the report and I contacted the primary author for clarification.
Unfortunately, the author did not provide a response.









In 1999, Quiroga, et al. [11], developed a procedure based on linearly referenced GPS data

that can be used to accurately measure both control delay and stopped delay. Algorithms were

developed which accurately detect when a GPS-equipped probe vehicle either begins or ends

acceleration or deceleration. More than 100 floating car travel time runs were made along two

coordinated corridors having a background cycle length of 150 seconds. In addition to

establishing the viability of this procedure for accurately determining stopped delay and control

delay, the following was discovered:

1. A linear relationship exists between stopped delay and control delay. However, the line
does not pass through the origin. It was found that control delay = (stopped delay + 19.3
seconds) x 1.04, which is quite different than the control delay = 1.3 x stopped delay
formulation provided in the Highway Capacity Manual. The authors caution that other
independent variables, such as length of the red interval, may be needed to properly
generalize this equation.

2. An average end-acceleration distance of 427 feet downstream of the stop bar was
established. An average begin-deceleration distance of 951 feet upstream of the stop bar
was also established, but this distance obviously depends on the extent of queuing at the
intersections.

3. Approximately 5% of the intersection control delay occurred after the vehicle crossed the
stop bar.

In 2001, Saito, et al. [5], estimated stopped delay using simulated vehicle images generated

by CORSIM and two image analysis methods: the gap method and the motion method. A

simulation duration of 15 minutes was used. The simple algorithms that were developed

produced promising results. The authors defined Percent Deviation using the following formula:

Percent Deviation = [Delay Estimated by Model Delay Estimated by CORSIM]/(Delay

Estimated by CORSIM) x 100

In 2004, Zheng, et al. [12], developed a methodology for using video image processing to

accurately detect queue lengths and phase failures on a signalized intersection approach. A

Trafcon video system was used to test the procedure on an actual intersection approach with a









field of view of about 18 vehicles. The camera was mounted 26 feet above the ground and was

oriented at a 30-degree downward angle. The video algorithm extracts stopped vehicle

information from the traffic stream, tracks the end of the queue, and identifies phase failures.

Zheng concludes that:

"The program based on this algorithm may provide reliable and accurate [phase] failure
detections in real time for many traffic management and operation purposes if the camera
that provides the video stream is correctly positioned to see the stop bar and a sufficient
number of queued vehicles".

We can safely assume that, if the camera cannot see a sufficient number of queued vehicles

(with a "sufficient number" obviously being to the end of the queue) then Zheng's technique will

provide erroneous results; hence, the need for the extension provided in this research.

In 2004, Hoeschen, et al. [13], developed a procedure for using travel time between

intersections (expressed as "segment delay") to approximate control delay. The approximation

was found to be much better than using stopped delay to estimate control delay, especially for

higher delay values. Control delay was approximated by subtracting mid-block delay from

segment delay. The authors cautioned that queue spillback from a downstream intersection or

non-recurring delay could negatively affect the results. The segment lengths for the research

varied between 14 mile and 1 mile in length. 300 feet was selected as the distance from the

upstream intersection at which most vehicles had accelerated to running speed. 300 feet was also

selected as the distance from the downstream intersection at which vehicles began decelerating.

Vehicle Reidentification via Inductance Loops

In 1999, Sun, et al. [14], examined the vehicle re-identification problem on freeways. A

vehicle waveform pair can be formed by using one downstream waveform and one upstream

waveform. The vehicle re-identification problem is to find the matching upstream vehicle from a

set of upstream vehicle candidates given a downstream vehicle.









Inductive loop detector manufacturers are incorporating the ability to monitor and output

vehicle inductance values (or waveforms). Detectors that output vehicle waveforms include

detectors manufactured by: Peek/Sarasota, Intersection Development Corporation (IDC), and

3M.

The authors concluded that solution of the vehicle re-identification problem has the

potential to yield reliable section measures such as travel times and densities. Implementation of

their approach used conventional surveillance infrastructure; 6' by 6' freeway inductive loops

spaced 1.2 miles apart on a 4 lane westbound stretch of freeway with no intervening ramps.

Typical 6' x 6' loops produce a less distinctive waveform that is more difficult to re-identify

compared with shorter (3.3') European loops. The 13 to 14 ms detector sampling period of most

detectors is also problematic in that it misses sharp corners of the waveform.

Previous approaches that utilized sequences (Bohnke and Pfannersstill, 1986) are suitable

for the case when sequences of vehicles are preserved from upstream to downstream. The

preservation of sequences occurs when there is very little lane changing and the speeds across all

traffic lanes are similar. The approach used in this study is suitable for cases where there is

significant difference in lane speeds. This approach also has the potential to yield partial

origin/destination demands and individual lane changing information.

This paper formulates and solves the vehicle re-identification problem as a lexicographic

optimization problem using goal programming. Goal Programming is an optimization method

wherein target values are set for each of the multiple objectives and then a single global

objective, which is the sum of the deviations from the target values over all objectives, is

optimized. Lexicographical Goal Programming is a goal programming procedure wherein the

multiple objectives are introduced in a specified hierarchical order. The lexicographic method is









a sequential approach to solving the multi-objective optimization problem where each objective

is ordered according to its importance. Multi-Objective Optimization is defined as the discovery

of optimum points x* within a feasible set x that are as good as can be obtained when judged

according to multiple criteria. A Pareto Set (a.k.a. an Efficient Frontier) is the optimum solution

for multi-objective problems in that it contains all points (efficient points) for which there does

not exist any other point that would be uniformly better on all objectives.

The results of the prior level of optimization constrain the feasible set for the current level

of optimization. A lexicographic method has advantages over the traditional weighted average

method in that the problem of specifying relevant weights when the multiple objectives are

measured in different units is avoided and, by introducing the multiple objectives sequentially,

the individual effect of each objective can be identified.

Five levels of optimization (multiple objectives) are used. The first three are implemented

as goal programs. They are used to reduce the feasible set by eliminating unlikely waveform

pairs.

* Level 1: travel time

* Level 2: vehicle inductance magnitude (the inductance magnitude is inversely proportional
to the height of the vehicle)

* Level 3: vehicle electronic length (derived from occupancy time)

Maximum tolerances must be set for each level and a minimum tolerance must also be set

for travel time. Level 4 uses a traditional weighted average utility function of the change in

inductance magnitude, lane changes, and change in vehicle speed between the upstream and

downstream detection points. Level 5 has a stochastic objective that is solved using Bayesian

analysis. Calibration of the algorithms was performed with training data.









This research shows that the direct measurement of section measures of traffic system

performance such as travel times and densities avoids the inaccuracies associated with estimating

such values from "point" speeds and occupancies. This research also shows that values of

"point" and section measures derived from freeway data differ significantly.

The authors also concluded that congestion causes more variability in the traffic stream

which translates into more mismatches. The authors also cautioned that, when a higher

percentage of trucks are matched (which often happens since they are longer and have more

distinguishable features), speed results could be biased.

In a 2000 paper, Palen, et al. [15], discussed three phases of Caltrans detector research

dealing with vehicle re-identification. Phase I initially used existing detectors with bivalent

output only. Bivalent Output is defined as a detector output wherein just the presence or

absence of a vehicle is reported. Vehicle lengths (calculated from loop-based time and distance

data) and headway sequences were used to match platoons of vehicles. Vehicle lengths can only

be calculated plus or minus 10% using conventional loop detection so additional sequence

information based on headway distributions was needed to obtain useful results. Since model

170 traffic signal controllers lack the computational power needed to carry out the matching

calculations for the sequence information, bivalent loop data was brought back to a web server

via a wireless Internet Protocol (IP) modem. A stretch of 1-80 near San Francisco currently uses

this technique to obtain performance measures.

Phase II used commercially available scanning detector cards to obtain loop signatures.

These signatures were used to match vehicles. This technique was applied to an intersection

approach in Irvine, California having a 2070 controller. This process is more accurate than the

Phase I process and loops can be spaced further apart.









Phase III examined new loop geometries.

In a 2001 study, Liu, et al. [16] used a vehicle re-identification algorithm developed at UC-

Irvine to estimate the average and total delay by movement during each cycle at a signalized

intersection, and these estimates were then fed to an on-line signal control algorithm to find the

optimal green splits. Vehicle re-identification based on inductive loop signatures was used to

estimate the delay. Knowing the prevailing free flow speed for the approaches, and the distance

between detector stations, the minimum travel time for each movement can be derived. The

delay of each vehicle was calculated by deducting this minimum travel time from the vehicle's

actual travel time.

The analysis was conducted at the Alton/Irvine Center Drive intersection in Irvine,

California with the microscopic simulation program Paramics used for online signal optimization

as a complementary module to the existing signal controller. Paramics provides a framework

that allows the user to customize many features of the underlying simulation model with access

provided through an Application Programming Interface (API). Inductance loops were used for

both vehicle detection and delay estimation in Paramics.

Thirty simulation runs were made for each scenario with each run comprising a 2-hour

period. The use of multiple simulation runs permits statistical evaluation. Three measures of

effectiveness were evaluated: total intersection delay, total throughput and average delay. The

average delay-based on-line control algorithms performed better than the off-line case for both

pre-timed and actuated signal control (as evidenced by a 10% reduction in delay).

In 2002 Sun, et al. [17], investigated the use of video cameras to improve the accuracy of

vehicle re-identification using inductance loops. In this research, color information from video

cameras was used to augment the inductive signature obtained from inductive loop detectors to









track individual vehicles. When inductive loop signatures alone are used, vehicles of the same

model or even different models on the same body frame can be mismatched. On the other hand,

the use of video alone can be sensitive to changes in illumination levels (night, dusk, dawn, rain,

glare, etc.)

The test section was located in one direction of a 4-lane arterial. The two lanes of arterial

traffic for the test section were treated separately; lane changing was ignored. Detector stations,

each of which consisted of a speed trap (double inductance loops), were located 425 feet apart.

A traditional method of vehicle re-identification is license plate matching. Other potential

methods of vehicle re-identification involve GPS, cellular, toll tags, or tracking beacons. Section

measures can also be obtained via video using tripwire systems or through vehicle tracking. The

advantages of using vehicle color are that it is not correlated with vehicle signatures (i.e.

represents an independent identification measure), it can be extracted from imperfect video

images, and it can be verified visually.

Linear feature fusion with six features was used in this study. The features used were: 1)

vehicle signature, 2) vehicle velocity (distance between loops divided by turn-on time), 3)

platoon traversal time (time between first and last vehicle in platoon crossing loop), 4) maximum

inductive amplitude (inversely proportional to the cube of the distance from the ground to the

vehicle undercarriage), 5) electronic length (length of metallic components only but includes the

length of the magnetic field generated by the loop), 6) RGB triplet (color). The combined

classifier score due to linear fusion is calculated using the following formula:

Dlinear = Ei1,n wi di

Where i is an index from 1 to 6 for the six features and di are the feature values. The fusion

weights (wi) are determined using an exhaustive search such that the re-identification accuracy is









maximized. The candidate upstream platoon that achieves the smallest D is matched to the

downstream platoon. A time window constraint with upper and lower bounds is applied to

identify candidate platoons.

The research concluded that the use of detector fusion provides system redundancy and

yields better results than the use of either inductive signature information or vehicle color

information alone. A re-identification rate of over 90% was obtained using multi-detector fusion

whereas the rate was 87% for inductive signature information alone and only 75% for color

alone.

The authors postulated that the results would be even better if the vehicle re-identification

system could be tied into the arterial's signal control system since this would allow the direct

estimation of lost time associated with starting and stopping. The tie-in would improve the

accuracy and possibly yield real-time estimates of startup delays and saturation flow rates. The

authors added that it is difficult to compute arterial travel times accurately using point measures

(speed, occupancy, counts) since lost times associated with starting and stopping are not

measured directly.

The authors provided the following definitions in the report:

* Point Traffic Parameters traffic parameters that pertain to a particular point on the
roadway (volume or flow, point speed, presence, occupancy)

* Section Traffic Parameters traffic parameters that pertain to a section of roadway (link
speed, travel time, origin/destination information)

* Platoon Matching a method of vehicle re-identification that matches groups of vehicles
rather than individual vehicles.

In 2002, Oh and Ritchie [18] used inductance loop signature data to track vehicles form

upstream approach loops to receiving lane loops at a signalized intersection. Features used in the

lexicographic optimization were maximum magnitude difference between front and back loops









(relates to vertical clearance), vehicle speed, and lane information. The matching rate was 32.5%

for vehicles turning right, 51.7% for thru vehicles, and 62.5% for vehicles turning left, for an

overall match rate of 46.7%. Left turns were eliminated from the analysis due to low absolute

volume.

Cluster analysis was used to determine LOS categories based on reidentification delay

(RD). Reidentification Delay is defined as the difference between the actual time required to

traverse vehicle reidentification stations at a signalized intersection and a base travel time (such

as that calculated from the speed limit). Two different aggregation methods were investigated,

cycle-length based average (CBA) and fixed time average (FTA). A fixed interval of 60 seconds

was used for FTA. K-means clustering, fuzzy clustering, and Self Organizing Map (2 layer

neural network) methods were used in the clustering analysis. Wilk's lambda was used to

compare the results:

Wilk's lambda = |W|/|B+W|

W = pooled within-group variance
B = between group variance

A lower Wilk's lambda value indicates better clustering. K-means clustering produced the

best results, with the most appropriate number of clusters being 5. When compared to ground

truth, reidentification delay errors were on the order of 26%

A rolling average RD based on 3 signal cycles was recommended to avoid signal control

related stability problems associated with single cycle delay reporting. A recommended RD

LOS classification system is presented with LOS I (excellent) through V (poor). The LOS table

stratification values are similar to those contained in the HCM if LOS F is eliminated. Slightly

different LOS stratification values are provided for right turn and thru movements.

Mean Absolute Percent Errors were calculated using the following formula:










MAPE = [Ei=1,N(ARDi-AADiAADAi) x 100 ]/N

MAPE = Mean Absolute Percent Error
ARDi =Average Reidentification Delay at time step i
AADi =Average Actual Delay at time step i
N = total number of time steps

In a 2003 paper, Coifman and Ergueta [19] presented an improved algorithm for vehicle

matching at a freeway inductive loop detector station having dual loops. This new algorithm,

which includes four separate tests, performed significantly better than older algorithms

developed in previous work by the authors. The algorithm should be applicable to any detector

technology capable of extracting a reproducible vehicle signature. In this study, vehicles were

matched based on length and lane changing was accounted for.

The algorithm matched between 35% and 65% of the vehicles, depending on lane. The

authors noted that other researchers have estimated that matching 20% of the population is

sufficient for travel time measurements. Matching percentage is improved as the speed

decreases. The report defined a False Positive as a collection of incorrect matches and Effective

Vehicle Length as Physical Vehicle Length plus Length of the Detection Zone. The algorithm is

attractive in that it utilizes existing surveillance equipment and performs well under congested

conditions.

In 2004, Coifman and Dhoorjaty [20] presented eight detector validation tests for freeway

surveillance. Five of these tests can be applied to single-loop detectors while all of the tests can

be applied to dual-loop detectors. The tests are used to compare the performance of different

detector models and to identify permanent or transient hardware problems such as crosstalk

between loops and shorts in the loop wire. Three of the tests could be applied to arterial loop

detectors and these tests could be incorporated into the controller software for continuous









monitoring. The authors discovered that some detector units stay on a fraction of a second after

the vehicle passes and some are prone to flicker (turning on and off multiple times as a vehicle

passes). A large variability in detector operation was noticed from one model to the next and, in

the case of one of the detectors, from one software revision to the next within the same model.

In a 2007 paper, Jeng, et al. [21] described an inductance loop based vehicle re-

identification algorithm (RTREID-2) that produced excellent results when compared to GPS

information from control vehicles.

Performance of Video Detection Systems

In 1999, Washburn and Nihan [22] evaluated the Mobilizer, a video image detection

system based on vehicle tracking developed by Condition Monitoring Systems. Preliminary

results indicated that the Mobilizer is capable of matching vehicles in successive fields-of-view

with a reasonable degree of accuracy and that the travel time estimates provided by the system

are statistically valid. Two sites were evaluated, one on an arterial and one on a freeway. For

both of these sites, a departing FOV (Field of View) was used. The arterial had 76% correct

matches while 78% of the freeway matches were correct. The system can be instructed to not

consider matches that fall outside of dynamic travel time ranges, ranges that are adjusted in real-

time by the system, however, the system does not currently utilize color information and the

system does not consider matches of vehicles that change lanes. The system was only evaluated

under free flow conditions.

In 2001, Grenard, et al. [23], evaluated various video detection systems (Autoscope,

VideoTrak and Odetics) for signalized intersections. They discovered that:

* The effective length of the detection zone increased from an average of 23.7 feet during the
day to an average of 67.7 feet at night, which could cause the signal to operate less
efficiently. The percentage increase in effective detection length at night due to headlight
glare ranged between 50% and 500%; this adds 2 seconds of detection time.









* False video detections became slightly larger at night with rain due to headlight glare.

* Video detection frequently only detects the headlights at night so the call is lost if the video
detection zone ends just a few feet in front of the stop bar. Extending the video detection
zone somewhat past the stop bar would help to remedy this situation, but at the expense of
detecting additional pedestrians or crossing/left turning traffic. This produces both safety
(due to missed calls) and efficiency problems. Illuminating the intersection eliminates this
problem.

* The video detection systems tested sometimes "stuck on" for substantial periods of time.

* During dawn and dusk, sunlight causes so much glare that the camera is often unable to
distinguish between the absence and presence of vehicles.

* Wet pavement does not significantly impact the likelihood of a TOL1 error (loop on when no
vehicle is present) but traffic volume does (probably due to spillover). Neither wet pavement
nor traffic volume significantly impact the likelihood of a T1LO error (loop off when vehicle
is present).

* Under base (optimal) conditions, the video detection system has a false detection rate of 2%
to 6% and a missed vehicle presence of between 7% and 8%

* The authors distinguished between Error, defined as video results compared to actual or
ground truth and Discrepancy, defined as video results compared to another type of
detection system (such as loops). Discrepant calls include false calls and missed calls
(discrepancies of less than 3/10 of a second were not recorded). Discrepant Call Frequency
is defined as the number of discrepant calls per cycle. Error Rate is defined as the ratio of
discrepant calls to true calls and Relative Error Rate is defined as the ratio of the error rate
to the average error rate.

* Under worst-case conditions (rain, night, wet pavement, average count, heavy camera
motion) video detection misses between 16% and 20% of vehicle presence time and indicates
false detection during about 40% of the vehicle absence time.

* The authors defined Activation Distance as the distance a vehicle is from the stop bar when
it is detected by the video detection system, and Blanking Band as a process used to remove
all discrepancies smaller than a user-defined value.

* Due to the imprecision of night detection, the authors recommended that video detection not
be used to provide dilemma zone protection.

* The authors cited past work in this area: MacCarley's 1992 evaluation of video detection
found that several conditions caused significant degradation in video detection performance:
non-optimum camera placement, day-to-night transition, headlight reflections on wet
pavement, shadows, fog, heavy rain with error rates of 20% to 40% for most tests performed.
MacCarley's 1998 evaluation of video detection found that several additional conditions
caused significant degradation in video detection performance: transverse lighting, low









lighting and vehicles that have a low contrast to the pavement. 65% of all vehicles were
detected correctly with an 8.3% false detection rate. 64.9% of all red-green transitions would
have been actuated correctly if video were used instead of properly functioning loops.
Middleton's 1999 evaluation of video detection found that video detection: 1.) consistently
over-counted by as much as 40% to 50% at night, 2.) at dawn and dusk sun angles produced
glare that caused undercount rates of 10% to 40%, 3.) undercounted by 6% to 8% during
heavy rain. The most consistent period of error was between midnight and 5:00 am.
Middleton and Parker's 2000 evaluation of video detection found that video detection: 1.)
over-counted both day and night during wet pavement conditions because of headlight
reflections, 2.) had reduced accuracy at night and when long shadows occurred.

The authors provided the following formulas for calculating detection errors:

Missed Detection Rate (MDR) = Number of Actual Detection Events Missed By Loop/Total
Number of Actual Vehicle Arrivals (discrete definition)

P(L=0|T=1) = D(L=0 & T=1)/D(T=1) where D=Duration, T=Ground Truth, L=Loop,
1=0n, 0=Off (continuous definition)

False Detection Rate (FDR) Number of False Detection Events Reported By Loop/Total
Number of Inductive Loop Events (discrete definition)

P(L=IT o0) = D(L=1 & T=0)/D(L=1) where D=Duration, T=Ground Truth, L=Loop,
1=0n, 0=Off (continuous definition)

P(L=1 T =) = D(L=1 & T=0)/D(T=0) where D=Duration, T=Ground Truth, L=Loop,
1=0n, 0=Off (revised continuous definition)

For the likelihood (probability) of a detection discrepancy the following formulas apply:

The probability of video detection being off when loop detection is on = P(V=01|L1)
D(V=0 & L=1)/D(L=1) where D=Duration, V=Video, L=Loop, 1=0n, 0=Off

The probability of video detection being on when loop detection is off = P(V=1 L-) =
D(V=1 & L=0)/D(L=0) where D=Duration, V=Video, L=Loop, 1=0n, 0=Off

For the likelihood (probability) of a detection error the following formulas apply:

The probability of video detection being off when a vehicle is present = P(V=01T=1)
P(L=I|T-1) x P(V=0|L=1) + P(L=0IT-1) x [1-P(V=1IL-0)]

The probability of video detection being on when a vehicle is not present = P(V=1IT-o) =
P(L=1IT o) x [1-P(V=0|L=1)] + P(L=0|T0) x P(V=I|L 0)

In 2002, Bonneson and Abbas [24] investigated the operation of Video Imaging Vehicle

Detection Systems (VIVDS) in Texas. It was estimated that about 10% of the intersections in









Texas were using VIVDS and that Texas DOT was installing VIVDS at about /2 of all newly

constructed intersections. They identified the following VIVDS manufacturers: Image Sensing

Systems (Autoscope system used by Econolite), Iteris (Vantage system used by Naztec and

Eagle), Peek Traffic Systems (VideoTrak system), Traficon, Nestor Traffic Systems and

Transformation Systems. A review of VIVDS product manuals revealed that these manuals do

not describe techniques for the effective use of delay, extend, or passage time settings in

conjunction with a VIVDS installation.

Their report made the following points:

Detection zones can be linked via Boolean logic functions (AND, OR, NOT, etc.)

VIVDS can provide reliable presence detection when the detection zone is relatively long
(say, 40 ft or more). However, its limited ability to measure gaps between vehicles
compromises the usefulness of several controller features that rely on such information
(such as volume-density control).

A VIVDS system is sometimes used to provide advance detection on high-speed
intersection approaches. However, some engineers are cautious about this use because of
difficulties associated with the accurate detection of vehicles that are distant from the
camera. Of those agencies that use a VIVDS for advance detection, the most
conservative position is that it should not be used to monitor vehicle presence at distances
more than 300 feet from the stop line.

The minimum camera height (in feet) for advanced detection is calculated using the
formula:

Ha= (xl+ x,)/R

Where xl is the distance in feet between the stop line and the upstream edge of the
detection, calculated as: xi= 1.47tbzV95, and:

x, = distance in feet between camera and stop line
R = distance-to-height ratio (17 in Texas)
Tbz = travel time from the start of the dilemma zone to the stop line (5 seconds)
V95 = 95th percentile speed in mph (= 1.07 x V85)

Table 4-2 in the report provides the resulting minimum required camera heights for

advanced detection. The required height varies between 24 feet and 36 feet.









* A camera's field of view is impacted by the following factors: camera height (distance
from ground to camera), camera offset (lateral distance from camera to the lane or lanes
being monitored), distance (longitudinal distance from the detection zone to the camera),
pitch angle (angle of downward "tilt" of the camera relative to the ground), and focal
length (which determines the relative size of objects in the camera's field of view).
Detection Design is defined as the selection of camera location and the calibration of its
field of view whereas Detection Layout involves locating detection zones, determining
the number of detection zones, and identifying the settings or detection features used with
each zone.

* The "10 ft to 1 ft" rule states that, if camera set up is optimal, one should be able to
extend out 10 feet for every 1 feet of camera elevation to a maximum distance of around
300 feet. However, Texas DOT staff indicated acceptable operations using 17 feet
instead of 10 feet.

* Detection accuracy will improve as camera height increases within the range of 20 to 40
feet. Increased height improves the camera's field of view of each approach traffic lane
by minimizing the adverse effects of occlusion. Three types of occlusion are present with
most camera locations: adjacent-lane, same-lane and cross-lane. Increasing camera
height tends to decrease call error, provided there is no increase in camera motion.
Cameras mounted above 34 feet may experience unacceptable camera motion unless
located on a stable pole. Adjacent-Lane Occlusion (Horizontal Occlusion) occurs when
the blocked and blocking vehicles are in adjacent lanes, which can result in false
detections in adjacent lanes. Table 4-1 of the paper provides minimum required camera
heights to reduce adjacent-lane occlusion. The required height depends on the lateral
offset, whether the offset is to the left or to the right, and the lane configuration, and
varies between 20 feet and 63 feet. The minimum required height is lowest for a camera
mounted in the center of the approach, 20 feet. Same-Lane Occlusion (Vertical
Occlusion) occurs when the blocked and blocking vehicles are in the same lane, which
can result in a low vehicle count. The extent of this problem increases as the distance
from the stop line increases. Same lane occlusion is associated with an increase in the
effective length of a vehicle. Consequently, passage settings must be reduced to yield
operation equivalent to that obtained with an inductance loop. Cross-Lane Occlusion
occurs when a vehicle crosses between the camera and the intersection approach being
monitored, which can result in false detections.

* The optimal field of view for a camera is one that has the stop line parallel to the bottom
edge of the view and in the bottom one-half of this view. The optimal field of view also
includes all approach traffic lanes. The focal length should be adjusted such that the
approach width, measured at the stop line, represents 90% to 100% of the horizontal
width of the view. The view must exclude the horizon.

* Detection accuracy is significantly degraded by glare from the sun and, sometimes, from
strong reflections from smooth surfaces. Sun glare typically causes problems for the
eastbound and westbound approaches. A larger pitch angle can reduce the impact of sun
glare and a camera equipped with an automatic iris (or electronic shutter) will minimize









the adverse effects of reflection. An infrared filter can also reduce the adverse effects of
glare. VIVDS processors have the ability to detect excessive glare or reflection and
automatically invoke maximum recall for the troubled approach. Detection Accuracy is
defined as the number of times that VIVDS reports detection when a vehicle is in the
detection zone, or reports no detection when a vehicle is not in the detection zone.

* Most VIVDS have separate image-processing algorithms for daytime and nighttime
conditions. The daytime algorithm searches for vehicle edges and shadows. During
nighttime hours, the VIVDS searches for the vehicle headlights and the associated light
reflected from the pavement. Research has found that the nighttime algorithm is less
accurate than the daytime algorithm and also has a tendency to place calls before the
vehicle actually reaches the detection zone. Intersection lighting can minimize the extent
of this problem.

* The detection design should avoid having pavement markings cross the boundaries of a
detection zone since camera movement combined with high-contrast images may confuse
the image processor and trigger false calls.

* The following equations are provided for determining the required length of a stop line
detection zone:

lsl = vq (MAH-PT)- lv

Iv* = (lv-lro) + xo(hv/he)

11s = length of stop line detection zone in feet
Vq = maximum queue discharge speed at the stop line (use 40 ft/sec)
MAH = Maximum Allowable Headway (use 3 seconds)
PT = controller Passage Time in seconds
l* = effective length of vehicle in feet
lv= length of design vehicle (use 16.7 feet)
Iro = distance from back axle to back bumper of design vehicle (use 4.3 feet)
x, = distance between the camera and the stop line in feet
hv = height of design vehicle (use 4.5 feet)
he = height of camera in feet

* The detection zone length should be approximately equal to the length of a passenger car
in order to maximize sensitivity. Stop line detection typically consists of multiple
detection zones. For reliable queue service, detection zones should extend at least 40 feet
from the stop line. Zone Location is defined as the distance between the upstream edge
of the detection zone and the stop line.

* The camera field of view should be established to avoid inclusion of objects that are
brightly lit in the evening hours, especially those that flash or vary in intensity. If these
sources are located near a detection zone, they can trigger false calls. The light from









these sources can also cause the cameras to reduce its sensitivity by closing its iris, which
results in reduced detection accuracy.

Each VIVDS detection zone has a directional mode that allows it to recognize calls only
for traffic moving in a specified direction. However, this mode appears to reduce the
sensitivity of the detection zone.

During daytime hours, swaying power lines, support cables or signal heads can trigger
false calls as they move into and out of the detection zone.

The performance of VIVDS is adversely affected by environmental conditions such as
fog, precipitation, and wind. Condensation and dirt buildup on the camera lens can
further degrade VIVDS operation.

Shadows can extend into a detection zone and trigger false calls or compromise the
VIVDS ability to detect vehicles.

Delay settings are sometimes used to reduce the frequency of false calls. For example, a
few seconds of delay is often set for stop line detection zones on the minor street
approach. The delay eliminates false calls at night caused by right-turning vehicles from
the major road whose headlights sweep across the detection zone. It also eliminates false
calls due to cross-lane occlusion caused by tall vehicles on the major road.

A lens adjustment module is an essential VIVDS-related installation device. It connects
to the back of the camera and is used during camera installation to adjust the camera's
zoom and focus settings. Having this device facilitates camera replacements or
adjustments. Enough room is needed in the controller cabinet to house the needed
VIVDS equipment. Standard RG-59 coaxial cable is good for up to a distance of about
500 feet for connecting the camera to the hardware in the controller cabinet.

Satisfactory operation of a VIVDS requires verification of the initial layout and periodic
on-site performance checks (at least every 6 months is recommended).

A review of some existing VIVDS installations in Texas indicated that there was more
than one discrepant call each cycle with about 1.8 discrepant calls per true call. About
80% of the discrepant calls averaged less than 2 seconds per call and were typically
associated with the VIVDS registering a call slightly before or after its true arrival or
departure time. Wholly missed or false calls were less frequent and often had a duration
in excess of 2 seconds. During approximately 20% of the signal cycles, a phase
experienced about 4 missed calls with the total duration of these missed calls being about
25 seconds per cycle.

In 2003, Oh and Leonard [25] obtained validation results for the PEEK VideoTrak 900

image processing system. The test site was on 1-75 in Atlanta. The test results showed huge









volume errors in some case, especially at night. The system also provided lower speeds than true

speeds at night. The farther the lane was from the camera, the more inaccurate was the count.

Signalized Intersection Queuing and Delay

In 1977, Riley and Gardner [26] investigated various techniques for measuring delay at

signalized intersections. Four possible techniques were listed:

Point Sample
1st Advantage: self-correcting, each sample is independent of the previous one
2nd Advantage: not dependent upon signal indications
Disadvantage: accuracy reduced when counts become high (an upward bias exists
such that an adjustment factor of 0.92 is recommended)

Input-Output (a.k.a. Interval Sample)
Disadvantage: field data must be corrected for vehicles that enter or leave the study
area between the input and output points (at driveways or cross streets)

Path Trace
Disadvantage: a very large sample of vehicles is needed to provide an estimate of
delay having reasonable confidence

Modeling

As part of their work, the authors concluded that; "Once the recommended field data

corrections have been made, stopped delay per vehicle multiplied by 1.3 will yield a good

estimate of approach delay per vehicle."

In 1984, Hurdle [2] proposed the use of delay models that take more account of variations

in travel demand over time. Hurdle noted that: "... any steady-state model that does not assume

completely uniform arrivals will predict that the queue length, and therefore the delay, approach

infinity as the v/c ratio approaches unity. This is, of course, the reason that systems with a high

v/c ratio take a long time to settle into a steady state; it simply takes a long time for such long

queues to form, particularly since vehicles keep leaking through the signal. As a result, one

seldom sees real delays as large as those predicted for high v/c ratios. This discrepancy is not a









result of faulty mathematics but of the unrealistic assumption that the system is in a steady state.

If vehicles continued to arrive at a rate v nearly equal to the capacity c, the giant queues really

would form, but in reality the peak period ends and v decreases long before a steady state is

reached. As a result, steady-state models are useful for predicting delays only at lightly loaded

intersections." Hurdle added: "...there is one group of models, the steady-state queuing models,

that work well when v/c is considerably less than one and another type, the deterministic queuing

models, that work well when v/c is considerably more than one. In between, there are

problems." He also stated: "What modeling approaches make very clear is that the development

of the queue is very dependent on the details of the arrival pattern ... more information about

arrival patterns must be provided than is now customary."

In 1992, Bonneson [27] developed a discharge headway model for signalized intersections

that was based on non-constant acceleration behavior. Bonneson mentions that, in 1977, Messer

& Fambro found that, except for the first position, driver response by queue position was fairly

constant at 1.0 second. The first driver experienced an additional delay of 2 seconds. Messer &

Fambro also found that the average length of roadway occupied by each queue position is about

25 feet. Bonneson found this distance to be 25.9 feet.

Bonneson used regression analysis to obtain an approximate equation for the Standard

Deviation (SD) of delay: SD = 0.42 x (mean delay).7. The Maximum Error (ME) in the

calculated delay at the 95% confidence interval is then: ME = 1.96 x SD = 0.82 x (mean

delay)07.

Bonneson concluded that the minimum discharge headway of a traffic movement is a

complex process that is dependent on driver response time, desired speed, and traffic pressure.

The discharge headway model developed in his research indicates that the minimum discharge









headway of a traffic movement is not reached until the eighth or higher queue position.

Bonneson also concluded that:


A rather strong inverse linear relationship exists between vehicle acceleration and stop
line speed.

For the driver acceleration model developed, the maximum acceleration ranges between 6
and 8 ft/sec/sec with an average of 6.63 (this is similar to a value of 6.0 found by Evans
and Rothery).

For the stop line speed model developed, stop line speed increases with queue position in
an exponential manner to a maximum value between 46.7 and 51.0 ft/sec with a median
value of about 49 ft/sec (33 mph).

Traffic pressure (vehicles per lane per cycle) is a significant factor (p=0.001) in reducing
discharge headways.

Based on the calibrated model, the start-up lost time for a typical through movement with
a common desired speed of 49 fps and a maximum acceleration of 6.63 ft/sec/sec is 3.67
seconds

Based on the calibrated model, the minimum discharge headway for a typical through
movement of an at-grade intersection with a common desired speed of 49 fps and a
nominal traffic pressure of 5 veh/ln/cycle is 1.81 seconds

The following formulas are provided in the report:

Briggs Models Based on Constant Acceleration

Calibrated Discharge Headway Model:

Headway of nth vehicle = hn = T + [2dn/A]1/2 [2d(n-1)/A]1/2
(if nd < dmax = Vq2/ 2A)

Headway of nth vehicle = hn = T + d/Vq
(if nd >= dmax)

Vq = desired speed of queued traffic (29.4 ft/sec)
d = distance between vehicles in a stopped queue (19.65 feet)
T = driver starting response time (1.22 seconds)
A = constant acceleration of queued vehicles (3.67 ft/sec/sec)
dmax = distance traveled to reach speed Vq
n = queue position









Bonneson Models Based on Non-Constant Acceleration

Calibrated Stop Line Speed Model:

Stop Line Speed for vehicle n = Vsi(n) = Vmax (1 enk)

k = -0.290 + 24.0/Vmax

Calibrated Discharge Headway Model:

Headway of nth vehicle = hn = (tau)Ni + T(d/Vmax)
+ 0.357[(Vsi(n) Vsi(n-i)/Amax] 0.0086v 0.23AGI

Calibrated Minimum Discharge Headway Model:

Minimum Headway = H = T + d/Vma,- 0.0086v 0.23AGI

Calibrated Start-Up Lost Time Model:

Start-Up Lost Time = Ks = 1.03 + 0.357Vmax/Amax

n = queue position
tau = additional response time for first queued driver (1.03 sec)
d = distance between vehicles in a stopped queue (25.25 feet)
T = driver starting response time (1.57 sec)
v = traffic pressure in vehicles per cycle per lane
Vmax = common desired speed of queued traffic in feet per second
Amax = maximum acceleration in feet per second per second
N1 = 1 for first queued vehicle, 0 otherwise
AGI = 1 for at-grade intersection, 0 for single point urban interchange

In 1997, Fambro & Rouphail [28] proposed a new set of delay equations that were, for the

most part, incorporated into the 2000 Highway Capacity Manual. The only difference is that the

formulas recommended for the d3 term were replaced by different formulas included in

Appendix F of Chapter 16 of the 2000 HCM.

Simulation (TRAF-SIM) data were used to validate the over-saturation and variable

demand component of the generalized delay model because of the difficulty in measuring over-

saturation delay in the field

The following parameters are defined in this study:










* I= parameter for variance-to-mean ratio of arrivals from upstream signal. Isolated signals
have the highest I value (I=1.0 Variance=Mean -Poisson Distribution). The I value
varies between 0.09 and 1.0 at coordinated intersections.

* The k value produces less delay for actuated signals with snappy extension intervals (down
to 2 seconds). The amount of the delay decrease depends on the degree of saturation, with
greater decreases experienced when the degree of saturation is low (toward 0.5) and no
decreases experienced when the degree of saturation is high (at 1.0)

* Including a T parameter in the generalized delay model to account for the duration of the
analysis period improves delay estimates under oversaturated conditions. Longer periods of
oversaturation and higher degrees of oversaturation result in longer delays. It is important to
note that part of the estimated delay during oversaturated conditions occurs after the analysis
period.

The following definitions are given in the report:

* Stopped Delay = the time an individual vehicle spends stopped in a queue while waiting to
enter an intersection.

* Average Stopped Delay = the total Stopped Delay experienced by all vehicles arriving
during a designated period divided by the total volume of all vehicles arriving during the
same period (used to determine LOS in 1985 and 1994 HCM).

* Signal Delay (a.k.a. Control Delay) = deceleration delay + queue move-up delay + Stopped
Delay + acceleration delay

The following formulas are provided in the report:

* Control Delay (delay per vehicle for each lane group) = dl (Uniform Delay) + d2
(Incremental Delay due to Random and Overflow Queues) + d3 (Incremental Delay due to
Oversaturation Queues at the start of the analysis period)

di = PF[0.5C 1-(g/C)}2]/[1-(g/C)min(X,1.0)]

PF = (1-P)fpA/[1-g/C] (from 2000 HCM)
X = v/c for lane group (aka degree of saturation)
C = average cycle length (seconds)
G = average effective green time (seconds)

d2 = 900T[(X-1) + {(X-1)2+8kIX/Tc}1/2]

I = upstream filtering/metering factor obtained from Exhibit 15-7 of 2000 HCM
k = incremental delay factor obtained from Exhibit 16-13 of 2000 HCM
c = capacity of lane group (vph)
T = duration of analysis period (hours)










d3 = (See Appendix F of 2000 HCM)

In 1997, Engelbrecht, Fambro, et al. [6] proposed a generalized delay model that handles

over-saturated conditions at signalized intersections. The delay equations calculate delays

consistent with the more accurate path-trace method of delay measurement rather than the less

accurate (but easier to carry-out) queue-sampling method. Delays estimated by the proposed

generalized model were in close agreement with those simulated by TRAF-NETSIM.

The path-trace method measures individual vehicle delays from arrival to departure, even

if the departure occurs after the end of the analysis period. Delay measurement using this

technique is typically complicated. However, advances in intelligent transportation system

technology may reduce the difficulty associated with this technique.

The queue-sampling method records the number of stopped vehicles at periodic intervals

(such as every 10 seconds), multiplies this by the length of the sampling period, and then divides

by the number of vehicles arriving during the analysis period.

For the path-trace method and queue count methods to be compatible, two conditions must

hold: 1.) There must not be a residual queue at the start of the analysis period, and 2.) Queue

counts must continue until all vehicles that arrived during the analysis period have cleared the

intersection. All vehicles joining the back of the queue after the end of the analysis period

should be excluded from this count.

TRAF-NETSIM calculates delay by subtracting the free-flow travel time from the actual

travel time to yield overall delay. However, the actual travel time includes not only intersection,

or control delay, but also some delay as a result of interactions between vehicles on the link

itself, or traffic delay. In the analysis, the authors decided to ignore this discrepancy, as it is very









difficult to separate control and traffic delay, and the error is assumed to be small, especially

under over-saturated conditions.

The following TRAF-NETSIM input values (representative of over-saturated conditions)

were analyzed:

Analysis Period (T) = 15 & 30 minutes
Cycle Length (C) = 60, 90, 120 seconds
Saturation flow (s) = 1800 & 3600 vphg
G/C ratio = 0.3, 0.5 & 0.7
Degree of Saturation (X) = 1.0, 1.1, 1.2, 1.3 & 1.4 (0.9 was also included)

The authors point out that equilibrium (in TRAF-NETSIM) can never be reached for over-

saturated conditions, as capacity is less than demand and outflow will always be less than inflow.

The initialization will terminate before equilibrium can be reached, leaving an initial queue of

unknown size. This queue will delay vehicles when it clears, increasing the delay experienced

by vehicles that arrive during the analysis period. Therefore, the authors decided to use 3 periods

in the analysis: an initial 60-second period with very low flow; the actual analysis period of

duration T; and a final period of duration T, again with very low flow (TRAF-NETSIM can not

handle zero flow). The first period is needed to initialize the network without transferring a

queue to the second period, the second period is the actual analysis period, and the third period

dissipates the over-saturation queue that built up over the second period.

Not all of the input scenarios yielded usable results. In some scenarios, the simulated

delays were incorrect because of queue spillback

In 2000, Tarko and Tracz [29] investigated uncertainty in saturation flow predictions and

concluded that standard errors reached 8 to 10%. They identified three primary sources of error:

temporal variance, omission of one or more capacity factors in the predictive model, and

inadequate functional relationships between model variables and saturation flow rates. The data









were collected on Polish highways but the authors conclude that the results should be

transferable to other countries.

Using data from over 1100 signal cycles, Tarko and Tracz discovered that the saturation

flow rate increases rapidly during the first 6 seconds of the green indication to a value of about

1400 pcphg (headway of 2.6 sec/veh), then slowly increases to a value of about 1600 pcphg

(headway of 2.2 sec/veh) after another 20 seconds. Past this 25 second mark the rate stabilizes.

This type of behavior occurred in all of the lanes investigated although the length of the periods

varied somewhat. Consequently, the length of the counting period has an effect on the saturation

flow rate that is obtained.

Tarko and Tracz also found that the percent of heavy vehicles in the traffic stream has an

effect on the headway of passenger cars, with the headway varying between 2.2 sec/veh when no

heavy vehicles are present to 2.6 sec/veh when the traffic stream is composed of 30% heavy

vehicles. Heavy vehicles also have longer headways than passenger cars, which is another factor

that reduces the saturation flow rate. Tarko and Tracz recommend the use of a Passenger Car

Equivalence (PCE) factor of 2.4, which is substantially higher than the value of 2.0 used in the

2000 Highway Capacity Manual or the 1.2 default factor used by CORSIM.

Tarko and Tracz proposed various predictive models for saturation flow that included the

following statistically significant independent variables: ratio of heavy vehicles, lane width,

turning radius (infinite for straight lanes), and lane location (near curb or middle). The authors

conclude by stating that: "Where possible, the saturation flow rates should be determined

through direct field measurement". This provides more support for the research at hand.

In 2002, Li and Prevedouros [30] studied three methods for describing the discharge

process of a standing queue at an approach of a signalized intersection. Method 1 (Ml) entails









measurements of headways based on the first 12 vehicles in a standing queue. Method 2 (M2 or

HCM Method) entails measurements of headways based on all vehicles in a standing queue.

Method 3 (M3) is the same as M2 except that arrivals which join the standing queue are

included.

According to the HCM, the saturation headway is estimated by averaging the headways

from the 5th vehicle to the last vehicle in a standing queue. The 2000 HCM suggests a base

saturation flow rate of 1900 pc/h/ln for thru lanes, which corresponds to a saturation headway of

1.895 seconds (3600/1900) and 1800 pc/h/ln (a 2 second saturation headway) for protected left

turn lanes. Start-Up Lost Time (SULT) is derived from the first four vehicles in a standing

queue. The 2000 HCM mentions typical observed values of between 1 and 2 seconds for thru

lanes.

Li and Prevedouros collected data on two lanes of a five-lane approach (3 thru lanes and a

dual left turn lane) of a signalized intersection in Honolulu, Hawaii. The outside thru lane and

the inside left turn lane were measured. These lanes were considered to be of 'ideal" design and

no queues with heavy vehicles were used in the analysis. A vehicle was considered to be

discharged when its rear axle passed the stop line. Observations containing fewer than four

vehicles at the end of a queue were not included.

Start-Up Response Time (SRT) was defined by the author's as the time from the beginning

of green to when the first vehicle's rear axle passes the stop line. The following relationship

between SRT and SULT was provided:

Start-Up Lost Time = SULT = SRT + 4*(H4-h)
Saturation Headway = h = (TN-T4)/(N-4)
Average Headway = Hi = (Ti Ti-4)/4
Where:
Ti = time when rear axle of vehicle i passes the stop line (To = SRT)









N = last vehicle in the queue

The saturation headways (h) derived by the three methods (Ml, M2 and M3) are

statistically different.

For thru movements:

h = 1.90 sec (s = 1895 pc/h/ln) for Ml, std dev = 0.21
h = 1.92 sec (s = 1875 pc/h/ln) for M2, std dev = 0.20
h = 1.98 sec (s = 1818 pc/h/ln) for M2, std dev = 0.22

The minimum headway was not reach until the 9th to 12th vehicle instead of the 5th

vehicle as implied by the HCM. If queue arrivals are included (M3), both the mean and

standard deviations of the headways increase after the 12th vehicle.

For protected left turn movements:

h = 2.04 sec (s = 1765 pc/h/ln) for Ml (1765/1895 = 0.931 LT factor), std dev = 0.23
h = 2.01 sec (s = 1791 pc/h/ln) for M2 (1791/1875 = 0.955 LT factor), std dev = 0.23

Headways decrease as queue position increases (motorists may be aware of the limited

green time and tailgate so as to not experience a phase failure). After the first 12 vehicles the

saturation flow rate remained well above 1800 pc/h/ln. Queues of medium length discharge

more efficiently than do short queues. After the 16th vehicle in the queue the saturation flow

rates of the left turn movement were larger than for the thru movement. The Start-Up Response

Time (SRT) for left turn movements (1.42 seconds) is less than for thru movements (1.76

seconds), indicating a heightened awareness of left turning drivers to the display of the green.

There was a high standard deviation of SRT for both movement types (0.61 for thru's and

0.74 for LT's), indicating a big variation amongst drivers. However, SRT was not sensitive to

queue length. The calculated SULT was well above the 1 to 2 seconds of the HCM (2.89 for

thru's and 2.38 for LT's under peak period conditions and 3.03 for thru's and 2.53 for LT's









under off-peak conditions.) As with the SRT's, the SULT's also have high standard deviations

(1.36 for peak thru's and 1.32 for peak LT's; 1.5 for off-peak thru's and 1.3 for off-peak LT's).

Linear regression models (one for thru movements and one for LT movements) were

developed that indicate a negative correlation between SULT and queue length (i.e. long queues

produced shorter start-up loss times).

Distribution tests showed that thru movement headways were lognormally distributed

without a shift and that LT headways were lognormally distributed with a shift of 1 second. SRT

was normally distributed for both movements.

In 2002, Cohen [31] used the Pitt car-following system to examine the effects of lane

changing and a heterogeneous vehicle mix on queue discharge headways.

In the Pitt car-following model, the first vehicle in the queue begins to move across the

stop line after the lost time (start-up delay) has expired. The second vehicle in the queue then

responds to the motion of the leader through the car-following system with no additional explicit

lost time added. The effect of lost time on subsequent vehicles is modeled through the

sluggishness of the car-following system.

Based on the results of the study, it can be concluded that trucks not only have longer

headways than cars, but they also increase the headways of the vehicles behind them. The closer

to the front of the queue that the truck is located, the greater the overall negative effect on queue

discharge. In addition, for trucks further back in the queue the major item affecting its

equivalency factor is its greater length whereas, for trucks near the head of the queue, the major

item is vehicle performance limitations. Queue Discharge Headway is defined as the difference

in stop line crossing times between each vehicle pair.









Lane changing also has a substantial effect on discharge headways, particularly if the lane

change takes place close to the stop line. For thru lanes with short adjacent turn lanes (where

lane changing is apt to take place) the saturation flow rate will be lowered on the basis of the

percentage of turns.

The results of the study also suggest that the start-up wave in a discharging queue will

slow down as it progresses upstream. Acceleration rates decrease as one progresses upstream in

the queue (each vehicle accelerates more slowly than its leader). Consequently, it takes longer

for gaps to open between pairs of vehicles in the queue and the presence of these gaps is the

necessary requirement for the follower to begin to move. Start-Up Wave (a.k.a. Green Wave,

Expansion Wave) is defined as the rate at which vehicles in the queue begin to move. (With

movement defined as the time at which a speed of 1 ft/sec is achieved.)

In addition, the study results indicate that the discharge headway distribution is almost flat

beyond the fifth vehicle in the queue, which is consistent with the HCM.

The author notes that the best approach for calibration of the Pitt car-following model is to

measure in the field the crossing times of both the front and rear of each vehicle in the queue as it

discharges across the stop line. These measurements allow the plotting of two curves, the front-

to-front time headway curve and the rear-to-front time spacing curve. Unfortunately, this type of

detailed data set is usually not collect in queue discharge studies.

The author explains that the NETSIM queue discharge mechanism is limited in that it is

based on the assumption that vehicles in a queue discharge from the intersection at equal time

headways (other than stochastic variations) subject to start-up delays applied to the first 3

vehicles in the queue. The effect of lane changing is ignored completely and the effect of

commercial vehicles is treated heuristically using vehicle equivalency factors.









In 2003, Mousa [32] presented a microscopic stochastic simulation model developed to

emulate the traffic movement at signalized intersections and estimate vehicular delays, including

acceleration and deceleration delay. By analyzing 48 cases with a fixed g/C ratio of 0.475, it was

found that the ratio of total delay to stopped delay is directly proportional to both the degree of

saturation and the approach speed, and inversely proportional to the cycle length. The effect is

greatest for degree of saturation and cycle length and least for approach speed. For the 48

simulated cases, the saturation flow obtained from simulation ranged from 1692 vph to 1807

vph, with an average value of 1770 vph and a standard deviation of 28 vph.

Approach speeds ranging from 30 to 50 mph and cycle lengths varying between 60 and

150 seconds were considered and tested in this study. Different levels of degree of saturation,

ranging between 0.5 and 0.9, are also considered. The ratio of total delay to stopped delay was

found to be between 1.5 and 3.0 with the minimum ratio resulting from the longest cycle length

(150 seconds) and the lowest degree of saturation (0.5) and the maximum ratio resulting from the

shortest cycle length (60 seconds) and the highest degree of saturation (0.9).

A sufficient length of approach was considered in the analysis to ensure that all

acceleration/deceleration delays incurred by individual vehicles were executed within the

simulated length.

In 2004, Rakha and Zhang [33] authored a paper that demonstrated the consistency that

exists between queuing theory and shock-wave analysis and that highlighted the common errors

that are made with regard to delay estimation using shock-wave analysis. The authors point out

that the main difference between shock-wave analysis and queuing models is the way vehicles

are assumed to queue upstream of the bottleneck. Queuing analysis assumes "vertical stacking"

of the queue whereas shock-wave analysis considers the horizontal extent of the queue.









Maximum queue reach (a.k.a. back of queue) can only be identified using shock-wave analysis.

The authors show that the size of the queue obtained from shock-wave analysis is the same as the

size of the queue obtained from deterministic queuing theory if the queuing theory value is

adjusted by a factored equal to total travel time divided by total delay.

In 2004, Perez-Cartagena and Tarko [34] demonstrated that, based on studies conducted in

Indiana, town size and lateral lane location (right-most lane or not) are important variables in

identifying the base saturation flow rate for a signalized intersection. Saturation flow rates were

estimated using the Headway Method and weighted regression analysis. The authors also

discovered that small communities tend to have considerably lower values of saturation flow

than large communities, indicating that drivers in large communities are more aggressive than

drivers in small communities. The reduction in saturation flow rate was about 8% for medium

size towns and 21% for small towns (as compared to large towns).

Kebab, et al. [35] developed an efficient field procedure for measuring approach delay at a

signalized intersection that segregated the delay by movement. The procedure produced good

results in comparison to ground truth obtained from video.

One section of a 2006 paper by Brilon, et al. [36] discussed variation in capacity that

occurs at signalized intersections due to "the randomness of driver behavior and interaction

between vehicles". The authors concluded that their stochastic concept of capacity "provides

better plausibility than the assumption of constant-value capacities" and that "the implications of

random capacities on delay distributions should be investigated by further research".

Probe Monitoring

The most promising alternative method for obtaining the type of globally applicable delay

estimates (estimates applicable to over-saturated as well as under-saturated conditions) addressed

in this paper is the use of probe vehicles. A considerable body of work is being conducted in this









area, including the potential use of cell phone data to track individual vehicles and the results of

the work are starting to show up in the literature.

A 2005 article by Jiang, et al. [37] examined the collection of signalized intersection delay

data using vehicles outfitted with global positioning system (GPS) technology. It was

determined that, compared to manually measured delays, the GPS approach provided the same

accuracy with considerably lower labor requirements.

A 2007 paper by Ko, et al. [38] also examined the collection of signalized intersection

delay data using vehicles outfitted with global positioning system (GPS) technology. Their

technique included algorithms for analyzing speed profiles and acceleration profiles in order to

automatically identify critical control delay points, such as deceleration onset points and

accelerating ending points. This automated process permits the analysis of large data sets and

provides consistent results. However, the approach experienced some difficulty in handling

over-capacity conditions and closely spaced intersections.

A 2007 paper by Comert and Certin [39] used probe vehicles to estimated queue lengths on

a signalized intersection approach. The best estimate of queue length was provided for high

volume, but under-saturated, conditions. The results are subject to sampling errors (a common

characteristic of probe use) and the procedure was not tested under congested conditions.

A 2007 Florida Department of Transportation report authored by Wunnava, et al. [40] of

Florida Atlantic University investigated cell phone tracking. The authors concluded that a host

of both technical and privacy issues need to be worked-out before probe vehicles can provide the

needed detail to accurately estimate approach delay:

... the team also found that the cell phone technology is not accurate in congested traffic
conditions, where the data is more important than in the free-flow traffic conditions, and
the accuracy decreases rapidly as the congestion increases... Additional issues remain such
as: (1) privacy of the cell phone users whose phone transmissions are being probed by the









cell companies for location data, (2) irregular and transient cell data for travel time and
speed computations, especially during congested traffic and severe weather conditions, (3)
limited capabilities of the travel time providers to follow changes by the cell companies in
their data formats and structures, and (4) incompatibility of data when switching from one
travel time provider to another.

If these issues, some of which are political in nature, cannot be addressed satisfactorily then

obtaining widespread delay information from probes may never occur.

Extending the Body of Knowledge

Although a number of researchers have investigated sampling techniques designed to

improve the estimation of travel time and delay along the through lanes of an arterial corridor

(such as through vehicle re-identification or the use of instrumented probes), the research effort

described herein is unique in that it attempts to estimate delay in a manner that is directly

applicable to the minor movements of the intersection as well as the major thru movements, and

it utilizes information from all approaching vehicles, not a restricted sample. In addition, none of

the previous research has dealt with the real-world problem of queues that extend beyond the

detection system for some period of time; either short-lived queues that occur during under-

saturated conditions because of spurts in activity or longer-lived, recurring queues that occur

during over-saturated conditions. This appears to be the only research that is attempting to

intelligently "estimate that which cannot be easily measured" with respect to intersection delay.

The basic problem that is being addressed is the need to establish a methodology that can

intelligently estimate delay associated with vehicles that are beyond the reach of the detection

system. This means obtaining reasonable estimates of vehicular delay even when queues are

long and multiple phase failures occur. The use of incomplete information, combined with a

concentration on over-saturated conditions, represent a deviation from the research conducted to

date.









CHAPTER 4
ESTIMATING NON-VISIBLE DELAY

This chapter describes the methodology that was established to predict non-visible delay

under conditions of limited information and the associated analysis procedure that was

developed. Variables important to the procedure are discussed and a series of new technical

terms relevant to the procedure are introduced (Objectives 1, 2 and 3).

Research activities were conducted using CORSIM (CORridor SIMulation) microscopic

traffic simulation software and TRAFVU (TRAFfic Visualization Utility) software that are

contained within the TSIS (Traffic Software Integrated Systems) software package. The

CORSIM software, which was developed by the Federal Highway Administration (FHWA),

consists of the FRESIM (FREeway SIMulation) component and the NETSIM (NETwork

SIMulation) component. TRAFVU is an object-oriented, graphics postprocessor for CORSIM

that displays traffic networks, animates simulated traffic and traffic controls, and reports

measures of effectiveness for the network under study.

The CORSIM runs made use of a very simple case, the intersection of 2 one-way streets,

each having a single approach lane. No trucks were placed into the traffic stream and no turns

were allowed. A random (Poisson) arrival pattern was set with arrival rates varying each 15-

minutes during a one-hour analysis time frame. The intersection was controlled by a 2-phase

semi-actuated traffic signal and delay data were collected and analyzed only for the actuated side

street approach. Goodness-of-fit testing using the chi-square technique was used to ensure that a

random (Poisson) arrival distribution was actually produced by CORSIM.

Data Analysis Programs

In order to obtain the data needed for analysis, a visual basic program called TSDViewer

[41] was developed which reads the output file of CORSIM and produces, on a second-by-









second basis, a variety of information pertaining to the number of vehicles crossing various

checkpoints and arriving and departing queues. TSDViewer automates the data collection

process from the CORSIM runs by reading CORSIM's output file (the .tsd file for CORSIM 5.1

and the .ts0 for CORSIM 6.0) and producing an Excel worksheet containing the following

information:

* The time at which each vehicle enters the approach link,
* The time at which each vehicle enters the delay zone,
* The speed of each vehicle when it enters the delay zone,
* The time at which each vehicle enters the Field of View (FOV),
* The time at which each vehicle arrives at the Back of Queue (BOQ),
* The time at which each vehicle departs the queue,
* The time at which each vehicle crosses the stop bar (leaves the link),
* The time at which each vehicle leaves the delay zone,
* The signal indication (red, yellow, or green) at each time point,
* If two queues exist simultaneously, the time at which vehicles arrive at the back of queue 2,
* If two queues exist simultaneously, the time at which vehicles depart queue 2,
* The number of vehicles experiencing 1 phase failure,
* The number of vehicles experiencing 2 phase failures,
* The number of vehicles experiencing 3 phase failures, and so on up to a maximum of 15

This information can be used to calculate, on a second-by-second basis, both queue length

and back of queue position. Stopped delay is then calculated using the queue length. An

example that shows the relationship between queue length and back of queue position is

provided in Figure 4-1. It is important to recognize that the back of queue is not itself a length,

but rather a position. As is shown in Figure 4-1, the value for the back of queue position can be

quite large even if the corresponding queue length is small.

A visual basic program named DTDiagram [42] was also developed as part of this

research. This program reads the CORSIM output file and produces trajectory information (a

series of time-distance points) for each vehicle. The data produced by DTDiagram is read by

BuckTRAJ [43], another visual basic program that was developed as part of this research to

calculate, for each vehicle, all of the components of control delay.









The programs developed allow the researcher to quickly simulate a variety of real-world

conditions in a relatively realistic manner and to accumulate the associated MOEs, such as delay.

The researcher can then compare "actual delay" obtained from CORSIM against the "predicted

delay" obtained from the techniques developed in this research. The use of simulation allowed

many different scenarios to be run in order to compare actual versus predicted delay, allowing us

to see how well our proposed delay estimation methodology performed. Essentially, micro-

simulation provided a source of verification against which our delay prediction methodology

could be developed and refined.

The pivotal task of the research was the creation of an automated analysis procedure that

can use the outputs of TSDViewer to produce queue and delay information that is required for

proper evaluation of candidate delay estimation procedures. The analysis procedure must be able

to, on a second-by-second basis, estimate the non-visible queue, add this queue to the visible

queue, calculate the associated stopped delay, and then compare the result to the "true" control

delay as calculated by CORSIM.

For the purposes of this study, stopped delay is defined as the delay experienced by

vehicles when they are at a complete stop (zero acceleration and zero velocity). Also for the

purposes of this study, a vehicle is considered queued when it comes to a complete stop (zero

acceleration and zero velocity). These are slightly more conservative definitions than those used

by CORSIM. CORSIM considers a vehicle to be stopped when its speed is less than 3

feet/second and considers a vehicle to be queued when its speed is less than 9 feet/second and its

acceleration is less than 2 feet/second/second. The zero-velocity-zero-acceleration complete stop

definition was chosen since it is easier to correlate with both video images of vehicle queues and









queues observed in the field. Much less discretion is needed to determine when a car stops than

when its acceleration and speed simultaneously fall below a certain set of values.

Control delay (Dc) is defined, both by CORSIM and in general, as the sum of initial

deceleration delay (DD), stopped delay (Ds), queue move-up delay (DMu), and final acceleration

delay (DA). Acceleration delay can be further subdivided into acceleration delay that occurs

prior to the stop bar (DA1) and acceleration delay that occurs after the stop bar (DA2). Figure 4-2

depicts the delay elements.

Total delay is defined as the sum of control delay, which is caused by the presence of the

traffic signal, and the delay associated with vehicular interactions that occur on the link (called

"interaction delay" in this study). Others have called this "cruise delay" or "traffic delay"

instead of interaction delay since it is the delay resulting from cruise speeds that are lower than

the free flow speed due to the presence of other traffic.

Ideally, we would like to have a tool that provides "accurate real-time measurement of

control delay". However, given the limitations of almost all detection systems, the best we can

hope for, and what has been developed in this research, is a procedure that provides a

"reasonably accurate real-time estimate of stopped delay". By applying an appropriate factor

(such as the commonly-used 1.3 value) or range of factors, we then scale-up the stopped delay

estimate to obtain a "reasonably accurate real-time estimate of control delay". Absent the

instrumentation of every vehicle, control delay cannot be accurately measured using current

vehicle detection systems for the following reasons:

1. Since vehicle detection systems are primarily used to allocate green time at a signal,
there is usually no detection in the departure lanes. Consequently, final acceleration
delay cannot be measured.

2. Queue lengths often extend beyond the limits of the detection system, especially
during peak hours. When this happens, we can only measure the stopped delay and









queue move-up time that occurs within the limits (or field of view) of the detection
system. Any stopped delay or queue move-up time that occurs outside the field of
view cannot be measured.

3. Motorists usually begin their initial deceleration far in advance of any signalized
intersection queue, often well beyond the field of view of the detection system. So,
most of the time, initial deceleration delay cannot be measured either.

In order to make use of existing detection systems it becomes necessary to measure that

portion of the delay that can be observed and then intelligently estimate what cannot be observed

(see Figure 4-3). The result is the methodology produced by this research, a methodology that

measures visible stopped delay; stopped delay that occurs within the Field of View (FOV) of the

detection system and then uses various analytical techniques to predict non-visible stopped

delay; stopped delay that occurs outside the FOV. The portion of the queue that is outside the

FOV is referred to in this research as the non-visible queue (see Figure 4-4) and the period of

time during which non-visible queues are present is referred to as the blind period.

During this research, a set of factors were identified that can be used to convert predicted

stopped delay to predicted control delay. Previous research by Mousa [32] suggests that the use

of a single 1.3 value is too simplistic. His simulation work suggests that the ratio of total delay

to stopped delay varies from a value of 1.5 to 3 depending mainly on cycle length and degree of

saturation. Figure 4-5 summarizes the relationship between this ratio and both the v/c ratio and

cycle length for over-saturated conditions.

For each CORSIM run, a certain Field of View (FOV) was assumed. Measured visible

locked-wheel stopped delay (delay occurring within this FOV) was added to the predicted non-

visible stopped delay to produce a total value for predicted stopped delay. This predicted value

was then compared to the actual value of locked-wheel stopped delay assuming an infinite FOV.

Finally, the predicted stopped delay was factored-up to obtain a predicted value for control delay.









This predicted control delay was then compared to the actual value of control delay, again

assuming an infinite FOV. As might be expected, these comparisons are more favorable when

traffic volumes are lower, or when the FOV is larger. In this case, queue lengths seldom go

beyond the FOV and most of the delay can be directly measured. Conversely, when traffic

volumes are higher, or the FOV is relatively short, the delay comparisons are less favorable

since, under these conditions, the queue frequently extends beyond the FOV requiring most of

the delay to be estimated.

CORSIM accumulates control delay on a link basis and, by necessity, the link numbering

changes at signalized intersections. The unfortunate result is that CORSIM's estimate of control

delay does not include any final acceleration delay that occurs past the stop bar. This forces the

development of an alternate measure of "control delay" to use as the CORSIM reference value.

This was accomplished by setting up a delay zone that begins well in advance of the intersection

and ends a few hundred feet downstream of the intersection. The location of the start and end

points for this delay zone were chosen carefully. The start point was set far enough in advance

of the intersection (upstream) so that all initial deceleration delay is accounted for, but not so far

in advance that a significant amount of pre-signal interaction delay occurs. Likewise, the end

point was set far enough past the intersection (downstream) so that all final acceleration delay is

accounted for but not so far past that a significant amount of post-signal interaction delay occurs.

The best location for the start point depends on the physical extent of the queuing that is

expected and was set in an iterative fashion. Given a fixed g/C ratio, the physical extent of the

queuing depends on both arrival volume and cycle length. For the range of variables considered

in this study, the location of the delay zone start point was located either 1600, 2600 or 3600 feet









in advance of the stop bar with corresponding CORSIM upstream link lengths of 2000, 3000, or

4000 feet used.

The best location for the end point was determined using the acceleration charts contained

in AASHTO's Geometric Design of Highways and Streets [44]. For example, using Exhibit 2-

24 in this AASHTO manual we see that, on level terrain, approximately 300 feet is required for

passenger cars to accelerate from a stop to 34 mph. Consequently, a delay zone that ends 300

feet past the stop bar is a reasonable configuration for a road with a posted speed limit of 35

mph. Since the link speeds used in our study were kept constant at 30 mph, 300 feet was chosen

as a reasonable downstream distance from the stop bar with a corresponding CORSIM

downstream link length of 1000 feet. The resulting delay zone length was either 1900 feet, 2900

feet, or 3900 feet.

If the start of the delay zone is positioned far enough upstream then all vehicles should

enter the delay zone at their free-flow speed (with free-flow speed being defined as the speed at

which the vehicle would travel had the signal not existed). The time it takes for a vehicle to

cover the length of the delay zone at its free-flow speed is defined as its free travel time. With

the delay zone boundaries properly established, the control delay is simply the difference

between the actual time it takes a given vehicle to traverse the delay zone and the vehicle's free

travel time. Although some interaction delay may occur near the start point and the end point of

the delay zone, it should be relatively minor in nature and should not significantly affect the

results.

For all CORSIM runs over-capacity conditions existed for at least a portion of the one-

hour analysis time frame, resulting in substantial levels of queuing. Such queues behave in a

manner consistent with shock-wave theory and when traffic volumes become very high in









relation to the capacity of the approach in question, vehicle re-queuing causes the formation of

one queue at the stop bar and another queue further upstream. The resulting simultaneous

queues are separated by vehicles moving between them, as is demonstrated in Figure 4-6. When

this occurs, it is often the case that vehicles arrive and depart both queues at the same time. The

analysis programs track both queues in order to provide accurate queuing information. In this

research, whenever there are two simultaneous queues, the queue closest to the stop bar is

referred to as queue 2 and the one furthest from the stop bar as queue 1. When either of the two

queues dissipates, the remaining queue is referred to as queue 1. The analysis programs were

designed to handle a maximum of two simultaneous queues since three simultaneous queues are

only present under extremely congested conditions, conditions for which almost any prediction

methodology would be grossly inaccurate.

Re-queuing events are associated with phase failures, which occur when a vehicle joins the

back of a queue and the next green interval is of insufficient duration to allow the vehicle to pass

through the intersection. Phase failures tend to occur under congested conditions, but can also

occur during uncongested conditions because of poor signal timings. Poor signal timings might

include insufficient maximum intervals, extension intervals that are too short for the detection

system, or even insufficient minimum intervals if the approach utilizes an upstream detection

system. Re-queuing is a necessary condition for the formation of simultaneous queues; however,

it is not a sufficient one. As shown in Figure 4-7, re-queuing may not result in the formation of

simultaneous queues.

Unusual or atypical events can also result in phase failures and associated re-queuing. For

example, a vehicle that does not respond in a reasonable time to the green indication (because it

is temporarily stalled, the driver is not paying attention, etc.) may cause an actuated approach to









gap-out prematurely, forcing this vehicle and all vehicles behind it to re-queue. CORSIM does

not model such atypical events, but they do occur periodically in the real world. As Courage &

Fambro [45] put it; "Simulation models introduce a stochastic element into the departure

headways based on a theoretical distribution. They are therefore able to invoke premature phase

terminations to some extent, but they do not deal with anomalous driver behavior".

A phase failure may be either "liberal" or "strict". A strict phase failure occurs when a

vehicle that was queued when the signal turned green is forced to re-queue when the signal turns

yellow then red. A liberal phase failure occurs when a vehicle joins the back of the queue during

the green indication but is forced to re-queue when the signal turns yellow then red. It should be

noted that the analysis process developed for this research recognizes both types of phase

failures, whereas CORSIM only reports strict phase failures.

It is worth noting that the number of vehicle re-queues is equal to the number of vehicle

stops if the first stop is ignored.

When the side street approach under investigation receives the red indication, vehicles

begin to queue at the stop bar. The time during which the entire queue is within the FOV and

can be "seen" by the detection system is referred to as the visible period.

Eventually, the queue fills-up the FOV and the detection system can no longer measure the

exact queue length. When this occurs, the system transitions from a visible period into a blind

period and the prediction process must begin for the non-visible queue. Figure 4-8 provides an

example of a blind period. During this blind period, vehicles attach themselves to the end of the

non-visible queue at some unknown rate, referred to as the actual arrival rate. The portion of the

blind period during which vehicles can attach themselves to the back of the non-visible queue,

but cannot leave the front of the non-visible queue since the signal has not yet turned green and









there are vehicles queued ahead of them, is referred to as the rising queue blind period (which

occurs from time T-7 to time T-34 in Figure 4-8).

Eventually the side street approach receives the green indication and vehicles on that

approach begin to cross the stop bar. The visible queue shrinks from the front until the last

vehicle in the FOV begins to move and the visible queue becomes zero. At this point, vehicles

can begin to depart the non-visible queue from the front while they continue to attach to the back

of the non-visible queue at the unknown rate. We refer to this portion of the blind period where

vehicles can both attach themselves to the back of the non-visible queue and leave the front of

the non-visible queue, as the falling queue blind period (which occurs from time T-34 to time T-

72 in Figure 4-8). The length of the non-visible queue is typically falling during this period since

vehicles almost always depart the front of the queue at a much faster rate than they arrive at the

back of the queue.

For example, assume a field of view (FOV) of 12 vehicles. When the visible queue

extends to a point where the 12th position is filled by a queued vehicle, the rising queue portion

of the blind period begins. After some period of time the signal turns green and, eventually, the

vehicle in position 12 moves forward. When this vehicle moves forward the rising queue portion

of the blind period ends and the falling queue portion of the blind period begins. After some

additional period of time, a gap of sufficient duration (such as 5 seconds) is encountered between

successive vehicles entering the FOV, signaling that the end of the queue has come into view.

When this happens, the blind period has ended (which occurs at time T-72 in Figure 4-8).

A review of the Figure 4-8 example reveals that the non-visible queue actually shrinks to

zero well before the end of the falling queue portion of the blind period (somewhere abound time









T-50). However, because of the limited FOV, we cannot be certain that the non-visible queue

has dissipated until time T-72.

Many blind periods may exist over a given analysis time frame, with the number of blind

periods depending on the number of times that the end of the actual queue goes out of, and then

comes back into, the field of view.

If a vehicle does not enter the queue FOV for some sufficiently long period of time (for our

Figure 4-8 example, 5 seconds), and if another queue does not fill the FOV prior to this 5-second

period, then the blind period is considered to have ended and the system returns to a visible state

where the actual queue length is known. When this occurs it is assumed that there no longer

exists a non-visible queue (i.e., the non-visible queue has been "flushed out"). However, if this

5-second headway does not occur before the FOV is once again filled with queued vehicles, then

the system transitions from one blind period into another with no intervening period of visibility.

When this happens, adjacent blind periods occur (see Figure 4-9). As one might expect, the

problem of estimating the length of non-visible queues and their associated delay becomes more

difficult (and, hence, more approximate) as the frequency of adjacent blind periods increases.

As we shall soon discover, the number of adjacent blind periods is an important variable

when attempting to predict the length of the non-visible queue and its associated stopped delay.

The non-visible delay estimation algorithm contained within our analysis software makes use of

two counters (labeled A and D for Ascending and Descending) that are tied to the rising

queue/falling queue status as shown in Figure 4-10.

One important variable in the queue formation/dissipation process is the average time it

takes a vehicle to depart the queue once the vehicle ahead of it has begun to move. This time,

referred to by Long [46] as the queue startup lag time (or by others, and in this research, as the









queue departure time), is 1 second in CORSIM. However, field studies by Long at 4 sites in

Florida involving 140 queues of at least 16 vehicles in length (for a total sample of 1893)

resulted in a slightly longer average startup lag time of 1.15 seconds with a standard deviation of

0.52 seconds. Long also references work by Herman, et al., in 1971 that indicated an average

startup lag time of 1.0 sec and work by Messer and Fambro in 1977 that produced an average

startup lag time of 1.1 sec. One must use the 1 second startup lag time when trying to replicate

CORSIM behavior, however, the 1.15 second value measured by Long would be applicable

when analyzing actual field data.

The necessary computations for carrying-out the delay estimation procedure were

incorporated into a software tool called "BuckQ". BuckQ is a visual basic application program

for Excel which reads the data provided by TSDViewer and produces a variety of useful

information based on this data. BuckQ provides, for a one-hour analysis time frame having four

15-minute periods, a second-by-second tabulation of items such as queue length, back of queue

position, stopped delay, move-up delay and control delay. It also provides a host of ancillary

capabilities, including automated calculation of: start-up-lost-time, saturation flow, and capacity

by cycle; HCM queuing and delay information by 15-minute period; and arrival type by 15-

minute period. In addition, BuckQ allows evaluation of arrival patterns using a chi-squared

goodness-of-fit test and provides extensive graphing capabilities. However, the most important

feature of BuckQ is its ability to accommodate second-by-second queue and delay prediction

procedures and its ability to compare the results of these procedures to CORSIM results. Using

BuckQ, delay prediction algorithms can be tested to see how well they perform and the results

presented in a graphical format.









The following information is compiled by BuckQ on a second-by-second basis for the

entire 3600-second (60-minute) analysis period:

* Length of queue 1
* Length of queue 2
* Actual stopped delay
* Back of Queue position for queue 1
* Back of Queue position for queue 2
* Length of visible queue 1 (constrained by FOV)
* Length of visible queue 2 (constrained by FOV)
* Visible stopped delay
* Visibility status
= 1 when there is a "rising queue blind period"
= -1 when there is a "falling queue" blind period"
= 0 when there is no blind period

Development, testing and refinement of the various software programs was carried out

using a large number of data sets covering a wide range of near-saturated and over-saturated

arrival patterns and three cycle lengths (80, 120 and 160 seconds). The extensive testing was

necessary to ensure that both programs functioned properly over a wide variety of conditions,

including grossly over-saturated conditions.

Prediction Algorithm for Non-Visible Delay

One of the central elements of this research is the development of a predictive algorithm

that determines a reasonable value for the delay associated with the non-visible portion of the

queue. The first component of the algorithm is an estimation technique that uses the rate of

arrivals into the FOV to estimate the arrival rate at the back of the non-visible queue.

Non-Visible Queue Estimation Technique

Estimated NVQ Length = f(vehicles entering FOV during blind period, length of the blind
period, departure rate)

This technique assumes that vehicles arrive at the back of the queue at a uniform rate

during the full extent of the blind period. The arrival rate is calculated using the number of









vehicles that enter the FOV during the blind period. For example, if the blind period last for 32

seconds and 8 vehicles enter the FOV, then the estimated arrival rate is 8 vehicles/32 seconds or

0.25 vehicles/second. All of these vehicles enter the FOV during the falling queue portion of the

blind period, a time when traffic is freely flowing thru the FOV.

Vehicles are also assumed to depart the non-visible queue at a constant rate of 1 vehicle

per second during the Falling Queue Blind Period. Since the departure rate is almost always

greater than the arrival rate, the non-visible queue shrinks in size and, if sufficient green time is

provided, eventually disappears during this period.

As discussed previously, the blind period ends when a 5 second (or greater) gap occurs

between vehicles entering the FOV since a gap of this size suggests that we have come to the end

of the non-visible queue of vehicles. The blind period thus gives way to a period of visibility

during which we know for sure what the true queue length is because we can observe it.

In reality, it may or may not be true that a 5 second headway signals the end of the blind

period. It may be that the last vehicle in the non-visible queue passed some time ago or,

conversely, it may be that there are more vehicles in the non-visible queue but that some

"sleeper" (a slow truck, someone fiddling with their radio, etc.) has allowed a large gap to form

in front of him or her. The use of a five-second headway is a reasonable compromise between

these two situations, at least when we are dealing with a stream of traffic composed solely of

passenger cars. In any event, given a limited field of view, selection of some reasonably prudent

headway value that is neither too long nor too short under most circumstances is the best that can

be done.

Initial experiments have verified that this particular technique does a good job of

estimating non-visible queues and delays when a period of visibility follows the blind period.









However, when traffic volumes intensify, it is often the case that the FOV fills with queued

vehicles without a 5-second headway being observed. In this case, "adjacent blind periods"

occur. The problem with adjacent blind periods is twofold: 1) The true number of vehicles that

arrived during the blind period is unknown because the FOV fills-up and all of the arrivals do not

come into the FOV, and 2) One never really knows where the true end of the queue is, forcing

non-visible queue length estimations to be made that depend on previous non-visible queue

length estimations. Additional adjustments are needed to handle adjacent blind periods.

When adjacent blind periods occur, the number of vehicles entering the FOV during the

blind period may substantially underestimate the number of vehicles that arrived at the back of

the non-visible queue during the blind period. A second "adjustment technique" is needed to

augment the initial "estimation technique" when this occurs.

Non-Visible Queue Adjustment Technique:

Adjusted NVQ Length = f(vehicles entering FOV during blind period, length of the blind period,
departure rate, adjacent blind period counter) = f(estimated arrival rate, departure rate, adjacent
blind period counter)

The adjacent blind period counter increments by a value of 1 whenever a blind period is

followed by another blind period, and resets to zero when a period of visibility occurs. The

estimated arrival rate is increased using an additive power function of the following form:

ARadj = ARest + [(ABPC + C)P]/X

Where: ARadj = Adjusted Arrival Rate
ARest = Estimated Arrival Rate
ABPC = Adjacent Blind Period Counter
C, P, X = Constants

The longer the end of the queue stays "out of view", the higher the ABPC becomes and the

more the adjusted arrival rate is increased in comparison to the estimated arrival rate. Extensive









testing suggests that the following constants provide good predictive abilities, even during highly

over-saturated conditions where some vehicles experience as many as six phase failures:

P= 0.4
C = 66
X =30

These constants can be varied to change the shape of the predicted cumulative delay curve.

Figure 4-11 is the base condition where P, C and X equal the values just listed. If P, the power

constant, is increased from 0.40 to 0.41 while holding C and X constant, the entire curve shifts

upward as shown in Figure 4-12. If C, the additive constant, is increased while keeping P and X

at their original values, then the curve both increases and flattens out. If X, the division constant,

is decreased while keeping P and C at their original values, then the tail end of the curve shifts

upward. The optimum combination of P, C and X that results in a predicted cumulative stopped

delay curve that most closely follows the actual cumulative stopped delay curve is obtained

through trial and error.

Non-Visible Queue Re-Adjustment Technique:

Re-Adjusted NVQ Length = f(vehicles entering FOV during blind period, length of the blind
period, departure rate, adjacent blind period counter, average headway, average free flow speed,
average vehicle length) = f(adjusted arrival rate, average headway, average free flow speed,
average vehicle length)

As a queue becomes longer the back of the queue propagates closer to its source of

arrivals. This tends to increase the effective arrival rate of vehicles at the end of the queue.

Hurdle [2] recognized this fact in his investigation of intersection delay:

"Another way of thinking about the model is to say that, in the model, vehicles do not line
up along the street but form a vertical stack at the stop line. The real queue is always
somewhat longer than the model predicts because the queue engulfs some vehicles that the
model assumes are still driving to the vertical stack at the stop line".









Figure 4-13 provides an example. In this example, an additional arrival effectively occurs

once every 60 seconds due to queue propagation. This adjustment becomes significant as

volume exceeds capacity and queues become extensive.

Examples

To demonstrate the analysis procedure, four examples based on a 120 second cycle length

were developed. Each example uses a different set of arrival rates that result in over-capacity

conditions at some point during the one-hour analysis time frame. Three runs replicationss) were

made for each example with a different random number set used for each of the three

replications: See Table B-29.

Tables 4-1 and 4-2 summarize the characteristics of these examples while Tables 4-3

through 4-5 summarize the predictive results.

The first column of each table lists the Random Number (RN) set that was used and the

second column provides an abbreviation of the file name that includes the 15-minute volumes

that were input into CORSIM. Considering the first row, random number set 1 was used and the

15-minute input volumes were 625 vph, 700 vph, 650 vph and 350 vph. The input volume for

the last 15-minute period was always set at a relatively low value so that all residual queues

would clear by the end of the one-hour analysis time frame. This ensured that all delay was

accounted for.

Because of the random fluctuation in arrivals, the arrival flow rates input into CORSIM

are, in almost every case, not the same as the arrival flow rates that actually enter the link. For

example, the 625, 700, 650, 350 vph input flow rates associated with random number set 1 (the

row 1 values) produce link entry flow rates of 640, 692, 628 and 364 vph. By the time these

entering vehicle reach the back of the queue, the arrival flow rates have changed once again to

the 676, 688, 652, 360 vph values shown in Table 4-1. It is these arrival at BOQ (Back of









Queue) volumes that are of interest because it is these volumes that contribute directly to the

formation of queues and the associated stopped delay. Arrival at BOQ volumes are also

provided for the hour as a whole and for the first 45 minutes of the hour (the portion of the hour

during which near or over capacity conditions exist).

Also provided in Table 4-1 are the approach capacity values for each 15-minute period;

along with the capacity value for the first 45 minutes of the hour. BuckQ automatically

calculates the capacity values by applying the methodology described in Chapter 16, Appendix H

of the Highway Capacity Manual [4] to traffic stream information obtained from CORSIM. In

order to calculate the capacity our analysis procedure determines, for each 15-minute period, the

needed intermediate variables such as queue discharge Headway (H), Start-Up Lost Time

(SULT), and effective green time (g). The Extension of Effective Green (EEG) is determined for

the first 45-minutes of the hour by minimizing the sum of the squared deviations between the

cycle-by-cycle capacity values calculated using the Highway Capacity Manual procedure and

actual cycle-by-cycle thruput. A review of Table 4-1 indicates that the calculated capacity

values show considerable variation. This is not surprising when one considers the substantial

degree of variation in driver behavior that has been incorporated into CORSIM, including

variations in driver aggressiveness associated with departing the queue (which affects both

SULT and H) and in making use of the yellow and all red change interval time (which affects the

EEG). All drivers do not behave the same and CORSIM correctly recognizes this.

Volume-to-capacity ratios are calculated for each 15-minute period and for the first 45

minutes of the hour. These values are also provided in Table 4-1. For individual 15-minute

periods, the v/c ratio varies from a low of 0.92 (RN set 2 for file 625_700_650_350) to a high of

1.24 (RN set 1 for file 725_700_700_350). For the first 45 minutes of the hour, the v/c ratio









varies from a low of 1.02 (RN set 2 for file 625_700_650_350) to a high of 1.12 (RN set 1, 2 or

3 for file 725_700_700_350).

A review of the average values shows that, for the first 45-minutes, both volume and v/c

ratio steadily increase as one moves down the table, while capacity remains constant at 644 vph.

The average volume increases from a low of 664 vph to a high of 722 vph while the average v/c

ratio increases from 1.03 to 1.12

The first section of Table 4-2 summarizes various values used for capacity analysis,

including cycle length, green time, queue discharge headway, saturation flow rate, and start-up

lost time. Our analysis procedure calculates these values on both a cycle-by-cycle basis and a

15-minute period basis as well as for the entire hour, but only the hourly values are presented

here. As the v/c ratio increases, the amount of green time (G) increases to its maximum setting

of 38 seconds, and the cycle length (C) increases to its maximum value of 120 seconds. This

makes sense for an actuated approach. The extension of effective green, start-up lost time, queue

discharge headway, and saturation flow rate all remain about the same as the v/c ratio increases,

which also seems reasonable. The overall average queue discharge headway of 1.81 seconds is

very close to the 1.80 CORSIM input value. However, the overall average start-up lost time

value of 2.7 seconds is significantly greater than the 2.0 second mean start-up delay input into

CORSIM. The difference is due to a definition inconsistency. CORSIM only applies the mean

start-up delay to the first vehicle, adding additional delay (of about 0.7 seconds) to subsequent

vehicles. In other words, CORSIM's mean start-up delay is not the same as start-up lost time.

The next section of Table 4-2 provides a quality control check on the results for actual

stopped delay and control delay during the one hour analysis time frame. This check is made by

comparing the values obtained from our analysis procedure to similar values found in the









CORSIM output report. Considering the delay definitions used in CORSIM, we would expect

CORSIM Stop Time to approximately equal the actual stopped delay obtained from our

procedure, and we would expect CORSIM Queue Delay to be slightly greater than the actual

stopped delay. This is true in every case. We would also expect CORSIM Delay Time to

approximately equal the actual control delay obtained from our procedure, and we would expect

CORSIM control delay to be slightly less than the actual control delay. Once again, this is true

in every case. As we might expect, the amount of both stopped delay and control delay increases

as the v/c ratio increases.

The final section of Table 4-2 summarizes, for the Poisson distribution, the chi-square

goodness-of-fit test results based on 20-second arrival intervals. During only one of the forty-

eight 15-minute periods examined (2% of the time) did the test statistic exceed the 95th

percentile reference statistic. CORSIM 6.0 appears to be generating truly random arrivals. It is

important to use 20-second arrival intervals when conducting this test since the use of longer

intervals reduces the number of available data points while the use of shorter intervals can give

rise to truncation effects that distort the results. The truncation effects arise because unsafe

headways of less than 1.5 seconds are rarely encountered within the CORSIM traffic stream.

Queue Prediction

Table 4-3 summarizes the queue prediction results for our analysis procedure as compared

to actual queues. Comparisons are made of average queue length, maximum queue length,

maximum back of queue position, and 98th percentile back of queue position. Figure 4-14

depicts actual queue length as a function of v/c ratio while Figures 4-15 through 4-17 compare

actual and predicted queue results for the average queue length, the maximum queue length, and

the 98th percentile back of queue, respectively. Figures 4-14 through 4-17 all demonstrate that,

as might be expected, queue length tends to increase linearly as a function of the v/c ratio. A









review of Figures 4-15 through 4-17 also indicates that our procedure is fairly good at predicting

all of these queues, with the amount of error increasing somewhat as the v/c ratio increases. The

procedures contained in the Highway Capacity Manual, provide information on the 98th

percentile back of queue. A review of Figure 4-17 indicates that the HCM procedures grossly

overestimate the 98th percentile back of queue.

Also provided in Table 4-3 is information on the number of (liberal) phase failures, the

percentage of cycles experiencing a phase failure, and the number of vehicle re-queues. Phase

failures are defined in relation to the cycle and, as such, are insensitive to the number of vehicles

involved. For example, a phase failure occurs for a given cycle if only one vehicle is forced to

re-queue, or if 100 vehicles are forced to re-queue. For this reason, the number of vehicle re-

queues is a much better indicator of the extent of congestion than the number of phase failures.

Figure 4-18 demonstrates that the number of vehicle re-queues tends to increase linearly as a

function of v/c ratio.

Stopped Delay Prediction

Table 4-4 summarizes the stopped delay prediction results for our analysis procedure as

compared to actual stopped delay. Figure 4-19 indicates that the procedure does a pretty good

job of predicting stopped delay over all v/c ratios.

Figure 4-20 shows the relative contribution of each segment of the prediction

methodology. For the examples under consideration, visible delay makes-up about 60% of total

stopped delay when the v/c ratio is near 1.02 but only 20% of total stopped delay when the v/c

ratio climbs to 1.12 This clearly demonstrates the need for this predictive procedure, at least for

the rather typical case where the cycle length is 120 seconds and the field of view is limited to 12

vehicles. The first step in the predictive process uses an estimated arrival rate based on vehicles

entering the field of view to predict the non-visible queue. This alteration increases the









percentage of captured stopped delay to about 80% when the v/c is near 1.02 and to about 30%

when the v/c is near 1.12. The results become reasonable for relatively low over-saturated v/c

ratios but not for the higher ratios. The second step in the predictive process uses an adjusted

arrival rate obtained from a power function adjustment that increases the estimated arrival rate

based on the number of adjacent blind periods. This alteration increases the percentage of

captured stopped delay to about 115% when the v/c is near 1.02 and to about 65% when the v/c

is near 1.12. The results are still reasonable for relatively low over-saturated v/c ratios, and are

greatly improved for the higher ratios, but the error for the higher ratios is still quite significant.

The third step in the predictive process adjusts the non-visible queue length and associated delay

due to queue propagation. This alteration has little or no affect on the percentage of captured

stopped delay when the v/c is close to one but increases the percentage of captured stop delay to

about 90% when the v/c is high. The results are now reasonable over all v/c ratios although a

slight upward bias of about 15% exists near the lower oversaturated v/c ratios and a slight

downward bias of about 10% exists near the higher v/c ratios. A tremendous improvement in

stopped delay estimation is clearly provided by our procedure. Figure 4-21 provides another

way of visualizing the final predictive results.

The maximum individual over-estimation of delay is 27% and the maximum individual

under-estimation is 17.5%. If the results are averaged over the three random number replicates,

as is documented at the bottom of 4-4, the maximum over-estimation is 13% and the maximum

under-estimation is 11%.

If we graph the sum of the Adjacent Blind Period Counter (ABPC) against stopped delay

(either actual or predicted) as shown in Figure 4-22, a strong linear relationship exists. This









provides rather strong support for our use of the ABPC as the explanatory variable in our arrival

rate adjustment process.

Control Delay Prediction

Table 4-5 summarizes the control delay prediction results as compared to actual control

delay. Figure 4-23 indicates that the analysis procedure also does a reasonably good job of

predicting control delay over all v/c ratios, even if we use a constant ratio of 1.3 to convert our

predicted stopped delay into predicted control delay. This conversion factor actually varies

somewhat by v/c ratio as shown in Figure 4-24. (Previous work has demonstrated that this factor

also varies by cycle length; but that is not of concern here since we have restricted our analysis to

a single cycle length.) Also included in Figure 4-23 is control delay as predicted by HCM

procedures. The HCM procedures tend to over-predict control delay for the lower over-saturated

v/c ratios.

Figures 4-25 and 4-26 provide two other ways of visualizing these comparisons between

actual control delay, predicted control delay, and HCM calculated control delay.

Control delay is composed of stopped delay, acceleration/deceleration delay, and queue

move-up delay. As shown in Figure 4-27, the percentage of stopped delay for our example

remains relatively constant at about 80% of the control delay. This is consistent with the fact

that the control delay/stopped delay ratio does not change much as the v/c ratio increases.

However, the percentage of queue move-up delay increases dramatically (more than doubles) as

the v/c ratio increases and the percentage of acceleration/deceleration delay falls

correspondingly. Recurrent cycle failures and extensive re-queuing associated with high v/c

ratios produces this steady and dramatic increase in queue move-up delay. Figure 4-28 provides

factors that convert "stopped delay plus queue move-up delay" to control delay. A review of this









figure reveals that there is much more variation in this new ratio than with a ratio based only on

stopped delay.

Variability Considerations

To investigate the degree of variability associated with the actual cumulative stopped

delay, and with the predicted stopped delay, ten replicate runs were made for the

700_725_625_350 volume pattern using the sets of random number seeds found in Table B-30.

The last number in the set produces vehicle behavior variation associated with various

driver aggressiveness characteristics, including driver response to the amber interval, the amount

of start-up lost time experienced by the first vehicle in the queue, the discharge headway of the

vehicle, and the free flow speed of the vehicle.

Table 4-6 provides a comparison between the actual 1-hour cumulative stopped delay and

the predicted stopped delay. A review of the embedded graph in this table shows that the

variation in the predicted stopped delay is very similar to the variation in the actual stopped

delay, with only of the 10 data points (the one associated with random number set 8) exhibiting a

somewhat unfavorable comparison. This similarity in variation provides some reassurance that

the prediction procedure is behaving appropriately. It is also encouraging to discover that, as is

shown in Table 4-6, the 95% confidence interval for the mean actual stopped delay includes the

mean predicted stopped delay.

Formal statistical testing was conducted to determine whether a significant difference

exists between the actual and predicted median stopped delay. The non-parametric Fisher Sign

Test, which does not require a symmetrical distribution, was used to test the null hypothesis that

the mean of the differences between the actual and predicted median delay is zero. Table 4-7

contains the test, which produces a p-value of about 0.11 The p-value is not significant so we

cannot reject the null hypothesis that the mean of the differences is indeed zero, which reinforces









the idea that the prediction procedure does a relatively good job of estimating the total

cumulative stopped delay.

Limitations to the Delay Prediction Procedure

Our analysis procedure includes a new technique for predicting delay on a signalized

intersection approach under conditions of limited information. Although the usefulness of the

technique is evident, limitations on the use of the technique should be understood. These

limitations include the following:


1. As the size of the field of view decreases, the accuracy of the technique also decreases.
Testing to date has concentrated on a field of view of 12 vehicles with additional runs
made at a field of view of 8 vehicles. Reasonable results are obtained with these fields of
view up to a v/c ratio of about 1.12 for the over-saturated periods. More testing is needed
to determine the maximum v/c ratio that can be accommodated with smaller fields of
view.

2. The current delay prediction technique can produce rather inaccurate delay forecasts if
"sleepers" are present at critical points in the non-visible queue. A "sleeper" is defined as
a motorist that does not exhibit normal car-following behavior within the queue; leaving
a large gap between his or her vehicle and the preceding vehicle in the queue. This type
of lethargic driver behavior can be caused by in-vehicle distractions or by simple
daydreaming. Under the current analysis methodology, the abnormally large gap
between vehicles caused by sleepers can result in a false conclusion that the end of the
queue has been reached. This causes the adjacent blind period counter to be lower than
desired which results in a correspondingly low adjusted arrival rate. The end result is an
underestimation of delay.

3. Our analysis procedure is essentially a queue prediction technique that uses predicted
queue length to calculate expected stopped delay. Consequently, by its very nature, the
procedure is relegated to directly predicting stopped delay, not control delay. The
emphasis on stopped delay makes sense when one considers the limited information
made available to the program. The program assumes no knowledge of various items
important in the direct calculation of control delay; including vehicle free flow speeds
and delay associated with both deceleration and acceleration most of which occurs
outside the field of view. Changing stopped delay to control delay requires the
application of a delay ratio. Typical delay ratios (such as the commonly used 1.30 value)
will need to be applied and there will be some inherent error in this factoring process.

4. If a motorist joins the queue and experiences delay but then, prior to entering the field of
view, becomes impatient and leaves the queue (known in the queuing literature as
"reneging"), the delay experienced by this motorist will not be accounted for. Any









"delay" associated with motorists that decided not to join the queue due to its excessive
length (known in the literature as "balking") would also not be accounted for.

5. The research to date has concentrated on random arrivals at an isolated intersection.
Some initial experimentation was conducted with platooned arrivals and, based on that
work, it is clear that the delay situation can change quite a bit depending on the relative
offsets of the upstream intersection and the intersection under study. This platoon
progression effect is well documented in the literature. Consequently, the analysis
procedure is less suitable for use on coordinated approaches, especially during under-
saturated or near-saturated conditions. For over-saturated conditions, platoon
progression effects on coordinated approaches tend to be minimized since all
approaching vehicle are forced to join the queue. The analysis procedure should perform
well under these conditions.

6. Work completed to date is based on a single micro-simulation tool and is subject to all
limitations and characteristics of the CORSIM software.

A final drawback is that the analysis procedure is still in the form of a research tool that is

oriented towards evaluating simulation runs. Converting the procedure to a practical engineering

tool that can be field implemented at a real intersection is an important extension that will require

additional effort.












T-I n


*TO Nrr rR


T-30 M c--~E


T-40 -4-


T-45 S 0


i- smarauf
7 STaP
E l- !L- iI-W E ] [-'II-

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1 a E" lf 1i
I I, I"*,


ZI- B BA


F STwPWt

i | 11


O EN 1-n-,%


Eli


|n


I0


LEGEND


SUMMARY STATiSTrCS
IMAX Q LENGTH 7
VAX BOCQ PfSITToFWN 10
VG. Q LENGTHm 1 4j.8

Figure 4-1. Queue relationships
Figure 4-1. Queue relationships


SIGNAL INDICATION
R RED
G = GREEN
VEHICLE OX
T TIME


Ho Q T^








DELAY ZONE


DEE


ED


CONTROL DELAY DECELERATION DELAY + STOPPED DEAY + QUEUE MOVEUP DELAY + ACCELERATION DELAY


Figure 4-2. Signalized intersection delay components


D


L


- STQPMR

H3


0











TRAFVU OUTPUT FROM
NETSIM (CORSIM)


= r_


D mE


12 VEHICLE
FIELD OF VIEW -
(FOV)


0




0

n
0

D









0


1 0 0 IZ E- I Z 1 1 3 = L-






MEASURE
DELAY
INSIDE FOV







ESTIMATE
DELAY
OUTSIDE FOV
I


Figure 4-3. Measured versus estimated delay


BIGGER FOV IS BETTER.
BUT ACHIEVABLE FOV DEPENDS ON
THE DETECTION SYSTEMS USED









0
J U


I| = VISIBLE VARIABLES
Q= NON-VISIBLE VARIABLES


QUEUE WITHIN FOV
(VISIBLE QUEUE)

12 VEHICLE
FIELD OF VIEW---
(FOV)




QUEUE BEYOND THE FOV
(NON-VISIBLE QUEUE)


VEHICLES LEAVI-NG FOV
N VEHICLES LEA.WfGQ


PRESENCE OF STATIONARY
VEHICLE AT END OF FOV

f: VEHICLES ENTERING FOV j


LENGTH OF TIME THAT QUEUE
EXTENDS BEYOND THE FOV


VEHICLES ARRIVING AT BOO


Figure 4-4. Visible and non-visible variables


XXI


C XXX_ ~


I













* 80 second cycle

* 120 second cycle

A 160 second cycle


o

o
. 1.25 2 0.502_A







.o The ratio of control delay to stopped delay decreases as the v/c ratio y = -0.3342x + 1.5652
1.3 .increases; the amount of the decrease varies with cycle length with R2 = 0.5042
1.20 -A A -
shorter cycles experiencing a more dramatic drcrease.

1.05




0.96 0.98 1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14 1.16
v/c Ratio During 1st 45 Minutes
vic Ratio During 1lst 45 Minutes


Figure 4-5. Relationship between v/c ratio and ratio of control delay to stopped delay


1.50


1.45


S 1.35










F-- STOFMP


T I Go -0-- |1C


r-4 M


CD rnrn rnrn rnJ 0-3
ii__ IIL.L _
v~CLE C


i-- rnWB r



a2 Cn-Ju
TADCKAoW5'


S 01 | =5


I sT0LLr
Ca, bJ~


Q0 =3
02=2


r-7 0


I EZWLI W W W

MT CUMLE


1EIGEND


X SIGNAL ~IICATKIN
R = RED
G GRHCLEEN

SVEHKILE #X


Figure 4-6. Re-queuing that results in simultaneous queues


01- 4










- T-wi I ^ R


r-4 *- I a w


o


1 T"a'l IP IL.lF


LE GE ND




rnl


I 3S 5 E 7
1sT am4r


w


r


[p~7~7;2


EE3


SIGNAL IN DICATION
R = RED
G GREEN
VEHICLE 0X


Figure 4-7. Re-queuing that does not result in simultaneous queues


EIKl


ww wI r 3
^------
i*T QUUE


fE .BTbPm-M

avana














1-4


S~mrnwrrnnE~E
a 4k IIaI r. I ~


P~tdDP ~Uc? I"~"I
BL"DPMM;A


T-25 M1
1 s1 Ia 14 rb I1I TI II11110I 1 I r
EEN WA1
ho Lem-rD FLuP,

QUSM POWWW~~ P~llw

BUNDPERICOUW


r1-4


LIM


I e WL'I 2


lM 21


131 6 1 II [Irr m 1 ID I E ?JI l L :
9 91RTG


TI


~~~sz ~ ~ ~ p ..sL~kL-LIIIu ~ d


I.Ll t


QyfLC r,4,,R-o,,ej
1: t aLr4:: U, 1 "L""


a GREEN
rn vwFrEzvx


r~e 'LfbClni NL ~0IBA L'aE :r.~1 OaJFI~~PrR CL' r .0
IM'd 111F. I PJ1 AI MI vlb CaE t a J
P.'d'EH tF,3 Vd F PAR, C I.-E Kj vi E bL UuL-k J
AN F.J" QIWA IJC ~~~PB~ CFbU~I O -


Figure 4-8. Example of a blind period


79, Mr-11M =7 IM


Mala


T-q -r-17












r, wQw


H BUMD3E pZL
T47 7


!aLb fo'gl] r1i ew

IgT GREu EN I PERM


wwwmmMIrj ini^j


- Urn.I
'.0-It'v


'rNFtit
r<->


H ',bv ,
r1i pn II u y- in 11 j- innnn
'" b-4--I ~~l|.HlE
YLf.I.~~I


L2! ~a1flhtI


t63; ~
o W mrI


MI


M


- pit m I


IJ


renPI I


a1


~L1!u!
fl w w w~~


bLjrut I
M^_ r a D B M "





I tft ILL
Ell Wrk fli W


a I






110 AM
.1UAL Ibll
II fMll


Lj


" rim


u]-,Ip.
I*.1.h chLL F'I rlic rR T .M ''h. 4An 'ntn.1Iidi .Irc:i a .c l U ,,rr 1 ; r i tM .' -2
i rIc lC ln i f \ .-L f lQ 4 ILJA DU I~ I .L i i PI Lq. I .P r i i I'n
I;,-I.Ir IHI qL9.r. bulMaint r.- pLJittj r -Jimio it lL:

J. Vt.I I Lr .. Lr t Mi To t 6 hiW 1 ; ).a 0 s'Ii I I --.


Figure 4-9. Example of adjacent blind periods


iwa.1t^


------~r-l ---~----- -------~ICCILh- -----------~-I--~


I


B


--


an nlmr













rnrrnB rr'


Fr-15 a


rf- r-"


r-r B E1 F3li1


mn


-f.-.
EE czp MEriDig


r~-


1


wi


I STRT DFR UtI QU LJEUEEtDi PEP
e.unw rA A -* wtgalA
CDDTRCEH = 0



| El W3BINgQEUEB.EL MPERI I'
'EHQEB CaMd AIvEAlT BQ ErJT
C4OT DEARTB NCON.-VILE QUELE
EKEIJTR 1i


IED MO RiSIM u EJE IBUND PERIO
VEHCLEB C AJ~1HVE AlT BOCI UT
eamaro TI DF*JF w"-rVlim "ajtL
DXJITER.bA RO* Ir t(iC-atu
COulNT a 0


FAMULLM JEUE BUMIPEIDD
WEHXlES D t AMIVoE AT BC AND
C.W EPjART NO VI mBLE UIEUE
c0M1jw -A HW CuWCM h
CUiTHER -D 4SCRE WibNFOf


............................--
I I y,
sir4MA

r--- r------
^ t*[j;~ ~~i
Ias num e


R4 ED
Q REE
[_.] ym IAt


Figure 4-10. Counters and queue status


I EE OFR IF Jh EimP l]


n-Eet8ie
man
Y.SKQVE


aWEl a
004


CNN
r"
0m









140WM


60000






%fl'h^E VISOB f


40012
ll l0 '-----------------------------





0 201 401 601 703 1001 1201 1401 1601 18~ 2001 2201 22401 201 2601 3001 3201 3401

TIME (SECI


Figure 4-11. Base case for P, C and X; stopped delay comparison














140000



12000D


100000


U



a
Di
-J
UI


60000


0 201 401 601 701 1001 1201 1401 1601 1801 2001 2201 22401 2601 2801 3001 3201 3401

TIME {SEC)


Figure 4-12. Effect of increasing the power constant on stopped delay comparison













I v- I- ___. -r-






- -l TOP-H 1 1
WWufnun



rE


1 (5 1 SC LW4 <) 4 h c F* fC) LT40 HM ASOO.CATID W TH OW
ft f AllEk IH.01Ya AT AVRAM
AVEPAGIE AVtAAft
it04X IFUOIN


18


lb 0.00!TPitfl SEC


.p.C"AVRKMA It It*ES *BOA 11
V*LH.XS INOU f. A 051AIE EQUAL
TOWS AWA= f*NADMAY
YovH *ERAGE* IT TAKES hBOu7k OE
WMVIE FCR st E. ELEUs TO
RAOPOCGATi A CMTMCl OF OeW
AVCRACE WADAik THE4REFM VWI
4ese#-T OrF AgT*CtkARrarn. ,EaW

SAO m*N. AJWlvAZ.VEtfRr SEO0O


R 00&7B VE'EC- 18 5 FTNM
I '[S 1 ECNAMW 42.4 FTSE"C5 its I BIaCArM


ADIXTNAH. AIWiV AL AvEja V L.ENOT
EVe iRBECEOA "AVE FEAM iAUKAVERAGE FSP


Figure 4-13. Queue propagation example


Z5i


"A-"f I I LtYa








100

90

80

70

60 -

50

0 -40
o "

30

20

10


01.
1.0


120 sec

ecI


I0


1.02


1.04


y = 520.6x- 492.0
R2 = 0.879


1.06 1.08 1.10
v/c Ratio During 1st 45 Minutes


J1..


M ly= 226.9x-221.9
-- R2 = 0.928


1.12


1.14


Figure 4-14. Actual vehicle queues


* Average Queue

* Maximum Queue

- 98th% BOQ Position

I


I + +


-aO00


_.wUj


Cc


y = 416.4x- 391.8
R = 0.915












0 Actual ....
35 120 sec

12 FOV Predicted
30
5o -

S25



8 20

Cn










1.02 1 04 1.04 1.07 1 07 1.07 1.08 1.09 1.10 1.12 1.12 1.12

v/c Ratio During 1st 45 Minutes


Figure 4-15. Average queue length comparison









80


70


60


S50
-

40





E
E
1 10
0
o

0 20


SOActual
12 FOV Predicteds
1012 FOV Predicted


1 02 1.04 1.04 1.17 1.07 1.07 1.08 1.09 1.10 1.12 1.12 1.12

v/c Ratio During 1st 45 Minutes


Figure 4-16. Maximum queue length comparison


I I


: : : :


::::


: : :











0 Actual

m12 FOV Predicted

o HCS+


1.02


1.04


1.07


1.07


1.08


1.09


110


1.12


1.12


rfl


1 12


v/c Ratio During 1st 45 Minutes


Figure 4-17. 98th percentile back of queue comparison


140



120



100



Z_ SO -
vr

o
0)


0a

3


a4
Pc
a)
(0


-








1500
1400
1300- Individual Data
1200 Points
Average Data
1100 -
10Points0 y =11066x 11170
R2 = 0.927
900 -- 120 sec cycle
)800 -
= 700
00

500
3400
>300

200 -
100 0

0

1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14
v/c Ratio During 1st 45 Minutes
Figure 4-18. Vehicle re-queuing










300


250

S1 2FOV
Predicted
200




V 150




100


50 --!|






1.02 1.04 1.04 1.07 1.07 1 07 1.08 1 09 1.10 1.12 1.12 1.12

vic Ratio During 1st 45 Minutes


Figure 4-19. Stopped delay comparison







200.0%
Readjusted Delay

i U0 Adjusted Delay
---------- 120 sec cycle -----------------------
0 Estimated Delay

L 0 Visible Delay












0.0% ------I I
1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14
v/c Ratio During 1st 45 Minutes
Figure 4-20. Stopped delay prediction, 12 FOV








200 I I I I I
190 12FOV- Predicted -----
180
170 Average Data -- --
160 .
150 Points
150
140----- ---
130____ ___
S120 -
S110 120 sec cycle
I 100
90
8 80
" 70 _
60
0 y 1.186x 22.91
40 R= 0.860


10 #
0 o

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200


Predicted Stopped Delay (sec/veh)


Figure 4-21. Comparison of actual and predicted stopped delay








200
190
180
170
160
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0


10000


20000


30000


40000


Sum of Adjacent Blind Period Counter


Figure 4-22. Adjacent blind period counter v. stopped delay


2, 20 sec cycl _


- y = 0.002x + 63.79
P2R = .8n Q 3
-g




__ lActual
y = U.u2Px + 45.01
A = 12 FOV
Predicted


02

a;
9
a.
a.
0
a


50000












300 -


250



200



D150



.100
a


5O
50



0


1 02


'120 e yl


. .- -


.. --- ----- __


1.04


1.04


1 07


1.07


1.07


1 08


1.09


1.10


1.12


vic Ratio During 1st 45 Minutes


Figure 4-23. Control delay comparison


0 Actual

*12 FOV Predicted (1.30)

oHCM


1.12


1.12








1.45


1.40 +-


* Individual Data
Points
* Average Data Points


-4 .5 .6 S


1 20 sec cycle







y -0.32x + 1.605
R2 = 0.714


1.00


1.02


1.04


z 1.35


1.30


1.25


1.20


1.15


1.12


1.14


Figure 4-24. Ratio of control delay to stopped delay


I I I
1.06 1.08 1.10
vic Ratio During 1st 45 Minutes













250



200



150






50
50



0


--20 sec cycle




HCM

16-H
agW


I I _i


1.00


1.02


1.04


1.06


1.08


1.10


1.12


v/c Ratio During 1st 45 Minutes


Figure 4-25. Graphical control delay comparison,











300.0% -










8 200.0% -
a.









00.0%










0.0%


- -------- ----- ----- ---- --- m'


- 120 sec cycle ---


--------]

---PNREDi

-----


----------
---------

----------

[-- ---- -- -- -


1.00


I - -- -
CTED ACTUAL





----- ACTUAL


1.02


1.04


1.06


vlc Ratio During 1st 45 Minutes


Figure 4-26. Control delay estimates


----


----


I


1.08


1.10


1.12


1.14


-


P


i


1


- - - -

-------- ------ ---


--A7-- ---------
-------- -----------
--- --- -- ----- -- -

---------------------
---------------------
--------------------

--------------------


------- -- -------


iw----- ---------

---------- ---------
----------- ----------
----------- ----------
------------ ----------
----------- ----------
----------- ----------
----------- ----------









100%
90%
80%
70%

60%
S50%
- 40%
0 30%
U
4-20%
E10%
0%


S (I t -N cc N- S. 0 N N N
0 0 0 0 4 0 0 M0 -
-v r: V r rfo r s T4 T-
v/c Ratio for First 45 Minutes


Figure 4-27. Control delay composition


SAccel/Decel Delay







1.30


1.25 120 sec cycle Individual Data Points-

1.20 -I e Average Data Points

1.15


l.1 y = -1.338x + 2.548
w R2 = Q0.893 -
1.05


1.00
1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14
vlc Ratio During 1st 45 Minutes


Figure 4-28. Ratio of control delay to stopped plus move-up delay







Table 4-1. Example summary- volume and capacity
File Name Random Arrival at Back of Queue Volumes Calculated Capacity and Volume-to-Capacity Ratio
15 min input volumes Number Set 15 min volumes 1st 45 minutes 1 hour 1st 15 minutes 2nd 15 minutes 3rd 15 minutes 4th 15 minutes 1st 45 minutes
v c vic
676 688 652 360 672 594 616 1.10 668 1.03 656 0.99 511 0.70 646 1.04
608 676 672 364 652 580 663 0.92 642 1.05 623 1.08 492 0.74 641 1.02
648 668 688 344 668 587 643 1.01 649 1.03 642 1.07 499 0.69 645 1.04
700 725 625 350vph 1 708 740 628 348 692 606 631 1.12 659 1.12 656 0.96 547 0.64 649 1.07
700 725 625 350vph 2 720 728 624 368 691 610 648 1.11 661 1.10 610 1.02 604 0.61 641 1.08
700 725 625 350vph 3 692 732 632 364 685 605 648 1.07 635 1.15 646 0.98 556 0.65 643 1.07
700 700 700 350vph 1 708 704 680 380 697 618 631 1.12 659 1.07 656 1.04 602 0.63 649 1.07
700 700 700 350vph 2 720 688 712 344 707 616 648 1.11 661 1.04 610 1.17 625 0.55 641 1.10
700 700 700 350vph 3 692 712 704 360 703 617 648 1.07 635 1.12 646 1.09 614 0.59 643 1.09
725 700 700 350vph 1 788 692 708 340 729 632 637 1.24 648 1.07 661 1.07 618 0.55 649 1.12
725 700 700 350vph 2 728 724 700 356 717 627 663 1.10 639 1.13 623 1.12 645 0.55 641 1.12
725 700 700 350vph 3 752 704 700 364 719 630 648 1.16 635 1.11 646 1.08 610 0.60 643 1.12
vph vph vph vph vph vph
Averages
644 677 671 356 664 587 641 1.01 653 1.04 640 1.05 501 0.71 644 1,03
700 725 625 350vph 707 733 628 360 689 607 642 1.10 652 1.13 637 0.99 569 0.63 644 1.07
700 700 700 350vph 707 701 699 361 702 617 642 1.10 652 1.08 637 1.10 614 0.59 644 1.09
725 700 700 350vph 756 707 703 353 722 630 649 1.16 641 1.10 643 1.09 624 0.57 644 1.12








Table 4-2. Example summary queue discharge, delay check and goodness-of-fit


___BuckQ Delay Check


Queue Discharge Data, Average Over All Cycles


File Name Random
15 min input volumes Number Set


Cycle Green
Length Time


Discharge Sat Flow Lost Ext.of
a/C ratio Headway Rate Time Green


Actual CORSIM CORSIM
Stopped Stop Queue
Delay Time Delay


CORSIM Actual CORSIM Goodness-of-Fit Test


Control Control Delay
Delay Delay Time


I I 4' I 7 4


700 725 625 350vph 1
700 725 625 350vph 2
700 725 625 350vph 3
700 700 700 350vph 1
700 700 700 350vph 2
700 700 700 350vph 3
725 700 700 350vph 1
725 700 700 350vph 2
725 700 700 350vph 3


C G g=G+EEG-SULT


118.5 35.7
116.7 34.7
117.6 35.6
118.8 36.6
119.1 37.1
118.9 36.9
119.8 37.6
119.5 37.5
120.0 38.0
120.0 38.0
120.0 38.0
120.0 38.0


S SULT EEGI ds =


2004 2.74 3.4
2009 2.73 3.4
1994 2.95 3.3
1999 2.56 3.4
2007 2.92 3.3
1980 2.60 3.3
1991 2.49 3.4
1998 2.71 3.3
1974 2.53 3.3
1985 2.65 3.4
2007 2.75 3.3
1971 2.52 3.3


(20-sec arrival intervals)
95% Ref. Statistic= 9.49


Chi-Square Test Statistic


I 4'


64.7
65.9
77.3
113.9
137.2
141.3
117.0
144.3
145.0
191.8
183.8
193.3


65.0
66.8
77.8
113.4
136.9
139.8
117.4
142.4
143.9
190.8
183.4
192.3


66.7
68.7
79.9
116.7
141.4
143.8
120.9
147.1
148.2
197.0
189.5
198.5


I -. I i


sec sec


sec/veh vphg sec sec


Averages
117.6 35.3 0.30 1.80 2002 2.8 3.4 69.3 69.9 71.8 82.5 88.4 87.5
700 725 625 350vph 118.9 36.9 0.31 1.80 1995 2.7 3.3 130.8 130.0 134.0 154.1 164.8 163.2
700 700 700 350vph 119.8 37.7 0.31 1.81 1988 2.6 3.3 135.4 134.6 138.7 159.8 170.5 169.1
725 700 700_350vph 120.0 38.0 0.32 1.81 1988 2.6 3.3 189.6 188.8 195.0 224.6 236.2 235.4


sec/veh


76.6 82.6 81.9
78.9 84.5 83.5
91.9 98.1 97.1
133.4 143.9 141.9
163.6 174.4 172.4
165.4 176.1 175.4
138.6 148.1 147.3
169.8 181.3 178.8
171.1 182.1 181.3
225.8 237.9 236.5
218.6 229.5 229.1
229.4 241.1 240.6


3.4 6.4 2.2
6.9 2.6 5.6 2.2
2.5 7.5 4.3 2.5
5.2 4.4 5.4 0.9
3.7 4.0 1.4 8.3
5.4 5.4 2.1 3.9
5.2 7.9 4.6 3.8
3.7 2.9 3.1 2.6
5.4 2.4 3.7 1.1
3.3 2.6 7.8 0.7
6.9 3.1 1.0 4.6
1.0 3.0 6.5 6.6


ALL 119.1 37.0


0.32 1.81 1993 2.7 3.3










Table 4-3. Queue prediction



File Name RNI Vol
15 min inputvol 4Seti45min v/c


I 12FOV


700 725 625 350
700 725 625 350
700 725 625 350
700_700_700_350
700 700 700 350
700 700 700 350
725 700 700 350
725 700 700 350
725 700 700 350


v
672
652
668
1 692
2 691
3 685
1 697
2 707
3 703
1 729
2 717
3 719


QUEUING


Average Queue Maximum Queue Maximum Back 98th Percentile
Length Length of Queue Position Back of Queue Position
A P %Err A P %Err A P %Err A P %Err HCM %Err


11 13 18% 32 38 19% 45 58 29%
11 10 -9% 37 34 -8% 45 47 4%
13 13 0% 37 35 -5% 45 53 18%
19 17 -11% 52 41 -21% 62 58 -6%
23 19 -17% 58 49 -16% 74 63 -15%
24 19 -21% 59 42 -29% 75 63 -16%
20 19 -5% 54 51 -6% 72 65 -10%
25 21 -16% 65 59 -9% 81 65 -20%
25 21 -16% 67 46 -31% 85 64 -25%
34 26 -24% 72 64 -11% 97 68 -30%
32 25 -22% 76 68 -11% 99 68 -31%
34 23 68% 75 51 -32% 103 63 -39%


58 41% 99 141%
45 2% 108 145%
52 24% 107 155%
58 -5% 117 92%
62 -11% 113 61%
62 -15% 123 68%
60 -5% 123 95%
63 -20% 114 44%
63 -24% 120 45%
69 -16% 139 70%
68 -28% 125 33%
63 -35% 128 32%


PHASE FAILURES


Actual
Phase
Fail


a


% of Actual
Cycles Vehicle
w/ PF Re-Q's


70%
63%
80%
80%
90%
90%
83%
93%
93%
97%
97%
97%


197
204
281
591
782
797
635
854
872
1308
1224
1334


vph veh veh veh veh veh veh veh veh
Cycles per Hour: 30.0
Averages
664 1.03 12 12 3% 35 36 1% 45 53 17% 42 52 22% 105 147% 21 71% 227
700 725 625 350 689 1.07 22 18 -17% 56 44 -22% 70 61 -13% 68 61 -11% 118 73% 26 87% 723
700 700 700 350 702 1.09 23 20 -13% 62 52 -16% 79 65 -18% 75 62 -17% 119 59% 27 90% 787
725700 700 350 722 1.12 33 25 -26% 74 61 -18% 100 66 -33% 91 67 -27% 131 44% 29 97% 1289


A = Actual
P = Predicted


I


----










Table 4-4. Stopped delay prediction


File Name
15 min input volumes


700_725_625_350vph
700_725_625_350vph
700_725_625_350vph
700_700_700_350vph
700_700_700_350vph
700_700_700_350vph
725_700_700_350vph
725_700_700_350vph
725 700_700_350vph


Random Volume
Number Set 1st 45 min


v/c
Ratio


1.04
1.02
1.04
1.07
1.08
1.07
1.07
1.10
1.09
1.12
1.12
1.12


Stopped Delay


Actual


BuckQ
Predicted


% of Time
Queue Not
Visible


Stopped Delay Prediction Steps


Visible


BuckQ
Estimated Adjusted Readjusted


t 1- ___ __


ds dsp


64.7
65.9
77.3
113.9
137.2
141.3
117.0
144.3
145.0
191.8
183.8
193.3


82.2
67.9
85.5
129.1
128.5
130.0
148.6
146.2
133.3
189.3
159.2
159.5


127.0%
103.0%
110.6%
113.3%
93.7%
92.0%
127.0%
101.3%
91.9%
98.7%
86.6%
82.5%


70%
63%
77%
83%
88%
88%
86%
92%
92%
98%
96%
99%


60%
57%
52%
36%
31%
30%
35%
29%
29%
22%
23%
22%


89%
80%
78%
56%
49%
47%
55%
47%
46%
36%
37%
36%


124%
101%
108%
93%
82%
82%
93%
79%
76%
66%
65%
63%


128%
104%
111%
113%
93%
92%
127%
101%
92%
99%
87%
83%


vph secs/veh


Averages
664 1.03 69 79 113% 70% 57% 82% 111% 114% 7057
700_725_625_350vph 689 1.07 131 129 99% 86% 32% 51% 86% 99% 35789
700_700_700_350vph 702 1.09 135 143 105% 90% 31% 49% 83% 107% 38873
725_700_700_350vph__ 722 1.12 190 169 89% 98% 22% 36% 65% 90% 48324


Sum of

Adjacent
Blind Period
Counter


ABPC
8131
2118
10921
30098
38508
38761
33220
41636
41762
48604
48170
48199


ALL


107% 82% 40% 61%


93% 107%








Table 4-5. Control delay prediction


File Name Random Volume v/c Control Delay/ Control Delay/
15 min input volumes Number Set 1st45min Ratio Stopped Delay Stop+QMU Delay
12FOV v vic
672 1.04 1.28 1.19
652 1,02 1.28 1.19
668 1.04 1.27 1.16
700 725 625 350vph 1 692 1.07 1.26 1.12
700 725 625 350vph 2 691 1.08 1.27 1.10
700 725 625 350vph 3 685 1.07 1.25 1.09
700 700 700 350vph 1 697 1.07 1.27 1.12
700 700 700 350vph 2 707 1.10 1.26 1.08
700 700 700 350vph 3 703 1.09 1.26 1.08
725 700 700 350vph 1 729 1.12 1.24 1.06
725 700 700 350vph 2 717 1.12 1.25 1.05
725 700 700 350vph 3 719 1.12 1.25 1.05
vph

Averages
664 1.03 1.28 1.18
700 725 625 350vph 689 1.07 1.26 1.10
700 700 700 350vph 702 1.09 1.26 1.09
725 700 700 350vph 722 1.12 1.25 1.05


Control Delay Percentage of Control Delay
1,XX 1.30 Stopped Stop & Q Queue Accel,/
Actual HCM BuckQ Pred BuckQ Pred Delay Move-Up Move-Up Decel.
dc dcH+ dcpx dcp3
82.6 160.5 194% 104.9 127% 106.9 129% 78% 84% 6% 16%
84.5 171.3 203% 87.1 103% 88.3 104% 78% 84% 6% 16%
98.1 163.2 166% 108.5 111% 111.2 113% 79% 86% 7% 14%
143.9 205.8 143% 163.1 113% 167.8 117% 79% 89% 10% 11%
174.4 191.7 110% 163.3 94% 167,1 96% 79% 91% 13% 9%
176.1 212.3 121% 162.0 92% 169.0 96% 80% 92% 12% 8%
148.1 199.6 135% 188.1 127% 193.2 130% 79% 90% 11% 10%
181.3 185.5 102% 183.7 101% 190.1 105% 80% 92% 13% 8%
182.1 193.0 106% 167.4 92% 173.3 95% 80% 92% 13% 8%
237.9 214.0 90% 234.8 99% 246,1 103% 81% 95% 14% 5%
229.5 207.4 90% 198.8 87% 207.0 90% 80% 95% 15% 5%
241.1 220.6 91% 198.9 83% 207.4 86% 80% 95% 15% 5%


I secs/veh secs/veh secs/veh


secs/veh secs/veh


88.4 165.0 187% 100.2 113% 102.1 115% 78% 85% 7% 15%
164.8 203.3 123% 162.8 99% 168.0 102% 79% 91% 12% 9%
170.5 192.7 113% 179.7 105% 185.5 109% 79% 92% 12% 8%
236.2 214.0 91% 210.9 89% 220.1 93% 80% 95% 15% 5%


1.26 1.11 142%


107% 110% 79% 89% 10% 11%










Table 4-6. Comparison of variation in actual and predicted stopped delay


Cumulative 1-Hour
Random Stopped Delay (sec)


Number Set

1
2
3
4
5
6
7
8
9
10

Mean
Std Deviation
CV
Std. Error
95% C.I.
Lower
Upper


Actual Predicted


68622
83364
85601
80081
59339
95345
94206
111432
66737
78859

82359
15441
0.19
4883
9571
72788
91929


77325
77713
78925
69056
57874
91536
78308
73012
67418
75952

74712
8836
0.12
2794
5477
69235
80189


120000



100000


80000



60000



40000


20000



0


Stopped Delay Variation

E Actual
1 2rrd 6itrd





-



















1 2 3 4 5 6 7 8 9 10


Random Number Set










Table 4-7. P-value determination for difference in median values
Fisher SignTest
For paired replicates

Null hypothesis: Differences between actual and predicted median delay is zero



Cumulative 1-Hour
Random Stopped Delay (sec)
Number Set Actual Predicted Difference Mu
RN X Y Z=Y-X u
1 68622 77325 8703 1
2 83364 77713 -5651 0
3 85601 78925 -6676 0
00 4 80081 69056 -11025 0
5 59339 57874 -1465 0
6 95345 91536 -3809 0
7 94206 78308 -15898 0
8 111432 73012 -38420 0
9 66737 67418 681 1
10 78859 75952 -2907 0
B= 2


From Reference Table with n = 10 and b = B = 2: p/2 = 0.0547, p = 0.1094









CHAPTER 5
THEORETICAL BOUNDS FOR DELAY ESTIMATION

This chapter describes the development of theoretical limits on the solution space for the

empirical delay prediction procedure (Objective 4).

The delay estimation procedure presented in the previous chapter begins by calculating an

"estimated arrival rate", which is actually the departure rate. Then, if the back end of the queue

is not visible, the procedure modifies the estimated arrival rate upward using a power function to

predict the real arrival rate. This power function adjusts the rate in a manner that, in essence,

varies with the amount of time during which the back end of the queue is not visible. A major

advantage of this approach is that the resulting estimated queues and associated delay can be

immediately calculated on a second-by-second basis, in real time. A major disadvantage of the

approach is that there is no relationship between the departure rate and the real arrival rate.

Under the right circumstances, errors can accumulate to the point that the delay estimation is no

longer reasonable. The potential for this is highest when the length of time that the end of the

queue is not visible covers most of the analysis time frame.

However, it is possible to calculate a set of theoretical upper and lower bounds on the

solution space by using information obtained at the end of the analysis period when the arrival

rate does equal the departure rate. In order to make any type of reasonable delay estimation, all

queues must dissipate prior to the end of the analysis time frame. Once this occurs, a calculation

of the arrival rate (which is equal to the departure rate) during the final portion of the analysis

time frame, the last 15 minutes of the hour, can be made. Knowing this final arrival/departure

rate and knowing the total number of vehicles that have crossed the stop bar during the entire

hour we can, by assuming a reasonable minimum peak hour factor, work backwards through the

period to identify arrival curves that serve as both lower and upper bounds. These theoretical









results can be used, in an ex post facto manner, to "bracket" the real-time delay estimation

procedure presented in the previous chapter. These bounds can also be used to identify an

independent "most probable" delay pattern by selecting an intermediate curve between the upper

and lower bounds that minimizes the maximum percent error between the estimate and the actual

delay.

Chapter 16 of the 2000 Highway Capacity Manual [4] contains a widely recognized and

well-accepted procedure for calculating per-vehicle control delay at signalized intersections. In

the 2000 HCM, this control delay has three components: dl (uniform delay), d2 (incremental

delay) and d3 (initial queue delay). Component d2 can be further subdivided into an over-

saturation element and a random delay element. The random delay element is based on a

coordinate transformation technique originally proposed by Whiting and refined for signalized

delay applications by Akcelik [47]. In 2007, Courage [48] demonstrated the relationships

between overflow delay, deterministic queue delay, incremental delay and initial queue delay.

Courage showed that overflow delay and deterministic queue delay (both of which can be

calculated using the area between the cumulative arrival curve and the uniform cumulative

departure curve) were each composed of initial queue delay and the over-saturation portion of

the incremental delay. The random portion of the control delay is not reflected in the cumulative

arrival and departure curves, nor is the portion of the control delay associated with acceleration

or deceleration. In addition, queue move-up delay is not explicitly depicted in the cumulative

arrival and cumulative departure curves although its effect is somewhat implied within the

general treatment of delay as the area between the curves. Appendix F of the 2000 HCM

discusses the relationship between the initial queue delay and deterministic queue delay. Five









specific arrival "cases" are discussed and the proper way to account for initial queue delay and

deterministic delay for each case is explained.

The theoretical delay literature is extended in this chapter through the development of a

theoretical framework for establishing the upper and lower bounds of the overflow delay given a

terminal arrival rate and a minimum Peak Hour Factor (PHF). The mathematical bracketing of

overflow delay using this type of information represents a new aspect of delay estimation.

Derivation of the Bounds

During a period of over-saturated flow on a signalized intersection approach, the

cumulative number of arrivals at the back of the queue exceeds the cumulative number of

departures from the stop bar, with resulting queue formation. Let us assume that over-saturated

flow begins immediately at the start of a one-hour observation period and that, at some point

near the end of the hour, it is replaced by a period of under-saturated flow that causes the queue

to dissipate before the hour expires. Let us also assume that the component 15-minute flow

rates follow a reasonable pattern that result in some minimum Peak Hour Factor (PHF). Figure

5-1 graphically depicts the analysis setting.

Both the cumulative arrival curve and the cumulative departure curve are monotonically

increasing functions. If we have enough information to construct both of these curves, then the

"delay" during the period can be found by simply calculating the area between the curves.

However, if we are dependent upon detection devices located at the intersection then, during

periods of over-saturated flow, we will only be able to measure the attributes of the departure

curve, not the arrival curve, since the end of the queue will be beyond our Field of View (FOV).

Under these circumstances we can still obtain, after the one-hour analysis period ends, a

reasonable estimation of the delay that occurred during the period. We cannot know with

certainty the delay that occurred because we have no direct knowledge of the shape of the arrival









curve. However, we can obtain an estimate of the most-likely amount of delay and can put limits

on the expected error associated with that estimate.

The delay estimation begins by measuring the following values: 1.) the total number of

vehicles that arrived during the analysis period; which also equals the number of vehicles that

departed during the analysis period since it is assumed that the overflow queue fully dissipates,

2.) the overflow queue clearance time, or the time point at which the cumulative arrival curve

and the cumulative departure curves intersect; which is also the time at which the overflow

queue is reduced to zero, and 3.) the total number of vehicles that have arrived when the

overflow queue clearance time was reached.

Using this information, the arrival rate during the last 15-minute period (period 4) of the

hour can be calculated:

AR4 = (CA6o-CAc)/(T60-Tc) (1)

Where: AR4 = Arrival Rate during period 4 (veh/sec)
CA60 = Cumulative Arrivals at time point 60 (end of the hour)
CAc = Cumulative Arrivals at overflow queue Clearance time point
T60 = Time point 60 (3600 seconds)
Tc = Time point when overflow queue Clears

In the example shown in Figure 5-2, the arrival rate is calculated to be:

AR4 = (575 veh 540 veh)/(3600 sec 3240 sec) = 0.0972 veh/sec

This can be converted to an hourly flow rate by multiplying by 3600 sec/hour:

V4 = (0.0972 veh/sec)(3600 sec/hour) = 350 veh/hr

The cumulative number of arriving vehicles at the beginning of the last 15-minute period is

calculated by multiplying this terminal hourly flow rate by the duration of the period and then

subtracting the resulting value from the cumulative number of arriving vehicles at time point 60:

CA45 = CA60 (AR4)(t4), or
CA45 = CA60 V4 (2)










Where: AR4 = Arrival Rate during period 4 (veh/sec)
V4 = Arrival Flow Rate during period 4 (veh/hr)
CA60 = Cumulative Vehicles at time point 60 (end of the hour)
CA45 = Cumulative Vehicles at time point 45
t4 = Duration of the 4th 15-minute time period (sec)

Continuing the Figure 5-2 example, the cumulative number of arrivals at the beginning of

the last 15-minute time period is calculated as:

CA45 = 575 veh (0.0972 veh/sec)(900 sec) = 487.5 veh

Given this value, we can now calculate the amount of overflow delay that occurs during

the last 15-minute period (see Figure 5-3):

OD4 = Area between Cumulative Arrival Curve and Uniform Departure Curve

OD4 = 0.5 (t4s)2(UDR4-AR4) = 0.5(T T45) 2(UDR4-AR4) (3)

Where: OD4= Overflow Delay during period 4 (veh-sec)
UDR4 = Uniform Departure Rate during period 4 (veh/sec)
AR4 = Arrival Rate during period 4 (veh/sec)
t4s = Duration of over-saturated flow during 4th 15-min time period (sec)

For our example, the overflow delay during period 4 is calculated to be:

OD4 = 0.5 (3240 sec 2700 sec)2(0.1667 veh/sec 0.0972 veh/sec) = 10,133 veh-sec

And the arrival rate in vehicles per hour during period 4 (V4) is calculated as:

V4 = (575 veh 487.5 veh)(4/hr) = 350 veh/hr

Calculating the overflow delay for the other three periods is not as straightforward. The

arrival rate during each period cannot be definitively established since one can only measure the

departure rate, not the true arrival rate, and since the extent of the queue is only visible to the end

of the Field of View. However, even with this limited information, one can still develop a "best

estimate" of the overflow delay. This is done by identifying both a "maximum reasonable delay"

arrival curve and a "minimum reasonable delay" arrival curve. Maximum and minimum delay









curves are then calculated which correspond to each of these arrival curves and a check

conducted to ensure that the delay estimated by the BuckQ analysis procedure falls within these

bounds. We can also use the theoretical bounds to establish an independent "best" estimate of

the overflow delay by construction an intermediate delay curve that minimizes the "maximum

percent error" in delay at each time point.

Two reasonable assumptions are required in order to bracket the estimated overflow delay

on both the low and high side. The first assumption is that the arrival rate observed during the

final 15-minute period is the lowest rate experienced during the hour. The second assumption is

that the PHF (Peak Hour Factor) is greater than or equal to some reasonable minimum value

(such as 0.75) that is specified in advance. The minimum PHF value can be easily obtained

through an examination of historical traffic counts for the approach under study.

A third assumption is also inherent in the proposed methodology; the assumption that the

arrival rate is constant over each 15 minute period. If the rival rate varies during a given 15-

minute period then the cumulative arrival curve will appear curvilinear in nature. This can be

problematic when dealing with the lower bound.

Derivation of the Upper Bound

Conservation of flow principals dictate that the average of the arrival flow rates during

each of the four 15-minute periods must equal the arrival rate over the entire 1-hour period:

(V1 + V2 + V3 + V4)/4 = CA60 (4)

Where: Vi = Arrival Flow Rate during period i (veh/hr)
CA60 = Cumulative Arrivals at time point 60 (veh)

Equation (4) constitutes the first constraint on the solution space for both the minimum and

maximum reasonable delay curves. Continuing our example, equation (4) becomes:


(V1 + V2 + V3 + 350 veh/hr)/4 = 575 veh/hr










V1 + V2 + V3 = 1950 veh/hr

Maximum overall delay is obtained when the highest 15-minute flow rates occur at the

start of the hour. Consequently, when identifying the maximum reasonable delay curve, the PHF

is defined as follows:

PHF = (Vi+V2+V3+V4)/[(4)Max(Vi,V2,V3,V4)]

PHF = (Vi+V2+V3+V4)/4V1 (5)

Equation (5) constitutes the second constraint on the solution space for the maximum

reasonable delay curve. Assuming a minimum PHF of 0.75 and continuing our example,

equation (5) becomes:

0.75 = (V1+V2+V3+350 veh/hr)/4Vi

3V1 = (V1+V2+V3+350 veh/hr)

2V1 V2 V3 = 350 veh/hr

Equations (4) and (5) cannot be uniquely solved since we have only 2 equations to solve

for 3 unknown variables (Vi, V2 and V3). However, an examination of the solution space for this

problem indicates that we can obtain an additional equation by attempting to set V2 as high as

possible (in a continued attempt to maximize delay). In this case, the upper limit for V2 is V1.

V2 cannot be greater than Vi or delay would not be maximized.

With V1 forming the upper limit for V2 we have the additional equation:

V1 = V2 (6)

We can now solve for all of the Vi's. Substituting equation (6) into equation (4) produces:

V1 + V1 + V3 + V4 = 4CA60

2V1 + V3 + V4 = 4CA60

V3 = 4CA60 V4 2V1 (7)










And substituting equations (6) and (7) into equation (5) produces:


PHF = (Vi +V1 + (4CA60 V4 2Vi) + V4)/(4Vi)

PHF = (Vi +V1 + 4CA60 V4 2V1 +V4)/(4Vi)

PHF = (4CA60)/(4Vi)

4V1PHF = 4CA60

V1 = CA60/PHF (8)

Substituting equation (8) into equation (6) produces:

V2 = CA60/PHF (9)

And substituting equations (8) and (9) into equation (4) yields:

CA60/PHF + CA60/PHF + V3 + V4 = 4CA60

V3 = 4CA60 2CA60/PHF V4

V3 = 2CA60 (2 1/PHF) V4 (10)

Continuing our example and utilizing equations (8), (9), and (10):

V1 = 575/0.75 = 766.7 veh/hr

V2 = 575/0.75 = 766.7 veh/hr

V3 = 2(575 veh/hr) (2 1/(0.75)) 350 veh/hr = 416.7 veh/hr

So, for our example, the cumulative arrival curve that produces the maximum reasonable

delay has quartile hourly flow rates of: 766.7 vph, 766.7 vph, 416.7 vph, and 350.0 vph. This

upper bound curve is depicted in Figure 5-4.

In this example, Vi was a feasible upper limit for V2, which results in maximum delay.

However, it is possible that Vi may not be a feasible upper limit for V2. This occurs when the

value of V4 is too high to allow Vi to equal V2 without violating the minimum PHF requirement.









To account for this possibility, equation (10) must be restricted so that V3 is greater than or equal

to V4. And since maximum delay occurs when V3 is minimized (which, in turn, maximizes V2

subject to the PHF constraint), V3 must equal V4. In other words, If V1 does not form the upper

limit for V2 then maximum delay will be obtained when V3 = V4, which is the minimum V3

given our initial assumption that V3 must be greater than V4. The value of V4 at which this

restriction occurs can be found by setting V3 equal to V4 in equation (10):

V3 = 2CA60 (2 1/PHF)- V3

2V3 = 2CA60 (2 1/PHF)

V3 = CA60 (2 1/PHF) = V4 (11)

For our example:

V3 = 575 (2 1/0.75) = 383.3 veh/hr

V4= V3= 383.3 veh/hr

Consequently, in our example, if V4 is less than 383.3 then Vi = V2 and equation (10) can

be used to calculate V3. Otherwise, V3 must be set equal to V4 and the remaining equations

solved accordingly. In general, V3 must be set equal to V4 if V4 > CA60 (2 1/PHF). If Vi does

not form the upper limit for V2 then we have the additional equation:

V3 = V4 (12)

We can once again solve for all of the Vi's. Substituting equation (12) into equation (4)

produces:

V1 + V2 + V4 + V4 = 4CA60

V1 + V2 + 2V4 = 4CA60

V2 = 4CA60 V1 2V4 (13)

And substituting equations (12) and (13) into equation (5) produces:









PHF = (V1+ (4CA60 Vi 2V4) + V4 + V4)/(4Vi)

PHF = (V1 + 4CA60 Vi 2V4 + 2V4)/(4Vi)

PHF = (4CA60 )/(4Vi)

4V1PHF = 4CA60

V1 = CA60/PHF (8)

This is the same result as before for Vi. Substituting equation (8) into equation (13)

produces:

V2 = 4CA60 CA60/PHF 2V4

V2 = (4 1/PHF)CA60 2V4 (14)

If, in our example, V4 was actually 385 instead of 350, then setting V1 = V2 and using

equation (10) would result in a value for V3 of:

V3 = 2(575 vph)(2 1/0.75) 385 vph = 381.6 vph

But this is not acceptable, since V3 = 381.6 would be less than V4 = 385, which violates

our original assumption that the last period must be the period with the lowest flow rate. Rather,

if V4= 385 vph, then V3 must be set equal to V4 and equation (13) used to solve for V2 (The

value of V1 does not change):

V2 = (4 1/0.75)(575 vph)- 2(385 vph) = 763.3

So, for this modified example, the cumulative arrival curve that produces the maximum

reasonable delay has quartile hourly flow rates of: 766.7 vph, 763.3 vph, 385.0 vph, and 385.0

vph.

Derivation of the Lower Bound

We previously discussed how conservation of flow principals dictate that the average of

the arrival rates during each of the four 15-minute periods must equal the arrival rate over the

entire 1 hour period:









(V1 + V2 + V3 + V4)/4 = CA60


Where:
Vi = Arrival Rate during period i (veh/hr)
CA60 = Cumulative Arrivals at time point 60 (veh)

For our example, equation (4) became:

(V1 + V2 + V3 + 350 veh/hr)/4 = 575 veh/hr

V1 + V2 + V3 = 1950 veh/hr

Minimum delay occurs when the vertical distance between the arrival curve and the

departure curve (the nominal queue length) is continually minimized, without the end of the

queue becoming visible. This happens when the nominal queue length equals the Field of View

(FOV). Under these conditions, the minimum value for Vi is:

V1 = [(UDRi)(ti) + FOV] x 4 periods/hr, or

V = C + 4FOV (15)

Where: V = Arrival Rate during period 1 (veh/hr)
UDR1 = Uniform Departure Rate during period 1 (veh/sec)
FOV = Field of View (veh)
tl = Duration of 1st 15-min time period (sec/period) = 900 sec/period
C = Capacity during period 1 (veh/hr)

V1 cannot be any lower than this value or the end of the queue would be visible at the end

of period 1 and no estimation of the delay associated with the overflow queue would be required.

If Vi equals this absolute lower bound, then we can continue to minimize delay by having V2

equal the following:

V2 = [(UDR2)(t2)] x 4 periods/hr, or

V2 = C2 (16)

This produces a cumulative arrival curve for period 2 that parallels the uniform departure

curve for period 2. Assuming a FOV of 12, we continue our example as follows:









V1 = [(0.1667 veh/sec)(900 sec/period) + 12 veh] x 4 periods/hr = 600 + 48 = 648 veh/hr

V2 = [(0.1667 veh/sec)(900 sec/period)] x 4 periods/hr = 600 veh/hr

We can now solve for all of the Vi's. Substituting equations (15) and (16) into equation (4)

produces:

C1 + 4FOV + C2 + V3 + V4 = 4CA60

V3 = 4CA60 C1 C2 4FOV V4 (17)

For our example:

V3 = 4/hr (575 veh) (600 veh/hr) (600 veh/hr) 4/hr (12 veh) 350 veh/hr = 702 veh/hr

So, for our example, the cumulative arrival curve that produces the minimum reasonable

delay has quartile hourly flow rates of: 648.0 vph, 600.0 vph, 702.0 vph, and 350.0 vph. This

lower bound curve is depicted in Figure 5-5.

When calculating the upper bound arrival curves, the minimum PHF is always maintained;

it represents a constraint on the solution space that is always in effect. However, this is not so

with the lower bound. Under lower bound conditions the PHF may or may not pose a constraint.

Substituting equations (15) and (16) into equation (5), and recognizing that V3 is the highest 15-

minute volume in this situation, the following is produced:

PHF = (Vi+V2+V3+V4)/[(4)Max(Vi,V2,V3,V4)] (5)

PHF = (Vi+V2+V3+V4)/4V3

PHF = (Ci + 4FOV + C2 + V3 + V4) / 4V3

PHF = (C + C2 + 4FOV + V3 + V4) / 4V3 (5B)

Substituting equation (17) into equation (5B) produces:

PHF = (C1 + C2 + 4FOV+ 4CA60 C1 C2 -4FOV -V4 +V4) / 4 (4CA60 -C1 2 4FOV V4)

PHF = 4CA60 / 4(4CA60 C1 C2 4FOV V4)

PHF = CA60 / (4CA60 C1 C2 4FOV V4) (18)










Continuing our example:

PHF = (575 veh/hr)/[(4(575 veh/hr) (600 veh/hr) (600 veh/hr) 4/hr(12 veh) 350 veh/hr]

PHF = 575 veh/hr / 702 veh/hr = 0.819

The actual peak hour factor is considerably larger than the minimum required value of

0.75. In this example, it was feasible for both Vi and V2 to meet their absolute minimum lower

bounds. However, it is possible that Vi may be able to meet its absolute minimum lower bound

while V2 cannot, or even that both Vi and V2 cannot meet their absolute minimum lower bounds.

This restriction occurs when the value of V4 is too low to allow Vi and/or V2 to meet their

absolute minimum lower bounds without either violating the minimum PHF requirement, the

conservation of flow equation, or causing the nominal queue length to shrink to a value that is

less than the FOV (thus eliminating the need for delay estimation).

If Vi and V2 are at their absolute minimum lower bound, then the maximum value for V4

can be calculated by setting V3 equal to its lowest possible bound which, as with V2, is parallel to

the cumulative departure curve:

V3 = C3 (19)

Substituting equation (19) into equation (17) yields:

C3 = 4CA60 C1 C2 4FOV V4

V4 = 4CA60 C1 C2 C3 4FOV (20)

Or, for our example:

V4 = 4/hr (575 veh) (600 veh/hr) (600 veh/hr) (600 veh/hr) 4/hr (12 veh)

V4 = 2300 veh/hr 1800 veh/hr 48 veh/hr

V4 = 452 veh/hr









The result is graphically depicted in Figure 5-6. This arrival curve produces the overall

minimum delay and has quartile hourly flow rates of: 648.0 vph, 600.0 vph, 600.0 vph, and

452.0 vph.

Once again, the PHF does not impose a constraint in this situation. Under conditions of

overall minimum delay, Vi is always the highest 15-minute volume, therefore:

PHF = (Vi+V2+V3+V4)/[(4)Max(Vi,V2,V3,V4)] = (Vi+V2+V3+V4)/4Vi (5)

PHF = (Ci + 4FOV + C2 + C3 + V4) / 4(Ci + 4FOV)

PHF = (Ci + C2 + C3 + 4FOV + V4) / 4(Ci + 4FOV) (21)

Continuing our example:

PHF=[600veh/hr+600veh/hr+600veh/hr+4/hr(12 veh)+452veh/hr]/4(600 veh/hr+4/hr(12 veh))

PHF = 2300 veh/hr / 2592 veh/hr = 0.887

The actual peak hour factor is once again considerably larger than the minimum required

value of 0.75 If Vi and V2 are at their absolute minimum lower bound, then the minimum value

for V4 can be calculated by setting V3 equal to its highest possible value while maintaining the

minimum required peak hour factor and preserving conservation of flow. Substituting equations

(15) and (16) into equation (4):

(Ci + 4FOV + C2 + V3 + V4)/4 = CA60

Ci + C2 + 4FOV + V3 + V4 = 4CA60

V4 = 4(CA60 FOV) Ci- C2 V3 (22)

Equation (22) is merely a rearrangement of equation (17). Substituting equations (15) and

(16) into equation (5) and recognizing that V3 has the highest arrival volume for this situation:

PHF = (Ci + 4FOV + C2 + V3 + V4) / 4V3

4PHFV3 = C + C2 + 4FOV + V3 + V4

4PHFV3 V3 = C1 + C2 + 4FOV + V4 (23)










Substituting equation (22) into equation (23) yields:

4PHFV3 V3 = C1 + C2 + 4FOV + 4CA60 C1 C2 4FOV V3

4PHFV3 = 4CA60

V3 = CA60 /PHF (24)

Now substituting equation (24) back into equation (22) gives:

V4 = 4CA60 C1 C2 4FOV CA60 /PHF

V4 = (4 1/PHF)CA60 C1 C2 4FOV (25)

Using the example values we obtain:

V3 = 575/0.75 = 766.7 veh/hr

and V4= (4-1/0.75)(575) 600 600 4(12) = 1533.3 1200 48

V4 = 285.3 veh/hr

So, V4=285.3 vph is the lowest possible V4 value that will allow both Vi and V2 to meet

their absolute minimum lower bounds (see Figure 5-7).

We have now examined the case where Vi, V2 and V3 are all at their minimum values, and

we have examined the case where Vi and V2 are at their minimum values but V3 is not. The next

arrangement of interest is when only Vi is at its minimum value. Substituting equation (15) into

equation (4) yields:

(C1 + 4FOV + V2 + V3 + V4)/4 = CA60

Solving for V2:

V2 = 4CA60 C1 4FOV V4 V3 (26)

For this situation, minimum delay is obtained when V3 is maximized, subject to the peak

hour constraint. Therefore:

PHF = (Vi+V2+V3+V4) / [4Max(Vi,V2,V3,V4)] = (Vi+V2+V3+V4)/4V3 (5)










Substituting equations (15) and (26) into equation (5) yields:

PHF = (C1 + 4FOV + 4CA60 C1 4FOV V4 V3 +V3 + V4)/4V3

PHF = (4CA60)/4V3

Solving for V3:

V3 = (CA60)/PHF (27)

Now substituting equation (27) back into equation (26) gives:

V2 = 4CA60 C1- 4FOV V4 (CA60)/PHF

V2= (4- 1/PHF)CA60- o 4FOV V4 (28)

We recognize that the highest possible value for V4 will occur when V2 is as low as

possible, which occurs when:

V2 = C2 (16)

Substituting equation (16) into equation (28) produces:

C2 = (4 1/PHF)CA60 C1 4FOV V4

Solving for V4:

V4= (4 1/PHF)CA6 C1 C2 4FOV (25)

This formula is consistent with the results obtained previously. We also recognize that the

lowest possible V4 will occur when V2 is as high as possible, which is when V2 = V3:

V2 = V3 (29)

Substituting equations (27) and (29) into equation (28) produces:

(CA60)/PHF = (4 1/PHF)CA60 C1 4FOV V4

Solving for V4:

V4 = (4 1/PHF)CA6o C1 4FOV (CA6o)/PHF

V4 = 4CA60/hr (CA6o)/PHF C1 4FOV (CA6o)/PHF










V4 = 4CA60/hr 2(CA60)/PHF C1 4FOV

V4 = 2CA60(2- 1/PHF) C1 4FOV (30)

Using our example values we obtain:

V4 = 2(575 veh/hr)(2-1/0.75) 600 veh/hr 4/hr(12 veh) = 118.7 veh/hr

So, V4= 118.7 vph is the lowest possible V4 value that will allow V1 to meet its absolute

minimum lower bound (see Figure 5-8). If V4 falls below the value given in equation (30) then

Vi (along with V2 and V3) will no longer be at its minimum value. For this situation, minimum

delay is obtained when V3 is maximized, subject to the peak hour constraint, and when V2 = V3.

Therefore:

PHF = (Vi+V2+V3+V4)/[(4)Max(Vi,V2,V3,V4)] = (V1+V2+V3+V4)/4V3 (5)

Substituting equation (29) into equation (5) yields:

PHF = (V1 + V3 + V3 + V4)/4V3

PHF = (V1 + 2V3 + V4)/4V3

4V3PHF = V1 + 2V3 + V4

4V3PHF 2V3 = V1 + V4

V = 4V3PHF 2V3 V4 (31)

Substituting equations (29) and (31) into equation (4) produces:

(4V3PHF 2V3 V4 + V3 + V3 + V4)/4 = CA60

4V3PHF/4 = CA60

V3 = CA60/PHF (27)

Substituting equation (27) into equation (29) yields:

V2 = CA60/PHF (32)









The value for Vi can be determined by substituting equations (27) and (32) into equation

(4), which produces:

(Vi + CA60/PHF + CA60/PHF + V4)/4 = CA60

V1 + 2CA60/PHF + V4 = 4CA60

V1 = 4CA60 2CA60/PHF V4

V1 = 2CA60(2 1/PHF) V4 (33)

Analysis of Bounds Summary

The results of the analysis of the bounds can be summarized as follows:

UPPER BOUND

V1 = CA60/PHF (8)

If V4 < CA60 (2 1/PHF) (11)


Then: V2
V3


CA60/PHF
2CA60 (2 1/PHF) V4


(9)
(10)

(11)

(14)
(12)


If V4> CA60 (2 1/PHF)


Then: V2
V3


CA60 (4 1/PHF) 2V4
V4


LOWER BOUND

If V4 = 4CA60 C1 C2 C3

Then:




If V4 < 4CA60 C1 C2 C3
And V4 >= (4 1/PHF)CA6o

Then:


- 4FOV

Vi = Ci + 4FOV
V2 = C2
V3 = C3
PHF = CA60 / (4CA60 C1 C2 4FOV V4)

- 4FOV
- C1 C2 4FOV

Vi = Ci + 4FOV
V2 = C2
V3 = 4CA60 C1 C2 4FOV V4
PHF = (Ci+ C2 + C3 + 4FOV + V4) / 4(Ci + 4FOV)


(20)

(15)
(16)
(19)
(18)

(20)
(25)

(15)
(16)
(17)
(21)









If V4 < (4 1/PHF)CA60 C1
And V4 >= 2CA60(2- 1/PHF)


4FOV
4FOV


Then: V1 = C1 + 4FOV
V2 = (4 1/PHF)CA60 C1
V3 = CA60/PHF


4FOV V4


If V4 < 2CA60(2- 1/PHF) C1 4FOV

Then: V1 = 2CA60(2 1/PHF) V4
V2 = CA60/PHF
V3 = CA60/PHF

For our example, the values are:

UPPER BOUND

V1 = 575 vph/0.75 = 766.7 vph

Is V4 = 350 vph < 575 vph (2 1/0.75) = 383.3 vph ?
YES

Then:


575 vph/0.75
2(575 vph)(2


766.7 vph
1/0.75)- 350 vph


416.7 vph


LOWER BOUND


Is V4 = 350 vph > 4CA6o C1
Is V4 = 350 vph > 4(575 vph)-
NO


- C2 C3 4FOV?
3(600 vph) 4/hr (12 veh) = 2300


1800- 48 = 452?


350 vph > (4 1/PHF)CA60 C1 C2- 4FOV?
350 vph > (4 1/0.75)575 vph- 2(600 vph) 4/hr(12 veh)?
350 vph > 1533.3 vph 1200 vph 48 vph = 285.3 vph?
YES

Then: Vi = C1 + 4FOV = 600 vph + 4/hr(12 veh) = 600 vph + 48 vph
V2= C2 = 600 vph
V3 = 4CA60 C1 C2- 4FOV V4
V3 = 4(575 vph) 2(600 vph)- 4/hr(12 veh) 350 vph
V3 = 2300 vph 1200 vph 48 vph 350 vph = 702 vph


Is V4= 350 vph< 2CA6o(2- 1/PHF) Ci
Is V4= 350 vph< 2(575 vph)(2- 1/0.75)


- 4FOV?
600 vph


4/hr(12 veh)?


(25)
(30)

(15)
(28)
(27)

(30)

(33)
(32)
(27)


Is V4
Is V4
Is V4


648 vph









Is V4= 350 vph< 766.67 vph 600 vph 48 vph = 118.7 vph?
NO

Derivation of Delay for Upper and Lower Bounds

Figure 5-9 shows the first two periods of the upper bound curve for our example. The

Overflow Delay for period 1 (OD1) is simply the area between the arrival and departure curves

within period 1. On the other hand, the Deterministic Queue Delay for period 1 (DQDi) is

composed of two elements: the in-period delay for period 1 (Dpi) and the out-of-period delay for

period 1 (Dcl). Both of these elements of the period 1 Deterministic Queue Delay are associated

with vehicles that arrive at the back of the queue during period 1, however, only the in-period

delay actually occurs during period 1, the out-of-period delay occurs during period 2.

For period 1, the in-period DQD equals the Overflow Delay, and can be calculated using the

following formulas:

CA15 = (Vi/3600 sec/hr)( Ti5 To) (34)

UCDi5 = (Ci/3600 sec/hr)( Ti5 To) (35)

ODi = Dpi = 0.5(T15 To)(CA15 UCD5i) (36)

Substituting equations (34) and (35) into equation (36) yields:

ODi = Dpi = 0.5(T15 To)[(Vi/3600 sec/hr)(Tis To) (Ci/3600 sec/hr)( T5i To)]

OD1 = 0.5(T5i To)(T5i To)(Vi Ci)/3600 sec/hr

ODi = Dpi = (Ti5 To)2(Vi Ci)/7200 sec/hr (37)

Where: CA15 = Cumulative Arrivals at time point 15 (veh)
UCDis = Uniform Cumulative Departures at time point 15 (veh)
OD = Overflow Delay during period 1 (sec)
C1 = Capacity during period 1 (veh/sec)
V1 = Arrival Rate during period 1 (veh/hr)
To = Time Point at Beginning of 15 minutes (sec) = 0 sec
T15 = Time Point at End of First 15 minutes (sec) = 900 sec









For our example:

CA15 = (766.7 veh/hr/3600 sec/hr)(900 sec) = 191.7 veh

C1 = 600.0 veh

ODi = Dpi = (900 sec)2(766.7 veh/hr 600 veh/hr)/7200 sec/hr

OD1 = Dp, = 18,750 veh-sec

The out-of-period portion of the DQD for period 1, which actually occurs in period 2, is

calculated using the following formulas. Accumulating departures:


UCDcl = UCD15 + (C2/3600 sec/hr)(Tc T15) (38)

A critical time point occurs when the last arriving vehicle during period 1 departs. This occurs

when:

UCDc = CA15 (39)

Where:
UCDci = Uniform Cumulative Departures at time point Ci (veh)
Ci = Capacity during period i (veh/sec)
Tci = Critical Time Point (Tcl is the critical time point at which the number of
Uniform Cumulative Departures = CA15)
CDci = Cumulative Departures at Critical Time Point Tci (sec)

Substituting equation (39) into equation (38) and solving for Tcl yields:

CA15 = UCD15 + (C2/3600 sec/hr)(Tcl) -(C2/3600 sec/hr)(T5s)

(CAi5 UCD15) + (C2/3600 sec/hr)(Ts5) = (C2/3600 sec/hr)(Tcl)

Tcl = 3600 sec/hr (CA15- UCD15)/C2 + T15 (40)

For period 1, the out-of-period DQD can be calculated using the following formula:

Dci = 0.5(Tci T5i)(CAi5 UCD5i) (41)

For our example:

Tc, = 3600 sec/hr (191.7 veh 150.0 veh)/600 veh/hr + 900 sec = 1150 sec









And:


Dci = 0.5(1150 sec 900 sec)(191.7 veh 150 veh) =

Dc, = 5208 veh-sec

Figure 5-10 shows the second and third periods of the upper bound curve for our example.

The Overflow Delay for period 2 (OD2) is still simply the area between the arrival and departure

curves within period 2. On the other hand, the Deterministic Queue Delay for period 2 (DQD2)

is now composed of four elements: the in-period over-saturation delay for period 2 (Dp2), the

out-of-period over-saturation delay for period 2 (Dc2), the in-period initial queue delay for period

2 (DIQA2) and the out-of-period initial queue delay for period 2 (DIQB2). All four components of

the period 2 Deterministic Queue Delay are associated with vehicles that arrive at the back of the

queue during period 2, however, only the in-period delay and in-period initial queue delay

actually occur during period 2, the out-of-period delay and out-of-period initial queue delay

occur during period 3. The in-period DQD for Period 2 can be calculated using the following

formulas:

Accumulating arrivals:

CA30 = (V2/3600 sec/hr)(T3o Ti5) + CA15 (42)

Accumulating departures:

UCD30 = (C2/3600 sec/hr)(T3o Ti5) + UCD15 (43)

By inspection we see that the bottom boundary of the area for Dp2 begins at point C15 and

is parallel to the departure curve. Defining UCD30A as the cumulative number of vehicles

obtained when this parallel boundary line reaches T30 (1800 sec), we have:


UCD30A = (C2/3600 sec/hr)(T3o T15) + CA15 (44)

The in-period over-saturation delay is then calculated as:










Dp2 = 0.5(T30 T15)(CA3o UCD30A) (45)

For our example:

CA30 = (766.7 veh/hr/3600 sec/hr)(1800 sec 900 sec) + 191.7 veh = 383.3 veh

UCD3o = (600 veh/hr/3600 sec/hr)(1800 sec 900 sec) + 150 veh = 300.0 veh

UCD30A = (600 veh/hr/3600 sec/hr)(1800 sec 900 sec) + 191.7 veh = 341.7 veh

Dp2 = 0.5(1800 sec 900 sec)(383.3 veh 341.7 veh)

Dp2 = (450 sec)(41.6 veh)

Dp2 = 18,750 veh-sec

The out-of-period over-saturation delay for period 2, which actually occurs in period 3, is

calculated using the following formulas. Accumulating departures:

UCDc2A = UCD30A + (C3/3600 sec/hr)(Tc2A-T30) (46)

A critical time point occurs when the last arriving vehicle during period 2 would have

departed had there not been an initial queue at the beginning of time period 2:

UCDc2A = CA30 (47)

Substituting equation (47) into equation (46) and solving for TC2A yields:

CA30 = UCD30A + (C3/3600 sec/hr)(Tc2A) (C3/3600 sec/hr)(T3o)

(CA3o UCD30A) + (C3/3600 sec/hr)(T3o) (C3/3600 sec/hr)(Tc2A)

TC2A = (3600 sec/hr)(CA3o UCD30A)/C3 + T30 (48)

For period 2, the out-of-period over-saturation delay can be calculated using the following

formula:

Dc2 = 0.5(Tc2A- T30)(CA3o UCD30A) (49)









For Figure 5-10 to be an accurate representation of the delay situation, the nominal queue

length at T30 must be greater than the nominal queue length at T15. If it is less, then both DP2 and

Dc2 are equal to zero. The nominal queue length at T30 is calculated as:

Q3 = CA30 UCD30 (50)

And the nominal queue length at T15 is:

Qi5 = CA15 UCD15 (51)

Consequently:

If Q30 > Q15 then equations (45) and (49) hold, otherwise Dp2 = Dc2 = 0

For our example:

Q3o = CA30 UCD30 = 383.3 veh 300 veh = 83.3 veh

which is greater than:

Qi5 = CA15 UCD15 = 191.7 veh 150 veh = 41.7 veh

Therefore, equations (45) and (49) hold:

Dp2 = 0.5(T30 T15)(CA30 UCD3oA) = 0.5(1800 sec 900 sec)(383.3 veh 341.7 veh)

Dp2 = 18,750 veh-sec

TC2A = (3600 sec/hr)(CA3o UCD30A)/C3 + T30

TC2A= (3600 sec/hr.)(383.3 veh 341.7 veh)/600 veh/hr + 1800 sec

TC2A = 2050 sec

Dc2 = 0.5(T2A- T30)(CA30 UCD3oA) = 0.5(2050 sec 1800 sec)(383.3 veh 341.7 veh)

Dc2 = 5208 veh-sec

An inspection of Figure 5-10 reveals that the in-period initial queue delay for period 2 is

represented by a trapezoid and a triangle. The trapezoid has a base of Tcl T15 and a height of

UCD30 CA15. The triangle also has a base of Tcl T15 but its height is UCD30A-UCD30.

Consequently:









DIQA2 = (Tc T15)(UCD30 CA15) + 0.5(Tc T15)(UCD30A- UCD30)

DIQA2 = (Tc1 T15)[(UCD30 CA15) + 0.5(UCD30A- UCD3o)] (52)

The total out-of-period delay for period 2, which actually occurs in period 3, is calculated

using the following formulas. Accumulating departures:

UCDc2 = UCD30 + (C3/3600 sec/hr)(Tc2 T30) (53)

Another critical time point occurs when the last vehicle arriving during period 2 departs:

UCDc2 = CA30 (54)

Substituting equation (54) into equation (53) and solving for Tc2 yields:

CA30 = UCD30 + (C3/3600 sec/hr)(Tc2) (C3/3600 sec/hr)(T3o)

(CA3o UCD3o) + (C3/3600 sec/hr)(T3o) =(C3/3600 sec/hr)(Tc2)

Tc2 = (3600 sec/hr)(CA3o UCD3o)/C3 + T30 (55)

For period 2, the total out-of-period delay can be calculated using the following formula:

DT2 = 0.5(T2 T30)(CA3o UCD3o) (56)

The out-of-period initial queue delay for period 2 is then calculated by simply subtracting

the out-of-period over-saturation delay from the total out-of-period delay:

DIQB2 = DT2 Dc2

DIQB2 = 0.5(Tc2 T30)(CA3o UCD3o)- Dc2 (57)

For Figure 5-10 to be an accurate representation of the delay situation such that equations

(52) and (57) apply, the nominal queue length at T30 must be greater than the nominal queue

length at T15. If it is less, then both DIQA2 and DIQB2 are calculated using different equations, as

we shall soon see for period 3. For our example the nominal queue length at T30 was previously

shown to be greater than the nominal queue length at T15. Therefore:

DIQA2 = (Tci T15)[(UCD3o CAi5) + 0.5(UCD30A UCD3o)]









DIQA2 = (1150 sec 900 sec)[(300 veh 191.7 veh) + 0.5(341.7 veh 300 veh)]

DIQA2 = (1150 sec 900 sec)[(300 veh 191.7 veh) + 0.5(341.7 veh 300 veh)]

DIQA2 = 32,292 veh-sec

Tc2 = (3600 sec/hr)(CA30 UCD30)/C3 + T30

Tc2 = (3600 sec/hr)(383.3 veh 300 veh)/600 veh/hr+ 1800 sec

TC2 = 2300 sec

DIQB2 = 0.5(Tc2 T30)(CA30 UCD30)- Dc2

DIQB2 = 0.5(2300 sec 1800 sec)(383.3 veh 300 veh) 5208 veh-sec

DIQB2 = 15,625 veh-sec

Figure 5-11 shows the third and fourth periods of the upper bound curve for our example.

The Overflow Delay for period 3 (OD2) is still simply the area between the arrival and departure

curves within period 3. On the other hand, since the queue at the end of the period is smaller

than the queue at the beginning of the period, the Deterministic Queue Delay for period 3

(DQD3) is now composed of the following two elements: the in-period initial queue delay for

period 3 (DIQA3) and the out-of-period initial queue delay for period 3 (DIQB3). Both components

of the period 3 Deterministic Queue Delay are associated with vehicles that arrive at the back of

the queue during period 3, however, only the in-period initial queue delay actually occurs during

period 3, the out-of-period initial queue delay occurs during period 4. The in-period DQD for

Period 3 can be calculated using the following formulas:

Accumulating arrivals:

CA45 = (V3/3600 sec/hr)(T45 T30) + CA30 (58)

Accumulating departures:

UCD45 = (C3/3600 sec/hr)(T45 T30) + UCD30 (59)









For our example:

CA45 = (416.7 veh/hr/3600 sec/hr)(2700 sec 1800 sec) + 383.3 veh = 487.5 veh

UCD45 = (600 veh/hr/3600 sec/hr)(2700 sec 1800 sec) + 300 veh = 450.0 veh

An inspection of Figure 5-11 reveals that the in-period initial queue delay for period 3 can

be calculated by taking the difference of two triangles. The larger triangle has a base of T45 T30

and a height of CA45 CA30. The smaller triangle has a base of T45 Tc2 and a height of UCD45

- CA30. Consequently:

DIQA3 = 0.5(T45- T30)(CA45 CA30) 0.5(T45- Tc2)(UCD45 CA30)

DIQA3 = 0.5 [(T45 T30)(CA45 CA30) (T45 Tc2)(UCD45 CA30)] (60)

The out-of-period initial queue delay for period 3, which actually occurs in period 4, is

calculated using the following formulas. Accumulating departures:

UCDc3 = UCD45 + (C4/3600 sec/hr)(Tc3 T45) (61)

A critical time point occurs when the last vehicle arriving during period 3 departs:

UCDc3 = CA45 (62)

Substituting equation (62) into equation (61) and solving for Tc3 yields:

CA45 = UCD45 + (C4/3600 sec/hr)(Tc3) (C4/3600 sec/hr)(T45)

(CA45- UCD45) + (C4/3600 sec/hr)(T45) = (C4/3600 sec/hr)(Tc3)

Tc3 = (3600 sec/hr)(CA45 UCD45)/C4 + T45 (63)

For period 3, the out-of-period initial queue delay can be calculated using the following formula:

DIQB3 = 0.5(Tc3 T45)(CA45 UCD45) (64)

For our example:

DIQA3 = 0.5[(T45 -T30)(CA45 CA30) (T45 Tc2)(UCD45 CA30)]

DIQA3=0.5[(2700sec-1800sec)(487.5veh-383.3veh)-(2700 sec-1800 sec)(450 veh-383.3 veh)]

DIQA3=0.5[(2700sec-1800sec)(487.5veh-383.3veh)-(2700 sec-2300 sec)(450 veh-383.3 veh)]










DIQA3 = 33,542 veh-sec

TC3 = (3600 sec/hr)(CA45 UCD45)/C4 + T45

TC3 = (3600 sec/hr)(487.5 veh 450 veh)/600 veh/hr. + 2700 sec

TC3 = 2923 sec

DIQB3 = 0.5(Tc3 T45)(CA45 UCD45)

DIQB3 = 0.5(2925 sec 2700 sec)(487.5 veh 450 veh)

DIQB3 = 4219 sec

For Figure 5-11 to be an accurate representation of the delay situation, the nominal queue

length at T45 must be less than the nominal queue length at T30. If it is greater, then both DIQA3

and DIQB3 are calculated as shown previously for period 2:

DIQA3 = (TC2 T30)[(UCD45 CA30) + 0.5(UCD45A UCD45)] (52B)

DIQB3 = 0.5(Tc3 T45)(CA45 UCD45)- DC3 (57B)

The nominal queue length at T45 is calculated as:

Q45 = CA45 UCD45 (65)

For our example:

Q45 = CA45 UCD45 = 487.5 veh 450 veh = 37.5 veh

Which is less than the previously calculated value for Q30 of 83.3 vehicles, therefore our

calculations are correct. In general:

If Qi+1 > Qi then equations (52) and (57) hold, otherwise equations (60) and (64) hold

Figure 5-11 shows that the Deterministic Queue Delay for period 4 (DQD4) is composed of

just one element, the initial queue delay (DIQ4). An inspection of Figure 5-11 reveals that this

delay can be calculated by taking the difference of two triangles. The larger triangle has a base









of Tc4 T45 and a height of CAc4 CA45. The smaller triangle has a base of Tc4 TC3 and a

height of CAc4 CA45. Consequently:

DIQ4 = 0.5(T4 T45)(CAc4 CA45) 0.5(Tc4 Tc3)(CAc4 CA45)

DIQ4 = 0.5(CAc4 CA45)(T4 T45 -Tc4+ Tc3)

DIQ4 = 0.5(CAc4 CA45)(Tc3 T45) (66)

For our example:

DIQ4 = 0.5(CAc4 CA45)(Tc3 T45) = 0.5(540 veh 487.5 veh)(2925 sec 2700 sec)

DIQ4 = 5906 veh-sec

It should be pointed out that the period 4 delay situation that is represented in Figure 5-11

corresponds to the Case III situation described in Appendix F of the 2000 Highway Capacity

Manual [4] whereas the period 3 situation represented in Figure 5-11 corresponds to CASE IV.

In addition, the period 2 situation represented in Figure 5-10 corresponds to Case V and the

period 1 situation represented in Figure 5-9 corresponds to Case II of Appendix F.

The total overflow delay for the one-hour analysis period is obtained by simply summing

the individual 15-minute period overflow delays. Inspection of Figures 5-9 through 5-11

indicates that

ODi = Dp1 (67)

OD2 = Dc1 + DP2 + DIQA2 (68)

OD3 = Dc2 + DIQB2 + DP3 + DIQA3 (69)

OD4 = Dc3 + DIQB3 + DIQ4 (70)

Therefore:

ODT = ODi+ OD2 + OD3+ OD4 (71)

ODT = Dpi + Dcl+ DP2 + Dc2 + DIQA2 + DIQB2 +DP3 + Dc3 + DIQA3 + DIQB3 + DIQ4 (72)









For our example:

OD1 = Dp1 = 18,750 veh-sec

OD2 = Dc1 + DP2 + DIQA2 = 5208 veh-sec + 18,750 veh-sec + 32,292 veh-sec

OD2 = 56,250 veh-sec

OD3 = Dc2 + DIQB2 + Dp3 + DIQA3 = 5208 veh-sec + 15,625 veh-sec + 0 veh-sec + 33,542 veh-sec

OD3 = 54,375 veh-sec

OD4 = Dc3 + DIQB3 + DIQ4= 0 veh-sec + 4219 veh-sec + 5906 veh-sec

OD4 = 10,125 veh-sec

ODT = ODi+ OD2 + OD3+ OD4

ODT = 18,750 veh-sec + 56,250 veh-sec + 54,375 veh-sec + 10,125 veh-sec

ODT = 139,500 veh-sec

The total overflow delay for the hour can also be obtained by summing all of the

deterministic queue delays.

DQD1 = Dp1 + Dce (73)

DQD2 = Dp2 + Dc2 + DIQA2 + DIQB2 (74)

DQD3 = Dp3 + Dc3 + DIQA3 + DIQB3 (75)

DQD4 = DIQ4 (76)

Therefore:

DQDT = DQD1 + DQD2 + DQD3 + DQD4 (77)

DQDT = Dpl+Dc +Dp2+Dc2+DIQA2+DIQB2+Dp+Dc3+DIQA3+DIQB3+DIQ4 (78)

For our example:

DQD1 = Dpi + Dce = 18,750 veh-sec + 5208 veh-sec

DQD1 = 23,958 veh-sec

DQD2 = DP2 + Dc2 + DIQA2 + DIQB2










DQD2 = 18,750 veh-sec + 5208 veh-sec + 32,292 veh-sec + 15,625 veh-sec

DQD2= 71,875 veh-sec

DQD3 = DP3 + DC3 + DIQA3 + DIQB3

DQD3 = 0 veh-sec + 0 veh-sec + 33,542 veh-sec + 4219 veh-sec

DQD3 = 37,760 veh-sec

DQD4 = DIQ4 = 5906 veh-sec

DQDT = DQD1 + DQD2 + DQD3 + DQD4

DQDT = 23,958 veh-sec+ 71,875 veh-sec+ 37,760 veh-sec+ 5906 veh-sec

DQDT = 139,500 veh-sec

The deterministic delay values can be changed to a "per-vehicle" basis by dividing the

deterministic queue delay for each period by the vehicles arriving during that period.

dqdi = DQD1/CA15 (79)

dqd2 = DQD2/(CA3o CA15) (80)

dqd3 = DQD3/(CA45 CA3o) (81)

dqd4 = DQD4/(CA6o CA45) (82)

dqdT = DQDT/CA60 (83)

where: dqdi = Per Vehicle Deterministic Queue Delay for Period i (T = Total)

For our example:

dqdi = DQD1/CA15 = 23,958 sec/191.7 veh

dqdi = 125.0 sec/veh

dqd2 = DQD2/(CA3o CA15) = 71,875 sec/(383.3 veh 191.7 veh)

dqd2 = 375.0 sec/veh

dqd3 = DQD3/(CA45 CA3o) = 37,760 sec/(487.5 veh 383.3 veh)










dqd3 = 362.5 sec/veh

dqd4 = DQD4/(CA60 CA45) = 5906 sec/(575 veh 487.5 veh)

dqd4 = 67.5 sec/veh

dqdT = DQDT/CA60= 139,500 sec/575 veh

dqdT = 242.6 sec/veh

The overflow delay for each period can be changed to a "per-vehicle" basis by dividing the

overflow delay for each period by the average of the vehicles arriving and departing during

that period.

odl = OD1/[(CA15 + UCD15)/2] (84)

od2 = OD2/[(CA3o + UCD3o)/2 (CA15 + UCD15)/2] (85)

od3 = OD3/[(CA45 + UCD45)/2 (CA30 + UCD30)/2] (86)

od4 = OD4/[CA60 (CA45 + UCD45)/2] (87)

odT = ODT/CA60 (88)

where: odi = Per Vehicle Overflow Delay for Period i (T = Total)

For our example:

od = OD1/[(CA15+UCD15)/2]=18,750 sec/[191.7 veh + 150 veh)/2]=18,750veh-sec/170.9 veh

od, = 109.8 sec/veh

od2 = OD2/[(CA30 + UCD30)/2 (CA15 + UCD15)/2]

od2 = 56,250 sec/[(383.3 veh + 300 veh)/2 (191.7 veh+ 150 veh)/2]

od2 = 56,250 sec/(341.7 veh 170.9 veh)

od2 = 329.3 sec/veh

od3 = OD3/[(CA45 + UCD45)/2 (CA30 + UCD3o)/2]

od3 = 54,375 sec/[(487.5 veh + 450 veh)/2 (383.3 veh+ 300)/2]









od3 = 54,375 sec/(468.8 veh 341.7 veh)

od3 = 427.9 sec/veh

od4 = OD4/[CA60- (CA45 + UCD45)/2]

od4 = 10,125 sec/[575 veh- (487.5 veh+ 450 veh)/2] = 10,125 veh-sec/(575 veh- 468.8 veh)

od4 = 95.3 sec/veh

odT= ODT/CA60 = 139,500 sec /575 veh

odT = 242.6 sec/veh

i. Delay Summary

The results of the overflow delay derivation can be summarized as follows:

PERIOD 1

DQD1 = Dp1 + Dc1 (73)

OD1 = Dpi (67)

Where: Dpi = (T15 To)2(Vi C1)/7200 sec/hr (37)

Dci = 0.5(Tci T15)(CA15 UCD15) (41)

PERIOD 2

DQD2 = Dp2 + Dc2 + DIQA2 + DIQB2 (74)

OD2 = Dcl + DP2 + DIQA2 (68)

Where, if_> 3oQs:

Dp2 = 0.5(T30 T15)(CA3o UCD30A) (45)

Dc2 = 0.5(Tc2A- T30)(CA3o UCD30A) (49)

DIQA2 = (Tc T15)[(UCD3o-CA15) + 0.5(UCD30A- UCD3o)] (52)

DIQB2 = 0.5(Tc2 T30)(CA3o UCD3o)- Dc2 (57)

Dci = 0.5(Tc T15)(CA15 UCD15) (41)









Or, if Q3o 15:

DP2 = 0

Dc2 = 0

DIQA2 = 0.5[(T30 T15)(CA30 CA15) (T30 Tci)(UCD30 CA15)] (60A)

DIQB2 = 0.5(Tc2 T30)(CA3o UCD3o) (64A)

Dc = 0.5(Tc T15)(CAi5 UCDi5) (49)

PERIOD 3

DQD3 = Dp3 + DC3 + DIQA3 + DIQB3 (75)

OD3 = Dc2 + DIQB2 + DP3 + DIQA3 (69)

Where, if045 < 30:

Dp3 = 0

Dc3 = 0

DIQA3 = 0.5[(T45 T30)(CA45 CA30) (T45 Tc2)(UCD45 CA30)] (60)

DIQB3 = 0.5(Tc3 T45)(CA45 UCD45) (64)

Dc2 = 0.5(T2A T30)(CA3o UCD30A) (49)

DIQB2 = 0.5(Tc2 T30)(CA3o UCD3o)- Dc2 (57)

Or, if 045 >Q30:

Dp3 = 0.5(T45 T30)(CA45 UCD45A) (45)

Dc3 = 0.5(TC3A- T45)(CA45 UCD45A) (49)

DIQA3 = (Tc2 T30)[(UCD45 CA3o) + 0.5(UCD45A UCD45)] (52B)

DIQB3 = 0.5(Tc3 T45)(CA45 UCD45)- Dc3 (57B)

Dc2 = 0.5(Tc2 T30)(CA3o UCD3o) (41)

DIQB2 = 0.5(Tc2 T30)(CA3o UCD3o)- Dc2 (57)









PERIOD 4

DQD4 = DIQ4 (76)

OD4 = DC3 + DIQB3 + DIQ4 (70)

Where: DIQ4= 0.5(CAc4 CA45)(Tc3 T45) (66)

Dc3 = 0

DIQB3 = 0.5(Tc3 T45)(CA45 UCD45) (64)


ALL PERIODS

DQDT= DQD1 + DQD2 + DQD3 + DQD4 (77)

ODT = ODi+ OD2 + OD3+ OD4 (71)

DQDT = ODT = DP1 + Dcl+ DP2 + DC2 + DIQA2 + DIQB2 +DP3 + DC3 + DIQA3 + DIQB3 + DIQ4 (72)

Using the above formulas we can, for a given capacity, construct a series of feasible delay

regions for each minimum Peak Hour Factor (PHF). Figure 5-12 shows an example set of

feasible delay regions for a minimum PHF of 0.75, while Figures 5-13 and 5-14 show similar

examples for minimum PHF's of 0.80 and 0.85, respectively. We can fit a series of quadratic

curves to the data with a rather high degree of correlation as is shown in the three figures. An

inspection of these figures provides some interesting information:

As we would expect, the amount of delay increases as the observed hourly arrival volume
(CA60) increases

As the observed hourly arrival volume (CA60) approaches capacity, the shape of the
feasible region morphs from triangular to bullet-shaped and the area between the
minimum delay curve and the maximum delay curve increases.

As the minimum PHF increases, the area between the minimum delay curve and the
maximum delay curve decreases. This makes sense since a higher peak hour factor
indicates lower variability in the 15-minute flow rates and thus lower variability in the
associated delay. We can provide tighter bounds on our solution space for delay when
we have higher minimum peak hour factors.









As the minimum PHF increases, the minimum observable arrival flow rate for period 4
(V4) increases. For example, when the PHF = 0.75 the minimum observable arrival flow
rate is theoretically zero whereas, when the PHF = 0.85, the minimum observable arrival
flow is approximately 275 vph.

The value of V4 at which the difference between the maximum delay curve and the

minimum delay curve is the greatest can be determined by setting equal to zero the first

derivative of the difference between the two curve formulas and solving for X (where X = V4).

For example, given a PHF of 0.75 and 585 for the value of CA60, the value of XMAX is calculated

as follows:

Delay Difference = (400.03-0.1727X-0.0007X2) (404.05-1.2365X+0.00106X2)

Delay Difference = 4.02 + 1.0638X 0.00176X2

d(Delay Difference)/dX = 0 at XMAX: 1.0638 2(0.00176)XMAX = 0

1.0638 = 0.00352 XMAX

XMAX = 302 veh/hr

So the maximum difference in delay occurs at a value of V4 = 302 vph. The associated

maximum delay difference is therefore:

Maximum Delay Difference = 4.02 + 1.0638XMAX 0.00176X2MAX

Maximum Delay Difference = 4.02 + 1.0638(302) 0.00176(302)2

Maximum Delay Difference = 156.7 sec/veh

We can also calculate the delay value associated with the maximum delay curve and the

delay value associated with the minimum delay curve at this point:

Maximum Delay = 400.03 0.1727X 0.0007X2

Maximum Delay = 400.03 0.1727(302) 0.0007(302)2

Maximum Delay = 284.0 sec/veh

Minimum Delay = 404.05 1.2365X + 0.00106X2










Minimum Delay = 404.05 1.2365(302) + 0.00106(302)2

Minimum Delay = 127.3 sec/veh

It can be shown that the following equation holds when we desire to have an intermediate

estimate that yields equivalent percentage errors when compared against both minimum and

maximum possible values:

Y = 2UL/(U+L) (89)

Where: Y = Estimate that yields equivalent percentage errors
U = Upper Value (in this case the Maximum Delay)
L = Lower Value (in this case the Minimum Delay)

Therefore, our delay estimate for the example would be:

Y = 2(284.0)(127.3)/(284.0 + 127.3) = 175.8 sec/veh

With a maximum potential percentage error of: (175.8 127.3)/127.3 = (284.0 -

175.8)/284.0 = 38.1% Although the maximum delay difference occurs towards the center of the

region, the highest percentage error occurs near the far right end of the region where the average

delay is least and the ratio of the delay difference to the average delay is greatest. Continuing

our PHF = 0.75 and CA60 = 585 example, at X = 492 the delay difference is:

Delay Difference = 4.02 + 1.0638X 0.00176X2

Delay Difference = 4.02 + 1.0638(492) 0.00176(492)2

Delay Difference = 93.3 sec/veh

While the minimum and maximum delay are:

Maximum Delay = 400.03 0.1727X 0.0007X2

Maximum Delay = 400.03 0.1727(492) 0.0007(492)2

Maximum Delay = 145.6 sec/veh

Minimum Delay = 404.05 1.2365X + 0.00106X2










Minimum Delay = 404.05 1.2365(492) + 0.00106(492)2

Minimum Delay = 52.7 sec/veh

Our delay estimate for this case would then be:

Y = 2(145.6)(52.7)/(145.6 + 52.7) = 77.4 sec/veh

And our maximum potential percentage error would be (77.4 52.7)/52.7 = (145.6 77.4)/145.6
= 46.9%.

Continuing these types of calculations, we can plot the maximum percentage error as a

function of the observed flow rate. For a minimum PHF of 0.75, this yields the set of curves

shown in Figure 5-15. Figures 5-16 and 5-17 provide a similar set of curves for a minimum PHF

of 0.80 and 0.85, respectively. These curves clearly show that the maximum percentage error of

the estimate increases as the observed flow rate increases. Looking at the PHF=0.75 graph, the

percentage error is only 20% at an observed terminal flow rate of 150 vph, whereas the

percentage error is close to 50% when the terminal flow rate rises to 350 vph. The curves also

show that the overall worst percentage error decreases as the PHF increases, from about 55% for

a PHF of 0.75 to approximately 35% for a PHF of 0.85.

Derivation of the Bounds with Visible Period 1 Queue

If the end of the queue remains visible during a long enough portion of period 1 such that

an arrival rate can be determined for the first 15 minutes of the hour, then the bound equations

can be simplified as follows.

Derivation of Upper Bound with Visible Period 1 Queue

Conservation of flow principals still dictate that the average of the arrival flow rates during

each of the four 15-minute periods must equal the arrival rate over the entire 1 hour period:

(V1 + V2 + V3 + V4)/4 = CA60 (4)

Where: Vi = Arrival Flow Rate during period i (veh/hr)









CA60 = Cumulative Arrivals at time point 60 (veh)

Equation (4) continues to constitute the first constraint on the solution space for both the

minimum and maximum reasonable delay curves. Using the previous example, equation (4)

becomes:

(V1 + V2 + V3 + 350 veh/hr)/4 = 575 veh/hr

V1 + V2 + V3 = 1950 veh/hr

With period 1 visible, the arrival rate for period 1 can be set equal to the capacity of period

1 for the purposes of overflow delay calculation. The overflow delay during period 1 equals zero

and there is no residual queue at the start of period 2. Maximum overall delay is obtained when

the highest 15-minute flow rate occurs during period 2. Consequently, when identifying the

maximum reasonable delay curve, the PHF is defined as follows:

PHF = (V1 + V2 + V3 + V4)/[(4)Max(Vi,V2,V3,V4)]

PHF = (V1 + V2 + V3 + V4)/4V2 (5C)

Equation (5B) constitutes the second constraint on the solution space for the maximum

reasonable delay curve. Given a minimum PHF of 0.75, equation (5B) becomes:

0.75 = (V1 + V2 + V3 + 350 veh/hr)/4V2

3V2 = (V1 + V2 + V3+ 350 veh/hr)

2V2 V1 V3 = 350 veh/hr

With a visible period 1 we know that:

V1 = C1 (6B)

Consequently, equations (4) and (5B) can be uniquely solved since we have 2 equations to solve

for 2 unknown variables (V2 and V3). Substituting equation (6B) into equation (4) produces:


C1 + V2 + V3 + V4 = 4CA60










2V1 + V3 + V4 = 4CA60

V3 = 4CA60 V4 V2 C1 (7B)

And substituting equations (6B) and (7B) into equation (5B) produces:

PHF = (C1 + V2 + ( 4CA60 V4 V2 C1) + V4)/(4V2)

PHF = (C1 + V2 + 4CA60 V4 V2 C1 + V4)/(4V2)

PHF = (4CA60 )/(4V2)

4V2PHF = 4CA60

V2 = CA60/PHF (8B)

Substituting equation (8B) into equation (7B) yields:

V3 = 4CA60 V4 CA60/PHF C1

V3 = CA60(4-1/PHF)- V4 C1 (10B)

Continuing our example and utilizing equations (6B), (8B), and (10B):

V1 = 600 veh/hr

V2 = 575/0.75 = 766.7 veh/hr

V3 = (575 veh/hr) (4 1/(0.75)) 350 veh/hr 600 veh/hr = 583.3 veh/hr

So, for our example, the cumulative arrival curve that produces the maximum reasonable

delay when visibility exists through period 1 has quartile hourly flow rates of: 600.0 vph, 766.7

vph, 583.3 vph, and 350.0 vph. This upper bound curve is depicted in Figure 5-18.

A minimum value exists for V4 if the minimum PHF is to be maintained. The minimum

V4 value is obtained when V3 is maximized. Since V3 cannot exceed V2, this occurs when:

V2 = V3 (29)

Substituting equations (6B) and (29B) into equation (5B) and solving for V2 produces:

PHF = (Vi + V2 + V3 + V4)/4V2










PHF = (C1 + V2 + V2 + V4)/4V2

4V2PHF = C1 + 2V2 + V4

4V2PHF 2V2 C1 + V4

2V2(2PHF 1) = C1 + V4

V2 = (C1 + V4) / 2(2PHF 1) (90)

Substituting equations (6B) and (29B) into equation (4) yields:

(V1 + V2 + V3 + V4)/4 = CA60

V1 + V2 + V3 + V4 = 4CA60

C1 + V2 + V2 + V4 = 4CA60

Ci + 2V2 + V4 = 4CA60

And then substituting in equation (90) and solving for V4 yields:

Ci + 2(Ci + V4) / 2(2PHF 1) + V4 = 4CA60

(Ci + V4) / (2PHF 1) + V4 = 4CA60 C1

(Ci + V4) + (2PHF 1)V4 = (4CA60 Ci)(2PHF 1)

Ci + V4 (1+ (2PHF 1))= (4CA60 C)(2PHF- 1)

C1 + 2PHFV4 = (4CA60 C1)(2PHF- 1)

2PHFV4= (4CA60 C1)(2PHF 1) Ci

V4 = [(4CA60 C1)(2PHF -1) C1] / 2PHF (91)

Continuing our example:

V4 = [(4(575 vph) 600 vph)(2(0.75) 1) 600 vph] / 2(0.75)

V4 = [(1700 vph)(0.5) 600 vph] / 1.5 = (850 vph 600 vph) / 1.5

V4 = 166.7 vph

The value of V4 can be no lower than this if the minimum PHF is to be maintained.









A maximum value also exists for V4 if the minimum PHF is to be maintained. The

maximum V4 value is obtained when V3 is minimized. Since V3 cannot be less than V4, this

occurs when:

V3 = V4 (12)

Substituting equations (6B) and (12) into equation (5C) and solving for V2 produces:

PHF = (Vi + V2 + V3 + V4)/4V2

PHF = (Ci + V2 + V4 + V4)/4V2

4V2PHF = Ci + V2 +2 V4

4V2PHF- V2 = C1 + 2V4

V2(4PHF- 1) = C1 +2V4

V2 = (C + 2V4) / (4PHF 1) (92)

Substituting equations (6B) and (12) into equation (4) yields:

(V1 + V2 + V3 + V4)/4 = CA60

V1 + V2 + V3 + V4 = 4CA60

C1 + V2 + V4 + V4 = 4CA60

C1 + V2 + 2V4 = 4CA60

And then substituting in equation (92) and solving for V4 yields:

C1 + (Ci + 2V4) / (4PHF 1) + 2V4 = 4CA60

(Ci + 2V4) / (4PHF 1) + 2V4 = 4CA60 C1

(Ci + 2V4) + 2(4PHF 1)V4 = (4CA60 C)(4PHF- 1)

Ci + 2V4 (1+ (4PHF 1)) = (4CA60 C1)(4PHF- 1)

C1 + 8PHFV4 = (4CA60 Ci)(4PHF 1)

8PHFV4= (4CA60 Ci)(4PHF 1) Ci









V4 = [(4CA60 C1)(4PHF 1) C1] / 8PHF (93)

Continuing our example:

V4 = [(4(575 vph) 600 vph)(4(0.75) 1) 600 vph] / 8(0.75)

V4 = [(1700 vph)(2) 600 vph] / 6 = (3400 vph 600 vph) / 6

V4 = 466.7 vph = V3

The value of V4 can be no higher than this if the minimum PHF is to be maintained. The

corresponding value for V2 can be obtained using equation (92):

V2 = (600 vph + 2(466.7 vph)) / (4(0.75) 1)

V2= 1533.4 vph /2

V2 = 766.7 vph

However, an additional constraint applies in that the value for V3 must be high enough so

that the end of the queue does not come within view by the end of the third period. In other

words:

V1 + V2 + V3> C1 + C2+ C3 +4FOV (94)

The minimum acceptable value of V3 is obtained when the equality holds for this equation:

600 vph + 766.7 vph + V3 = 600 vph + 600 vph+ 600 vph + 4/hr(12 veh)

1366.7 vph + V3 = 1848 vph

V3 = 481.3 vph

And the corresponding value for V4 is obtained via conservation of flow:

(600 vph + 766.7 vph + 481.3 vph + V4)/4 = 575 veh

(600 vph + 766.7 vph + 481.3 vph + V4)/4 = 575 veh

1848 vph + V4 = (575 veh)(4/hr)

V4 = 452 vph









This is less than the previously calculated value of 466.7 vph and is therefore the true

minimum value of V4.

Derivation of Lower Bound with Visible Period 1 Queue

Conservation of flow principals continue to dictate that the average of the arrival rates

during each of the four 15-minute periods must equal the arrival rate over the entire 1-hour

period:

(V1 + V2 + V3 + V4)/4 = CA60 (4)

Where:

Vi = Arrival Rate during period i (veh/hr)
CA60 = Cumulative Arrivals at time point 60 (veh)

For our example, equation (4) became:

(V1 + V2 + V3 + 350 veh/hr)/4 = 575 veh/hr

V1 + V2 + V3 = 1950 veh/hr

Minimum delay occurs when the vertical distance between the arrival curve and the

departure curve (the nominal queue length) is continually minimized, without the end of the

queue becoming visible. This happens when the nominal queue length equals the Field of View

(FOV). Under these conditions, the minimum value for V2 is:

V2 = [(UDR2)(t2) + FOV] x 4 periods/hr, or

V2 = C2 + 4FOV (15B)

Where:

V2 = Arrival Rate during period 2 (veh/hr)
UDR2 = Uniform Departure Rate during period 2 (veh/sec)
FOV = Field of View (veh)
t2 = Duration of 2nd 15-min time period (sec/period) = 900 sec/period
C2 = Capacity during period 2 (veh/hr)









V2 cannot be any lower than this value or the end of the queue would be visible at the end

of period 2 and no estimation of the delay associated with the overflow queue would be required.

Assuming a FOV of 12, we continue our example as follows:

V1 = Ci= 600 veh/hr

V2 = [(0.1667 veh/sec)(900 sec/period) + 12 veh] x 4 periods/hr = 600 + 48 = 648 veh/hr

We can now solve for V3. Substituting equation (15B) into equation (4) produces:

C1 + C2 + 4FOV + V3 + V4 = 4CA60

V3 = 4CA60 4FOV C C2 V4

V3 = 4(CA60- FOV) C- C2- V4 (17B)

For our example:

V3 = 4/hr (575 veh 12 veh) (600 veh/hr) (600 veh/hr) 350 veh/hr = 702 veh/hr

So, for our example, when the first period is visible the cumulative arrival curve that

produces the minimum reasonable delay has quartile hourly flow rates of: 600.0 vph, 648.0 vph,

702.0 vph, and 350.0 vph. This lower bound curve is depicted in Figure 5-19.

Analysis of Bounds Summary with Visible Period 1 Queue

The results of the analysis of the bounds can be summarized as follows when the first

period is a visible period:

UPPER BOUND

Vi= Ci

V2 = CA60/PHF (8B)

V3 = CA60 (4 1/PHF) V4- C1 (10B)

LOWER BOUND

Vi= C

V2 C2+ 4FOV (15B)










V3 = 4(CA60 FOV) C1 C2 V4 (18B)

For our example, the values are:

UPPER BOUND

Vi = 600 vph

V2 = 575 vph/0.75 = 766.7 vph

V3 = 575 vph (4 1/0.75) 350 vph- 600 vph = 583.3 vph

V4 = 350 vph

LOWER BOUND

Vi = 600 vph

V2 = 600 vph + 4/hr(12veh) = 648 vph

V3 = 4/hr(575 veh 12 veh) 600 vph 600 vph 350 vph = 702 vph

V4 = 350 vph

Derivation of Delay with Visible Period 1 Queue

The calculation of Overflow Delay and Deterministic Queue Delay proceeds as before,

with the exception that there is no Overflow Delay or Deterministic Queue Delay during period 1

(DOi = DQD = 0). The results are summarized as follows:

PERIOD 1

DQD = OD = 0

PERIOD 2

DQD2 = DP2 + Dc2 (73B)

OD2 = DP2 (67B)

Where: Dp2 = (T30 T15)2(V2 C2)/7200 sec/hr (37B)

Dc2 = 0.5(Tc2- T30)(CA30 UCD30) (41B)









PERIOD 3

DQD3 = Dp3 + Dc3 + DIQA3 + DIQB3 (75)

OD3 = DC2 + Dp3 + DIQA3 (95)

Where, if Q45 < Q30:

Dp3 = 0

Dc3 = 0

DIQA3 = 0.5[(T45 T30)(CA45 CA30) (T45 Tc2)(UCD45 CA30)] (60)

DIQB3 = 0.5(Tc3 T45)(CA45 UCD45) (64)

Dc2 = 0.5(Tc2 T30)(CA3o UCD3o) (49)

Or, if 045 >Q30:

Dp3 = 0.5(T45 T30)(CA45 UCD45A) (45)

Dc3 = 0.5(TC3A- T45)(CA45 UCD45A) (49B)

DIQA3 = (Tc2 T30)[(UCD45 CA30) + 0.5(UCD45A UCD45)] (52)

DIQB3 = 0.5(Tc3 T45)(CA45 UCD45)- Dc3 (57B)

Dc2 = 0.5(Tc2 T30)(CA3o UCD3o) (41)

PERIOD 4

DQD4 = DIQ4 (76)

OD4 = Dc3 + DIQB3 + DIQ4 (70)

Where: DIQ4= 0.5(CAc4 CA45)(Tc3 T45) (66)

Dc3 = 0

DIQB3 = 0.5(Tc3 T45)(CA45 UCD45) (64)

ALL PERIODS

DQDT= DQD1 + DQD2 + DQD3 + DQD4 (77)









ODT = ODi+ OD2 + OD3+ OD4 (71)

DQDT = ODT = Dp2 + Dc2 +DP3 + DC3 + DIQA3 + DIQB3 + DIQ4 (72B)

Derivation of the Bounds When Queue is Visible During Three Periods

If the end of the queue remains visible during three of the four 15-minute analysis periods

such that an arrival rate at the back of the queue can be determined for three of the four periods,

then the bounds converge to a single unique solution. Conservation of flow principals still

dictate that the average of the arrival flow rates during each of the four 15-minute periods must

equal the arrival rate over the entire 1 hour period:

(V1 + V2 + V3 + V4)/4 = CA60 (4)

Where:

Vi = Arrival Flow Rate during period i (veh/hr)
CA60 = Cumulative Arrivals at time point 60 (veh)

Consequently: V1 = 4CA60 V2 V3 V4, or
V2 = 4CA60- V1 V3 V4, or
V3 = 4CA60- V1 V2 V4, or
V4 = 4CA60- V1 V2 V3

In general: Vi = 4CA60 EjVj<>i

The calculation of Overflow Delay and Deterministic Queue Delay proceeds as before.

Derivation of the Bounds When Analysis Time Frame is Greater Than One Hour

The analysis procedure can be expanded to a time frame greater than one hour. However,

to do so we must replace the Peak Hour Factor, which is based on four 15-minute periods, with a

newly defined "Peak Period Factor" that is consistent with the actual number of 15-minute

periods under consideration. For example, the Peak Period Factor for a 5-period analysis time

frame (P5F) would be calculated as follows:

P5F = (V1 + V2 + V3 + V4+ V5) / [(5)Max(Vi,V2,V3,V4,V5)] (96)

In general: PNF = (ENVi) / (N)Max(Vi)










The conservation of flow equation would continue to apply for the expanded number of

periods. For 5 periods it would be:

(V1 + V2 + V3 + V4+ V5) / 5 = CA5 (97)
Where:

Vi = Arrival Rate during period i (veh/analyis timeframe)
CA5 = Cumulative Arrivals at the end of the last (5th) period (veh/analysis timeframe)

In general: (EVi) / N = CAN

Equation (97), the conservation of flow equation, constitutes the first constraint on the

solution space for both the minimum and maximum reasonable delay curves.

Note that both Vi and CA are expressed in terms of vehicles per analysis time frame. When

the analysis time frame is not exactly one hour, as is the case here, Vi must be divided by the

analysis time frame (atf) to obtain the period flow rate in vehicles per hour. If Vi = 766.7

vehicles/analysis time frame then the equivalent hourly flow rate would be 766.7 vehicles per

analysis time frame / 1.25 hours per analysis time frame = 766.7 veh/atf/ 1.25 hr/atf = 613.3

vehicles/hour.

Derivation of the Five Period Upper Bound

Maximum overall delay is obtained when the highest 15-minute volume occurs at the start

of the analysis time frame. Consequently, when identifying the maximum reasonable delay

curve, P5F is defined as follows:

P5F = (V1 + V2 + V3 + V4 + Vs)/[(5)Max(Vi,V2,V3,V4,Vs)]

PHF = (V + V2+ V3+ V4+ V5)/5V1 (96B)

Equation (96B) constitutes the second constraint on the solution space for the maximum

reasonable delay curve.









Equations (96B) and (97) cannot be uniquely solved since we have only 2 equations to

solve for 4 unknown variables (Vi, V2, V3 and V4). However, an examination of the solution

space for this problem indicates that we can obtain additional equations by attempting to set V2

and V3 as high as possible (in a continued attempt to maximize delay). In this case, the upper

limit for V2 and V3 is V1. V2 and V3 cannot be greater than Vi or delay would not be

maximized. With Vi forming the upper limit for V2 and V3 we have the additional equations:

V1 = V2 (98)

V1 = V3 (99)

We now have 4 equations and 4 unknowns and we can solve for all of the Vi's.

Substituting equations (98) and (99) into conservation of flow equation (97) produces:

V1 + V1 + V1 + V4 + V5 = 5CAs

3V1 + V4 + V5 = 5CA5

V4 = 5CA5 V5 3V1 (100)

Substituting equations (98), (99) and (100) into peak period factor equation (96) and

recognizing that Vi will have the largest value when delay is maximized:

P5F = (Vi + V1 + V1 + (5CA5 V5 3Vi) + Vs)/(5Vi)

P5F = (3V1 + 5CA5 V5 3V1 + Vs)/(5Vi)

P5F = (5CA5)/(5Vi)

5V1P5F = 4CA5

V1 = CAs/P5F (101)

Substituting equation (101) into equations (98) and (99) produces:

V2 = CAs/P5F (102)

V3 = CAs/P5F (103)









And substituting equations (101), (102) and (103) into conservation of flow equation (97) yields:

CAs/P5F + CAs/P5F + CAs/P5F + V4 + V5 = 5CA5

V4 = 5CA5- 3CAs/P5F V5

V4 = CA5 (5 3/P5F) V5 (104)

Continuing our example and utilizing equations (101), (102), (103) and (104):

V1 = 575/0.75 = 766.7 veh/atf

V2 = 575/0.75 = 766.7 veh/atf

V3 = 575/0.75 = 766.7 veh/atf

V4 = (575 veh/atf) (5 3/(0.75)) 350 veh/atf = 225.0 veh/atf

However, this solution violates our initial requirement that V4 (225 vpatf) be greater than

or equal to V5 (350 vpatf). Consequently, in this case, we must re-work our solution with

equation (99) eliminated, replaced with:

V4 = V5 (105)

Substituting equations (98) and (105) into conservation of flow equation (97) produces:

V1 + V1 + V3 + V5 + V5 = 5CAs

2V1 + V3 + 2V5 = 5CA5

V3 = 5CA5 2V5 2V1 (106)

And substituting equations (98), (105) and (106) into peak period factor equation (96) produces:

P5F = (V1 + V1 + (5CA5 2V5 2V1) + V5 + Vs)/(5Vi)

P5F = (2V + 5CA5 2V5 2V1 + 2V)/(5Vi)

P5F = (5CA5)/(5Vi)

5V1P5F = 5CA5

V1 = CAs/P5F (101)









Substituting equation (101) into equations (98) produces:

V2 = CAs/P5F (102)

These are the same equations for Vi and V2 that were previously obtained. However,

substituting equations (101), (102) and (105) into equation (97) now yields:

CAs/P5F + CAs/P5F + V3 + V5 + V5 = 5CA5

V3 = 5CA5 2CAs/P5F 2V5

V3 = CA5 (5 2/P5F) 2V5 (107)

Continuing our example and utilizing equations (101), (102), (104) and (107):

V1 = 575/0.75 = 766.7 veh/atf

V2 = 575/0.75 = 766.7 veh/atf

V3 = (575 veh/atf) (5 2/(0.75)) 2(350 veh/atf) = 641.7 veh/atf

V4 = 350 veh/atf

This is an acceptable solution. So, for our example, the cumulative arrival curve that

produces the maximum reasonable delay has period flow rates of: 766.7 vpatf, 766.7 vpatf,

641.7 vpatf, 350 vpatf and 350 vpatf. This upper bound curve is depicted in Figure 5-20.

Dividing by 1.25, the length of the analysis time frame in hours, converts these values into

hourly flow rates: 613.3 vph, 613.3 vph, 513.3 vph, 280 vph and 280 vph

In this example, V1 was a feasible upper limit for V2 but was not a feasible upper limit for

V3. However, it is possible that V1 may be a feasible upper limit for both V2 and V3. This occurs

when the value of Vs is low enough to allow V3 to equal Vi without forcing V4 to be lower than

Vs. The value of V5 at which this restriction occurs can be found by setting V4 equal to V5 in

equation (104):

V4 = CA5 (5 3/P5F) V4









2V4 = CA5 (5 3/P5F)

V4 = CA5 /2(5 3/P5F) = V5 (108)

For our example:

V4 = 575 /2(5 3/0.75)

V4 = V = 287.5 veh/atf

Therefore, in our example, if V5 is less than 287.5 then Vi = V3 and equation (104) can be

used to calculate V4. In general, equation (104) can be used to calculate V4 if Vs < (CAs /2)(5 -

3/P5F). If V5 > CA5 /2(5 3/P5F) then V4 must be set equal to Vs and the remaining equations

solved accordingly. This will yield an acceptable answer as long as Vi can serve as an upper

limit for V2, which occurs if V5 is not too high. If Vi does not form the upper limit for V2 then

we have the additional equation:

V3 = Vs (109)

And we must re-work our solution with equations (98) and (99) eliminated. Substituting

equations (105) and (109) into equation (97) produces:

V1 + V2 + V5 + V5 + V5 = 5CA5

V1 + V2 + 3V5 = 5CA5

V2 = 5CA5 3V5 V1 (110)

And substituting equations (105), (109) and (110) into conservation of flow equation (97)

produces:

P5F = (Vi+ (5CAs 3V5 Vi) + Vs + Vs +Vs)/(5Vi)

P5F = (V + 5CA5 3V5 Vi+ 3Vs)/(5Vi)

P5F = (5CA5)/(5Vi)

5ViP5F = 5CA5









V1 = CA5/P5F (101)

This is the same equation for Vi that was previously obtained. However, substituting

equations (101), (105) and (109) into equation (97) now yields:

CA5/P5F + V2 + V5 + V5 + V5 = 5CA5

V2 = 5CA5 CA5/P5F 3V5

V2 = CA5 (5 1/P5F) 3V5 (111)

If we modify our example such that V5 is actually 450 instead of 350, then setting Vi = V2

and using equation (107) would result in a value for V3 of:

V3 = 575 vpatf(5 2/0.75) 2(450 vpatf) = 441.7vpatf

But this is not acceptable, since V3 = 441.7 would be less than V4 = V5 = 450, which

violates our original assumption that the last period must be the period with the lowest flow rate.

Rather, if V5 = 450, then V3 must be set equal to Vs and equation (111) used to solve for V2 (The

value of Vi does not change):

V2 = 575 vpatf (5 1/0.75) 3(450 vpatf) = 763.3 vpatf

So, for this modified example, the cumulative arrival curve that produces the maximum

reasonable delay has period flow rates of: 766.7 vpatf, 763.3 vpatf, 450.0 vpatf, 450.0 vpatf and

450.0 vpatf. Or, expressed as hourly flow rates: 613.3 vph, 613.3 vph, 360.0 vph, 360.0 vph

and 360.0 vph.

In the original example, Vi is a feasible upper limit for V2 but in the modified example it is

not. The value of Vs is too high in the modified example to allow V2 to equal Vi without forcing

V3 to be lower than Vs. The value of Vs at which this restriction occurs can be found by setting

V3 equal to Vs in equation (105):

V3 = CA5 (5 2/P5F) 2V3









3V3 = CA5 (5 2/P5F)

V3 = CA5/3(5 2/P5F) = V5 (112)

For our original example:

V3 = 575 /3(5 2/0.75)

V3 = V5 = 447.2 vpatf

Consequently, if Vs is less than 447.2 then Vi = V2 and equation (105) can be used to

calculate V3. In general, equation (105) can be used to calculate V3 if Vs > CA5 /2(5 3/P5F)

and V5 < CA5 /2(5 3/P5F). If Vs > CA5 /3(5 2/P5F) then V3 must be set equal to Vs and the

remaining equations solved accordingly. Equation (109) must be used to solve for V2 when this

occurs since V2 no longer equals V1.

Derivation of the Five Period Lower Bound

Minimum delay occurs when the vertical distance between the arrival curve and the

departure curve (the nominal queue length) is continually minimized, without the end of the

queue becoming visible. This happens when the nominal queue length equals the Field of View

(FOV). Under these conditions, the minimum value for Vi is:

V1 = [(UDRi)(ti) + FOV] x 5 periods/atf, or

V1= C + 5FOV (113)

Vi cannot be any lower than this value or the end of the queue would be visible at the end

of period 1 and no estimation of the delay associated with the overflow queue would be required.

If Vi equals this absolute lower bound, then we can continue to minimize delay by having V2 and

V3 equal their respective capacities:

V2 = C2 (114)

V3= C3 (115)









This produces a cumulative arrival curve for periods 2 and 3 that parallels the uniform

departure curve for these periods. We continue our ongoing example as follows:

V1 = [(0.1667 veh/sec)(900 sec/period) + 12 veh] x 5 periods/atf = 600 + 60 = 660 veh/atf

V2 = [(0.1667 veh/sec)(900 sec/period)] x 5 periods/hr = 600 veh/atf

V3 = [(0.1667 veh/sec)(900 sec/period)] x 5 periods/hr = 600 veh/atf

We can now solve for V4. Substituting equations (113), (114) and (115) into conservation

of flow equation (97) produces:

C1 + 5FOV + C2 + C + V4 + V5 = 5CA5

V4= 5CA5 C1 -C2 C 5FOV -V5 (116)

For our example:

V4=5/atf (575 veh)-(600 veh/atf)-(600 veh/atf)-(600 veh/atf)-5/hr (12 veh)-350 veh/atf=665
veh/atf

The resulting P5F is found by substituting equations (113), (114) and (115) into equation (96):

P5F = (C1 + 5FOV + C2 + C3 + V4 + V5) / (5V4)

And then substituting in equation (114) for V4:

P5F = (Ci + 5FOV + C2 + C3 + 5CA5 Ci C2 C3 5FOV V5 + V5) / (5V4)

P5F = (5CA5) / (5V4)

P5F = CA5 / V4 (117)

P5F = 575 vpatf/ 665 vpatf = 0.865

Which is greater then the minimum required value of 0.75

So, for our example, the cumulative arrival curve that produces the minimum reasonable

delay has period flow rates of: 660 vpatf, 600 vpatf, 600 vpatf, 665 vpatf, and 350 vpatf. This

lower bound curve is depicted in Figure 5-21. Dividing by 1.25, the length of the analysis time









frame in hours, converts these values into hourly flow rates: 528 vph, 480 vph, 480 vph, 532

vph and 280 vph

If V1, V2 and V3 are all at their lower limit, then the maximum value for V5 can be

calculated by setting V4 equal to its lowest possible value. As with V2 and V3, V4's lowest

possible value occurs when it parallels its cumulative departure curve:

V4= C4 (118)

We can now solve for he maximum value of V5. Substituting equation (118) into equation

(116) produces:

C4 = 5CA5 C1 C2 C3 5FOV V5

V5 = 5CA5 C1 C2 C3 5FOV C4

For our example:

V5 = 5/atf (575 veh)-(600 veh/atf) (600 veh/atf) (600 veh/atf) 5/hr (12 veh) 600
veh/atf

Vs = 415 veh/atf

If V1, V2 and V3 are all at their lower limit, then the minimum value for Vs can be

calculated by setting V4 equal to its highest possible value while maintaining the minimum

required PHF and preserving conservation of flow. Recognizing that V4 will have the highest

flow rate for this situation:

P5F = (Vi + V2 + V3 + V4 + Vs)/(5V4) (119)

Substituting equations (113), (114) and (116) into the peak period equation (119) yields:

P5F = (C1 + 5FOV + C2 + C3 + V4 + Vs)/(5V4)

5P5F V4 = C1 + 5FOV + C2 + C3 + V4+ V5

5P5F V4 V4= C1 + 5FOV + C2 + C3 + V5

V4 = (C1 + 5FOV + C2 + C3 + V5) / (5P5F 1) (120)










Substituting equation (120) into equation (116) produces:

(C + 5FOV + C2 + C3 + Vs) / (5P5F 1) = 5CA5 C1 C2 C3 5FOV

V5 = 5CA5 C1 C2 C3 5FOV (C + 5FOV + C2 + C3 + Vs) / (5P5F

V5 (5P5F 1) = (5P5F 1) (5CAs C C2 C3 5FOV)- C 5FOV -

5P5F V5 V5 = (5P5F 1) (5CAs C1 C2 C3 5FOV) C1- 5FOV -

5P5F V5 = (5P5F 1)(5CA5 CC2 C3 5FOV) + (5CA5 C 5FOV C2

5P5F Vs = (5P5F 1 + 1) (5CAs Ci C2 C3 5FOV) 5CA5

V5 = [(5P5F) (5CA5 C C1 2 C3 5FOV) 5CA5] / 5P5F

Vs = (5CAs C1 C2 C3 5FOV) CA5 / P5F

V5 = 5CA5 C1 C2 C3 5FOV) CA5 / P5F

V5 = CA5 (5-1/P5F) C1 C2 C3 5FOV


- V

-1)

C2 C3 V

C2 C3 V5

- C3) 5CA5










(121)


For our example:

V5 = 5/atf (575 veh)-(600 veh/atf)-(600 veh/atf)-(600 veh/atf)-5/hr (12 veh)-[575 veh / 0.75]

V5 = 1015 veh/atf- 766.7 veh/atf

Vs = 248.3 veh/atf

The corresponding value of V4 is found by inserting this value for Vs into formula (120):

V4= (600 vpatf+ 5 (12) + 600 vpatf + 600 vpatf + 248.3 vpatf) / (5(0.75) 1)

V4= (2108.3 vpatf) / 2.75

V4= 766.67 vpatf

If V1, V2, V3 and V4 are all at their lower limits then the value of Vs is fixed due to
conservation of flow. Substituting equations (113), (114), (115) and (118) into equation (96)
yields:


(Ci + 5FOV + C2 + + C C4 + V) / 5

V5= 5CA5 5FOV- C1 C2 C3 C4


CA5


(122)









For our example:

V5 = 5/atf(575 veh) 600 vpatf 600 vpatf 600 vpatf 600 vpatf

V5 = 475 vpatf

The corresponding P5F value is obtained by substituting equations (113), (114) and (115)

into equation (96B):

P5F = (V1+ V2+ V3+ V4+ V5)/5V1 (96B)

P5F = (Ci + 5FOV + C2 + C3 + C4 + V5)/5(C1 + 5FOV)

And then substituting in equation (122):

P5F = (Ci + 5FOV + C2 + C3 + C4 +5CA5 5FOV Ci C2 C3 C4)/5(C1 + 5FOV)

P5F = (5CA5)/5(Ci + 5FOV)

P5F = CA5/(C + 5FOV) (123)

For our example:

P5F = 575 vpatf /660 vpatf

P5F = 0.871

If Vi and V2 are at their lower limit, then the minimum value for Vs can be calculated by

setting V3 and V4 equal to their highest possible values while maintaining the minimum required

PHF and preserving conservation of flow. Recognizing that V4 will need to have the higher flow

rate to minimize delay:

P5F = (V1 + V2 + V3 + V4+ V5)/(5V4) (124)

Substituting equations (113) and (114) into equation (124) yields:

P5F = (C1 + 5FOV + C2 + V3 + V4 + Vs)/(5V4)

5P5F V4 = C + 5FOV + C2 + V3 + V4+ V5

5P5F V4 V4= C + 5FOV + C2 + V3 + V5









V4 = (C1 + 5FOV + C2 + V3 + V5) / (5P5F 1) (125)

Substituting equations (113) and (114) into equation (97) yields:

(Ci + 5FOV + C2 + V3 + V4+ Vs) / 5 = CA5

C + 5FOV + C2 + V3 + V4+V5 =5CA5

V3= 5CA5- C -C2-5FOV- V -V4

Substituting in equation (125) produces:

V3 = 5CA5 C C2 5FOV V5 [(Ci + 5FOV + C2 + V3 + Vs) / (5P5F 1)]

(5P5F 1)V3 = (5P5F 1)5CAs (5P5F 1)C1 (5P5F 1)C2- (5P5F 1)5FOV
(5P5F 1)Vs C1- 5FOV C2 V3 V5

(5P5F 1)V3 = 25P5F CA5 5CA5 5P5F C1 + C1 5P5F C2 + C2- 25P5FFOV + 5FOV
5P5F V5 + V5 C 5FOV -C2 V3 V

(5P5F 1)V3 = 25P5F CA5 5CA5 5P5F C1 5P5F C2- 25P5FFOV 5P5F V5 V3

(5P5F 1)V3 + V3 = 25P5F CA5 5CA5 5P5F C1 5P5F C2- 25P5FFOV 5P5F V5

5P5F V3 V3 + V3 = 25P5F CA5 5CA5 5P5F C1 5P5F C2- 25P5FFOV 5P5F V5

5P5F V3 = 25P5F CA5 5CA5 5P5F C1 5P5F C2 25P5FFOV 5P5F V5

V3 = (25P5F CA5 5CA5 5P5F C1 5P5F C2 25P5FFOV 5P5F Vs) / 5P5F

V3 = 5CA5 CA5 / P5F C1 C2 5FOV V5

V3 = CA5 (5 1/P5F) C1 C2- 5FOV V5 (126)

Substituting equation (126) into equation (125) produces:

V4 = (Ci + 5FOV + C2 + CA5 (5 1/ P5F) Ci C2 5FOV V5 + Vs) / (5P5F 1)

V4 = CA5 (5 1/P5F) / (5P5F 1)

V4 = CA5 (5 1/P5F) / (5P5F 1)

This can be simplified by showing that (5 1/ P5F) / (5P5F 1) = 1/P5F:

(5 1/ P5F) / (5P5F 1) = 5 /(5P5F 1) 1/ P5F/(5P5F 1)

(5 1/ P5F) / (5P5F 1) = 5 /(5P5F 1) 1/ [P5F(5P5F 1)]










(5 1/ P5F) / (5P5F 1) = 5 P5F / [P5F (5P5F 1)] 1/ [P5F(5P5F 1)]

(5 1/ P5F) / (5P5F 1) = (5 P5F 1) / [P5F (5P5F 1)]

(5 1/ P5F) / (5P5F 1) = 1 / P5F

Therefore:

V4= CA5 / P5F (127)

Continuing our example:

V4= 575 / 0.75

V4 = 766.7 vpatf

And using equation (126):

V3 = 575 (5 1/0.75) 600 600 5(12veh) V5

V3 = 2108.3 vpatf 1260 vpatf- Vs

V3 = 848.3 vpatf V

The value of V5 is minimized when V3 is maximized. The maximum value of V3 is

constrained by the PHF equation:

P5F = (V1 + V2 + V3 + V4 + Vs)/(5V3)

P5F = (C1 + 5FOV + C2 + V3 + V4 + Vs)/(5V3)

5P5F V3 = Ci + 5FOV + C2 + V3 + V4+ V5

5P5F V3 V3 = C+ 5FOV + C2 + V4 + V5

V3 = (C + 5FOV + C2 + V4 + Vs) / (5P5F 1)

Substituting in equation (127) we obtain:

V3 = (C1+ 5FOV + C2 + CAs /5P5F + Vs) / (5P5F 1)

And substituting in conservation of flow equation (126) produces:

CA5 (5 1/ P5F) C1 C2- 5FOV V5 = (C1 + 5FOV + C2 + CA5 / P5F + Vs) / (5P5F 1)









CA5 (5 1/ P5F) C1 C2- 5FOV = (C1 + 5FOV + C2 + CA5 / P5F + V5) / (5P5F 1) + V5

(5P5F-1)[CA5(5 -1/ P5F)-Cl- C2- 5FOV] = C, + 5FOV +C2 +CA5 /P5F + V5 + V5 (5P5F 1)

V5 (1+5P5F-1) = (5P5F 1)[CA5(5 1/ P5F) -C1 -C2 5FOV] C1 5FOV C2- CA5 / P5F

5P5F V5= (5P5F 1)[CA5(5 1/ P5F) -C1 C2 5FOV] C1 5FOV C2- CA5 / P5F

V5 = {[CA5(5 1/ P5F) C1 C2 5FOV] (5P5F 1)- C1 5FOV C2 CA5 / P5F}/ 5P5F

V5 = {[5CA5 CA5/ P5F C1 C2 5FOV] (5P5F 1) C1 5FOV C2 CA5 P5F}/ 5P5F

Vs= (25P5FCAs- 5CA5- 5P5FC1 5P5FC2- 25P5FFOV 5CA5 + CAs/P5F + C1 + C2
+ 5FOV C 5FOV C2- CAs/P5F)/5P5F

V5= (25P5FCA5 5CA5 5P5FC 5P5F C2- 25P5F FOV 5CA5)/ 5P5F

V = 5CA5- CA5/5P5F C1 C2- 5 FOV CA5/5P5F

V = 5CA5- 2CA5/5P5F C1 C2- 5 FOV

V5= CA5(5 2/P5F) Ci- C2- 5FOV (128)

Substituting this equation into equation (126) produces the formula for V3:

V3= CA5 (5 1/ P5F) C C2- 5FOV CA5(5 2/P5F) + C1 + C2 + 5FOV

V3 = CA5 (5 1/ P5F) CA5(5 2/P5F)

V3 = CA5 [(5 1/ P5F) (5 2/P5F)]

V3= CA5 (5 1/P5F 5 + 2P5F)

V3= CA5 (- 1/ P5F + 2/P5F)

V3 =CAs (1/P5F)

V3= CA5 /P5F (129)

For our example V3 and Vs are calculated as follows:

V3= 575 / 0.75

V3= 766.7 vpatf

Vs = (575)(5 2/0.75) 600 600 5(12)









V5= (575)(2.33)- 1260

Vs= 81.7 vpatf

So we see that, if V1 and V2 are at their lower limit, then the minimum possible value for

Vs is 81.7 vpatf.

If V1 is held to its lower limit, then the minimum value for Vs can be calculated by setting

V2, V3 and V4 equal to their highest possible values while maintaining the minimum required

PHF and preserving conservation of flow. Recognizing that V4 will still have the highest flow

rate for this situation:

P5F = (V1 + V2 + V3 + V4+ Vs)/(5V4) (124)

Substituting equation (113) into peak period factor equation (124) yields:

P5F = (V1 + V2 + V3 + V4 + Vs)/(5V4)

5P5F V4 = V1 + V2 + V3 + V4 + V

5P5F V4 V4= V1 + V2 + V3 + Vs

V4= (Vi+ V2 + V3 + Vs) / (5P5F 1) (130)

Substituting equation (113) into conservation of flow equation (97) yields:

(Ci + 5FOV + V2 + V3 + V4+ Vs) / 5 = CA5

C1+5FOV+V2+V3+V4+V5 = 5CA5

V3= 5CA5 -C1 5FOV- V2- V5 V4 (131)

The value of V5 is minimized when V3 is maximized. The maximum value of V3 is

constrained by the PHF equation:

P5F = (V1 + V2 + V3 + V4 + Vs)/(5V3)

P5F = (C1 + 5FOV + C2 + V3 + V4 + Vs)/(5V3)

5P5F V3 = C1 + 5FOV + C2 + V3 + V4+ V5









5P5F V3 V3 = C1+ 5FOV + C2 + V4 + V5

V3 = (C1+ 5FOV + C2 + V4 + Vs) / (5P5F 1)

Substituting in equation (127) we obtain:

V3 = (C + 5FOV + C2 + CA5 (5 1/ P5F)/(5P5F 1) + Vs) / (5P5F 1)

V3 = (C1+ 5FOV + C2 + CA5 / P5F + Vs) / (5P5F 1)

And substituting in conservation of flow equation (126) produces:

CA5 (5 1/ P5F) C1 C2- 5FOV V5 = (C1 + 5FOV + C2 + CA5 / P5F + Vs) / (5P5F 1)

CA5 (5 1/ P5F) C C2- 5FOV = (CI + 5FOV + C2 + CA5 / P5F + V5) / (5P5F 1) + V5

The value of V5 is minimized when V3 and V4 are maximized. The maximum value of V4 was

provided previously as equation (127):

V4= CAs / P5F

For V3 to be maximized it must also share this P5F-constrained value:

V3= CA5 / P5F (132)

Substituting equation (113), (127) and (132) into conservation of flow equation (97) and solving

for V2 we obtain:

(Vi + V2 + V3 + V4+ Vs) / 5 = CA5

(Ci + 5FOV + V2 + 2CA5/ P5F + Vs) / 5 = CA5

C1 + 5FOV + V2 + 2CA/ P5F + V5 = 5CA5

V2 = 5CA5 C1 5FOV 2CA5 / P5F V5

V2 = CA5(5 2/P5F) C1 5FOV V5 (133)

The value of Vs is minimized when V2 is maximized. The maximum value of V2 is constrained

by the PHF equation:

P5F = (V1 + V2 + V3 + V4 + Vs)/(5V2)

P5F = (C1 + 5FOV + V2 + V3 + V4 + Vs)/(5V2)










5P5F V2 =C1 + 5FOV + V2 + V3 + V4+ V5

5P5F V2 V2= C1 + 5FOV + V3 + V4 + V5

V2 = (C1 + 5FOV + V3 + V4 + Vs) / (5P5F 1)

Substituting in equations (127) and (130) we obtain:

V2 = (C1 + 5FOV + CAs/ P5F + CAs/ P5F + Vs) / (5P5F 1)

V2 = (C1 + 5FOV + 2CA5 / P5F + Vs) / (5P5F 1) (134)

And substituting in conservation of flow equation (133) produces:

CA5(5 2/P5F) C1 5FOV V5 = (C + 5FOV + 2CA5 / P5F + Vs) / (5P5F 1)

CA5(5 2/P5F) C1 5FOV = (Ci + 5FOV + 2CA5 / P5F + Vs) / (5P5F 1) + V5

(5P5F 1) [CA5(5 2/P5F) C1 5FOV] = C1 + 5FOV + 2CA5 / P5F + V5 + V5 (5P5F 1)

V5 + V5 (5P5F 1) = (5P5F 1) [CA5(5 2/P5F) C1 5FOV] C1- 5FOV 2CA5 / P5F

V5 + 5P5F V5 V5 = (5P5F 1) [5CA5 2 CA5/P5F C1 5FOV] C 5FOV 2CA5 / P5F

5P5FV5 = 25 P5F CAs 10CA5 5P5F C 25 P5F FOV
5CA5 + 2CA5/P5F + C1 + 5FOV C1-5 FOV 2CA5/P5F

5P5F V5 = 25 P5F CA5 -15CA5 5P5F C1- 25 P5F FOV

V5 = 5CA5 -3CA5 /P5F C 5FOV

V5 = CA5(5-3/P5F)- C- 5FOV (135)

Continuing our example:

V5 = 575(5 3 /0.75) 600 5(12)

V = 575(1)- 600- 60

Vs = 85 vpatf

And using equation (132):

V2 = (600 + 5(12) + 2(575) / 0.75 85) / (5(0.75) 1)

V2= (600+ 60 + 1533.3 85) / 2.75










V2= 2108.3 / 2.75

V2 = 766.7 vpatf

This is obviously not a feasible solution since V5 is negative. In this particular case, the

value for V2 cannot be maximized with respect to its peak period factor. Once again, using

equation (134):

V2 = 575 (5 2/0.75) 600 5(12) Vs

V2 = 575 (2.33)- 660 V5

V2 = 1341.7-660 -V5

V2 = 681.7 V5

And since V5 is minimized when V2 is maximized:

V2 = 681.7 vpatf

Vs = 0 vpatf

So we see that if Vi is at its lower limit, then the minimum possible value for Vs is 0,

which occurs when V2 = 681.7 vpatf. If Vi is not held to its lower limit, then the minimum

value for V5 can be calculated by setting V1, V2, V3 and V4 equal to their highest possible values

while maintaining the minimum required PHF and preserving conservation of flow.

Recognizing that V4 will continue to have the highest flow rate for this situation:

P5F = (V1 + V2 + V3 + V4+ Vs)/(5V4) (124)

P5F = (V1 + V2 + V3 + V4 + Vs)/(5V4)

5P5F V4 = V1 + V2 + V3 + V4 + V5

5P5F V4 V4= V1 + V2 + V3 + V5

V4= (V1 + V2 + V3 + Vs) / (5P5F 1) (136)









Rearranging conservation of flow equation (97) yields:

(Vi + V2 + V3 + V4+ Vs) / 5 = CA5

Vl+V2+V3 +V4+ V5 =5CAs

V2= 5CA5 V- V3- V V4 (137)

The value of V5 is minimized when V3 and V4 are maximized. The maximum values of V3 and

V4 were provided previously as equations (127) and (132):

V4= CA5 / P5F

V3 = CA5 /P5F

For V2 to be maximized it must also share this P5F-constrained value:

V2= CA5 / P5F (138)

Substituting equation (127), (132) and (138) into conservation of flow equation (97) and solving

for Vi we obtain:

(Vi + V2 + V3 + V4+ V) / 5 = CA5

(V1 + 3CAs/P5F + Vs) / 5 = CA5

Vi + 3CA5/P5F + V5 =5CA5

V1 = 5CA5 3CA5 / P5F V5

Vi = CA5(5 3/P5F) V5 (139)

For our example Vi equals:

Vi = 575 (5 3/0.75)- V5

Vi = 575 (1.00)-V5

V1 = 575 Vs

And since Vs is minimized when Vi is maximized:

V1 = 575 vpatf









However, this is not a feasible solution for this example since Vi must be greater than C1 +

5FOV (which is 600 + (5)12 = 660 vpatf) or the end of the queue will be visible during the first

period. As we discovered previously, V2 cannot reach its maximum peak period factor value of

CA5 / P5F for this example without causing Vi to drop to a value that is too low.

In general, for V2 to reach its maximum peak period factor constrained value while

maintaining a minimum non-negative value for Vs (i.e. zero), the value of Vi must satisfy

equation (139) and conservation of flow must be maintained. Since Vi must continue to equal

Ci+ 5FOV for the queue to remain non-visible, this places a maximum value on Ci of:

C1 + 5FOV = CA5(5 3/P5F) V5

Ci <= CA5(5 3/P5F) 0 5FOV

Ci <= CA5(5 3/P5F) 5FOV (140)

For our example, the maximum value that C1 can be if V2 is to be maximized is:

C1 = 575(5 3/0.75) 5(12)

Ci= 515

And since conversation of flow must be maintained:

(Vi + V2 + V3 + V4+ V) / 5 = CA5

V1 + CA5 /P5F + CA5 / P5F + CA5 / P5F + V5 = 5CA5

V1 + 3CA5 / P5F + V5 = 5CA5

Ci + 5FOV + 3CA5 / P5F + V5 = 5CA5

C1 = 5CA5 3CA5 / P5F 5FOV V5

C1 = CAs(5 3/ P5F) 5FOV V5

For our example:

515 = (575)(5 3/0.75)- 5(12) -V5









Vs= (575)(1)- 60- 515


V5= 0

Which checks.

Five Period Analysis of Bounds Summary

For the 5 period case the results of the analysis of the bounds can be summarized as

follows where the Vi's are expressed in terms of vehicles per analysis time frame:


UPPER BOUND


V1 = CA5/P5F


If V5 < (CA5 /2)(5 3/P5F)




If V5 > (CA5 /2)(5 3/P5F)
And Vs < (CA5 /3)(5 2/P5F)




If V5 > (CA5/3)(5 2/P5F)


Then: V2
V3
V4



Then: V2
V3
V4


Then: V2
V3
V4


CA5/P5F
CA5 /P5F
CA5(5 3/P5F)


CA5/P5F
CA5 (5 -
V5


CA5 (5 -
V5


2/P5F) 2V5



1/P5F) 3V5


(101)

(108)
(102)
(103)
(104)

(108)
(112)
(102)
(107)
(105)

(112)
(111)
(109)
(105)


If V5 = 5CA5- Ci C2 C3 C4- 5FOV
Then:


IfV5 < 5CA5- C1 C2- C3
And V5 >= CA5 (5-1/P5F) -


V =
V2=
V3=
V4=
P5F


C1 + 5FOV
C2
C3
C4
= CA5 / (C1 + 5FOV)


- C4- 5FOV
C1 C2 C3 5FOV


Then: Vi = C1 + 5FOV
V2 = C2


LOWER BOUND


(122)
(113)
(114)
(115)
(118)
(123)

(122)
(121)

(113)
(114)









V3= C3 (115)
V4 =5CA5-Ci C2 C3 5FOV V5 (116)
P5F = CA5 / V4 (117)

If V5 < CA5 (5 1/P5F) C1 C2 C3 5FOV (121)
And V5 >= CA5(5 2/P5F) C1 C2 5FOV (128)

Then: V1 = C1 + 5FOV (113)
V2 = C2 (114)
V3 = CA5(5-1/P5F) C1 C2 5FOV V5 (126)
V4 = CA5 /P5F (127)
P5F = CA5 / V4 (117)

If V5 < CA5(5 2/P5F) C1 C2 5FOV (128)
And V5 >= CA5(5 3/PHF) C 5FOV (137)

Then: V1 = C1 + 5FOV (113)
V2 = CA5 (5 2/P5F) C 5FOV V5 (133)
V3= CAs/P5F (132)
V4= CA5 / P5F (127)
P5F = CA5 / V4 (117)

If V5< CA5(5 3/PHF) Ci- 5FOV (137)

Then: V1 = CA5 (5 3/P5F) V5 (139)
V2 = CA5 /P5F (138)
V3= CA5 /P5F (129)
V4= CA5 / P5F (127)
P5F = CA5 / V4 (117)
Ci <= CA5(5 3/P5F) 5FOV (138)

For our example, the values are:

UPPER BOUND

V1 = 575/0.75
V1 = 766.7 vpatf
(Vi = 766.7/1.25 = 613.3 vph)

Is V5 = 350 < (575 /2)(5 3/0.75)?
Is V5= 350 < (575 /2)(1)?
Is Vs= 350 <287.5? NO

Is V5 = 350 < (575 /3)(5 2/0.75)?
Is V5= 350 < (191.7)(2.33)?
Is V5= 350 <447.2? YES









(Vs = 350/1.25 = 280 vph)


V2 = CA5/P5F = 575/0.75
V2 = 766.7 vpatf
(V2 = 766.7/1.25 = 613.3 vph)
V3 = 575(5 2/0.75) 2(350) = 575(2.33) 2(350)
V3 = 641.7 vpatf
(V3 = 641.7/1.25 = 513.3 vph)
V4 = 350 vpatf
(V4 = 350/1.25 =280 vph)

Is V5> (575 /3)(5- 2/0.75)?
Is V5 > (191.7)(2.33)?
Is V5 > 447.2? NO

LOWER BOUND

Is V5 = 350 = 5(575)- 600 600 600 600 5(12)?
Is V5 = 350 = 2875- 2400- 60?
Is V5 = 350 = 415? NO

Is V5= 350 < 5(575)- 600- 600- 600- 600- 5(12)?
IsV5 = 350 < 415? YES
AND
Is V5 = 350 > 575 (5 1/0.75) 600 600 600 5(12)?
Is V5 = 350 > 2108.3 1800 60?
Is Vs = 350 > 248.3? YES
(V5 = 350/1.25 =280 vph)
V1= 600 + 5(12)
V = 660 vpatf
(V = 660/1.25 =528 vph)
V2 = C2
V2 = 600 vpatf
(V2 = 600/1.25 =480 vph)
V3 = C3
V3 = 600 vpatf
(V3 = 600/1.25 =480 vph)
V4 =5(575)-600-600 600 5(12)- 350
V4= 2875- 1800-60-350
V4 = 665 vpatf
(V4= 665/1.25 =532 vph)
P5F = 575 / 665
P5F = 0.865

Is V5 = 350 < 575 (5 1/0.75)- 600 600 600 5(12)?
IsV5=350< 2108.3 1800-60?









IsV5 = 350< 248.3? NO

Is V5 = 350 < 575(5 2/0.75) 600 600 5(12)?
Is Vs= 350 < 1341.7- 1200- 60?
Is Vs= 350 < 81.7? NO

Is Vs= 350 < 575(5- 3/0.75)- 600- 5(12)?
IsV5=350< 575-600-60?
IsV5=350< -85? NO (Not Feasible)

Only Feasible If:
Ci <= 575(5 3/0.75)- 5(12)
Ci <= 575- 60
C1 <= 515 (but Ci = 600)

Generalized Analysis of Bounds Summary

The conservation of flow equation and the peak period factor equation can be generalized

to any number of periods as follows:

(ENVi)/N= CAN

(ENVi ) / [N Max(Vi)] = PNF

where N is the number of periods in the analysis time frame and the Vi's are expressed in terms

of vehicles per analysis time frame. The corresponding analysis of bounds results can be

generalized as well:


UPPER BOUND

V1 = CAN/PNF

If VN < (CAN /(N-3))(N (N-2)/PNF)
Then: VN-3 = CAN/PNF
VN-2 = CAN/PNF
VN-1 = CAN(N (N-2)/PNF) (N-4)VN


If VN > (CAN /(N-3))(N (N-2)/PNF)
And VN < (CAN/(N-2))(N (N-3)/PNF)

Then: VN-3 = CAN/PNF









VN-2 = CAN(N (N-3)/PNF) (N-3)VN
VN-1 = VN

If VN > (CAN /(N-2))(N (N-3)/PNF)
Then: VN-3 = CAN(N (N-4)/PNF) (N-2)VN
VN-2 = VN
VN-1 = VN
LOWER BOUND

If VN = NCAN (EN-1Ci) NFOV
Then: VN-4 = CN4 + NFOV
VN-3 = CN-3
VN-2 = CN-2
VN-1 = CN-1
PNF = CAN / (CN-4 + NFOV)

If VN < CAN(N- (N-5)/PNF) (EN-Ci) NFOV
And VN >= CAN (N (N-4)/PNF) (EN-2Ci) NFOV

Then: VN-4 = CN-4 + NFOV
VN-3 = CN-3
VN-2 = CN-2
VN-1 = CAN (N (N-5)/P5F) (EN-2Ci) NFOV VN
PNF = CAN / VN-1

If VN < CAN (N (N-4)/PNF) (EN-2Ci) NFOV
And VN >= CAN(N (N-3)/PNF) (EN-3Ci) NFOV

Then: VN-4 = CN-4 + NFOV
VN-3 = CN-3
VN-2 = CAN (N (N-4)/P5F)- (EN-3Ci) NFOV VN
VN-1 = CAN /PNF
PNF = CAN / VN-I

If VN < CAN(N (N-3)/PNF) (EN-3Ci) -NFOV
And VN >= CAN(N (N-2)/PHF) (EN-4Ci) NFOV

Then: VN-4 = CN-4 + NFOV
VN-3 = CAN (N (N-3)/PNF) (EN-4Ci) NFOV VN
VN-2 = CAN / PNF
VN- = CAN / PNF
PNF = CAN / VN-1

If VN < CAN(N (N-2)/PHF) (EN-4Ci ) NFOV

Then: VN-4 = CAN (N (N-2)/PNF) (EN-Ci) VN









VN-3 = CAN / PNF
VN-2 = CAN / PNF
VN- = CAN / PNF
PNF = CAN / VN-1

C1 <= CAN(N (N-2)/P5F) NFOV

And these equations can be further generalized to the following:

UPPER BOUND

V1 = CAN/PNF

Forj = 1:
When VN > VNLL = (CAN/2)[N (N-2)/PNF]

Then:

For 1 < k
For k = N-l: Vk = 2VNUL VN

For k = N: Vk = VN


Forj = 2 to N-3:
When VN> VNLL = (CAN/j)(N (N
And VN < VNUL = (CAN/(j +1))[N -

Then:

For 1 < k
For k= N-j:

For N >= k > N-j:

Forj = N-2:
When VN > VNLL = (CAN /(N-2))(N

Then:

Fork = 1:

For k = 2:

For N >= k > 2:


j)/PNF)
(N-j-1)/PNF]


CAN /PNF

VNUL (j +1) -jVN

VN


-2/PNF)



Vk = CAN/PNF

Vk = (N-1)VNUL- (N-2)VN

Vk=VN


(141)


(142)


(143)

(144)

(145)


(146)
(147)


(143)

(148)

(145)


(149)


(143)

(150)

(145)










LOWER BOUND

Forj = 1 to N-2:

Ifj 1
And VN = VNUL = NCAN (EN- Ci)

Then:

Fork= 1:

For N-1 > k > 1:


-NFOV

PNF = CAN / (Cl + NFOV)

Vk = Ck+ NFOV

Vk= Ck


Otherwise:
When VN < VNUL = CAN (N (j-1)/PNF) (EN- Ci) NFOV
And VN > VNLL = CAN (N (j)/PNF) (EN1 Ci) NFOV

Then: PNF = CAN VN-I

Fork = 1: Vk = Ck+ NFOV

For
For k= N-j: Vk= VNUL + Ck- VN

For N-1 > k > N-j: Vk = CAN /PNF


(151)

(152)

(153)

(154)


(155)
(156)

(157)

(153)

(154)

(158)

(159)


Forj = N-l:

IfVN < VNUL = CAN (2/PNF) Ci -NFOV (160)

And Ci <= CAN(N (N-2)/P5F) -NFOV (161)

Then: PNF = CAN / VN-1 (152)

Fork= 1: Vk= VNUL+ Ck- VN (153)

For N- > k > 1: Vk= CAN/PNF (159)

Historical Peak Hour Factors

This theoretical bracketing procedure for estimated delay is dependent upon the ability to

identify a minimum peak hour factor for each approach under consideration. Fortunately,









information on peak hour factors is routinely collected as part of the data collection effort for

most intersection evaluations. Consequently, historical peak hour factors are rather easy to

identify, at least for intersections that are not over-saturated. Appendix B contains a sample of

historical PHF information for various locations in Jacksonville, Florida.

Tarko and Perez-Cartagena [49] proposed the following prediction model for the Peak

Hour Factor (PHF) based on time of day, population, and peak hour volume:

PHF = 1 exp (-2.23 + 0.435 AM + 0.209 POP 0.258 VOL)

Where: AM = 1 if AM, 0 Otherwise
VOL= Peak Hour Volume (1000's/hour)
POP = Population (1,000,000's)

Applying this equation to our four examples and assuming that we are dealing with the

weekday PM peak hour at an intersection that is situated in a city of 1,000,000 people yields the

following results:

625_700_650_350vph: PHF = 1 exp (-2.23 + 0 + 0.209(1.0) 0.258 (581.25/1000)) = 0.89

700_725_625_350vph: PHF = 1 exp (-2.23 + 0 + 0.209(1.0) 0.258 (600/1000)) = 0.89

700_700_700_350vph: PHF = 1 exp (-2.23 + 0 + 0.209(1.0) 0.258 (612.5/1000)) = 0.89

725_700_700_350vph: PHF = 1 exp (-2.23 + 0 + 0.209(1.0) 0.258 (618.75/1000)) = 0.89

These expected peak hour factors are well above the 0.80 minimum PHF assumed in our

analysis. If the AM peak hour were under consideration, the PHF would fall to a value of 0.82,

which is still above the minimum assumed value.

Hellinga and Abdy [50] investigated the variability of peak hour traffic volumes and the

Peak Hour Factor (PHF) at 10 urban locations in Waterloo and Kitchener, Ontario, Canada. The

average PHF for their 10-location urban data set varied between 0.88 and 0.94 with an overall









average PHF of 0.92. Their minimum PHF varied between 0.47 and 0.87 with an average

minimum PHF of 0.78

One fortunate aspect of the use of peak hour factors is that low peak hour factors (factors

below a value of about 0.80) are typically encountered only on low volume approaches where

queues tend to remain small and delay can be directly measured. As volumes rise on an

intersection approach, the associated peak hour factor tends to rise as well.

Through the use of minimum historical peak hour factors we can develop a reasonable set

of lower and upper bounds for the overflow delay. After making the necessary modifications

discussed in the next chapter, these bounds can be used to bracket the results of our delay

prediction procedure. The required historical peak hours are readily available or can be easily

derived from archived traffic count information.

Limitations to the Theoretical Bracketing Procedure

The peak hour factor based technique for theoretically bracketing delay represents a novel

approach for keeping delay estimates within reasonable limits. Although the usefulness of the

technique is evident, limitations on the use of the technique should be understood. These

limitations include the following:

1. The technique assumes that the flow rate remains constant within each 15-minute
period, which results in a piecewise linear cumulative arrival curve. This constant
arrival rate assumption is made by many analysis procedures, including those
contained within the Highway Capacity Manual. If the cumulative arrival curve is
actually curvilinear then the bounds, especially the lower bound, may be incorrect.

2. The technique also assumes that the arrival rate observed during the final 15-minute
period is the lowest rate experienced during the analysis time frame. One could
conceive of circumstances where this would not be the case, especially for a long
analysis time frame of greater than an hour.

3. If the end of the queue remains beyond the field of view for more than four 15-minute
periods then a peak period factor will need to be used to establish the upper and lower
bounds instead of the peak hour factor. Historical peak hour factors are readily
available and, as was previously discussed, there even exists an equation to predict the









peak hour factor given volume, population and time-of-day information. However,
since the concept of a peak period factor is introduced in this research, no information
on peak period factors is directly available. Fortunately, historical peak period factors
can be easily derived from archived traffic count information.

4. If, for some reason, an unusual level of peak period flow occurs such that the
minimum peak period factor is violated, then the upper bound will be incorrect.
Extreme peak period flows could be due to some unusual event, such as a serious
accident, a special activity in the area or a weather-related incident.

















600




500




. 400
a


S300




200




100




0


300 600 900 1200 1500 1800 2100 2400 2700 3000 3300

Time (sec)


Figure 5-1. Cumulative arrival-departure curves and overflow delay





















































3000 3300


Time (sec)


Figure 5-2. Critical time and volume points for period 4


C)
.2
C-

g 550


E
o


500 -





450 L





400 -
2700


3600















Oversaturated Period


Overflow Queue
Clearance Time


3000 3300


Time (sec)


Figure 5-3. Overflow delay in period 4


650





600

5,

a
" 550

E'

OF 5


2700


OD = Overflow DelayI

487,5



:1414Uniform Delay















600




500




2 400



.C
S300




200




100




0


0 300


X4 = 350 vph





Maximum Cumulative Arrival Curve
----------------- ~~~~ ~C rye_________--------^


V1=766.7, V2=766.7, V3=416.7, V4=350
PHF =0.75


Uniform Cumulative Departure Curve
(Capacity)=C1=C2=C3=C4=600 vehlhour


600 900 1200 1500 1800 2100 2400


3000


Time (sec)

Figure 5-4. Maximum reasonable cumulative arrival curve


Op -














600


Minimum Cumulative Arrival Curve

500 V1=648, V2=600, V3=702, V4=350 \

I
3_ __PH F =0.75_ _ _





12 FOV ~



Uniform Cumulative Departure Curve
(Capacity) = C1 =C2=C3 =C4=600 veh/hour
200 I
12 FOV


100 Parallel from 900 to 1800 sec
I

0 0

0 300 600 900 1200 1500 1800 2100 2400 2700 3000 3300 3600

Time (sec)
Figure 5-5. Minimum reasonable cumulative arrival curve





















600





500





. 400
c
,-
0)


E 300





200





100





0


0 300 600 900 1200


1500 1800 2100 2400 2700 3000

Time (sec)


3300 3600


Figure 5-6. Minimum overall reasonable cumulative arrival curve










700




600 _X4 = 285.3 vph]

Minimum Cumulative Arrival Curve

V1=648, V2=600, V3=766.7, V4=285.3
500 --PHF =0.75




5 400


S12 FOV
E 300-

Uniform Cumulative Departure Curve
(Capacity) = C1 =C2=C3 =C4=600 vehlhour
200
12 FOV



100 I Parallel from 900 to 1800 sec




0
0 300 600 900 1200 1500 1800 2100 2400 2700 3000 3300 3600
Time (sec)


Minimum reasonable cumulative arrival curve (minimum V4 for minimum V1 and V2)


Figure 5-7.















600




500



a,
S400




E 300




200




100




0


Time (sec)


Figure 5-8. Minimum reasonable cumulative arrival curve (minimum V4 for minimum Vi)


4 = 120 vph


Minimum Cumulative Arrival Curve

V1=647, V2=766.5, V3=766.5, V4=120
PHF =0.75













0 Uniform Cumulative Departure Curve
(Capacity) = C1 =C2=C3 =C4=600 vehlhour
12 FOV









0 300 600 900 1200 1500 1800 2100 2400 2700 3000 3300 36















350



300



c 250
o


D 200


E
o 150



100



50




0
0


300 600 900 1200 1500
1Time T(se)
Time (sec)


Figure 5-9. Period 1 delay for the upper bound










500.0


400.0'



u 350.0
o


. 300.0


E
O 250.0


100.0 -
900

T15


Figure 5-10. Period 2 delay for the upper bound


1200 1500 1800 Tc 2100 T 2400

Time (sec) T3


270
T45











600.0


550.0


o
S500.0

I,

3 450.0
E


400.0


250.0 I I I
I I I I


200.0 -I
1800 2100 2400 2700 3000 3300 3600

FTO3 1c2 1c2 3 Time (sec) 45 T3


Figure 5-11. Period 3 and period 4 delay for the upper bound










500
480
460
440
420
400
380
360
340
320
" 300
I 280
260
| 240
220
a 200
0 180
160
140
120
100
80
60
40
20
0


-


Total Cumulative Arrivals, CA60 (vph)













575

---595-
486 500 I I


Observed Arrival Flow Rate (V4)


Figure 5-12. Reasonable overflow delay region for 600 vph capacity and 0.75 minimum PHF











500
480
460
440
420
400
380
360
340
320
300
S280
260
| 240
0
220
S 200
0 180
160
140
120
100
80
60
40
20
0


Total Cumulative Arrivals, CA60 (vph)













519 550 585 59
525
i''''= ^^^ ^

----- \-I ^ Is Is I I I I < ^ I,----
______ I I I^^ ^ I^^
------------------------------ I IIII-------- -.---------------E ---- ^Sc- 1EH f ---- --- ------- ---- ^^------------
I^ ^ IS Is I I 75
III IIII ^^' ^^^ 55 I^^ 55-


Cumulative Arrivals/Departures When Visibility Returns (CA4)

Figure 5-13. Reasonable overflow delay region for 600 vph capacity and 0.80 minimum PHF










500
480
460
440
420
400
380
360
340
320
300
" 280
260
" 240
0
S220
1 200
0 180
160
140
120
100
80
60
40
20
0


Total Cumulative Arrivals, CA60 (vph)


Observed Arrival Flow Rate (V4)


Figure 5-14. Reasonable overflow delay region for 600 vph capacity and 0.85 minimum PHF


585
5515 595'
6r.6 676
__________________51 ^^ ^S sJ^ ^^ ^ ___













100%


90%

80%

70%


60



STotal Cumulative Arrivals, CA75
Lu 550
50%
585
S40%

30%
Total Cumulative Arrivals, CA60
20%

10%

0%
0 100 200 300 400 500 600

Observed Flow Rate, V4 (vph)


Figure 5-15. Maximum delay estimation error for 0.75 minimum PHF
























0
LU
5o50o
550 575 585
S550
40%
595

30%
525 Total Cumulative Arrivals, CA60
20% -

10%

0%
0 100 200 300 400 500 600
Observed Flow Rate, V4 (vph)


Figure 5-16. Maximum delay estimation error for 0.80 minimum PHF












100%


90%

80%

70%

2 60%
LJ
W
E 50%

40%
575 585


20% 565 595

10% Total Cumulative Arrivals, CA60

0% 1.
0 100 200 300 400 500 600

Observed Flow Rate, V4 (vph)


Figure 5-17. Maximum delay estimation error for 0.85 minimum PHF


















































0 300 600 900 1200 1500 1800 2100

Time (sec)

Figure 5-18. Maximum reasonable cumulative arrival curve with period 1 visible


V4 = 350 vph


Maximum Cumulative Arrival Curve

Vi=600, V2=766.7, V3=583.3, V4=350 00
PHF 0.75









Uniform Cumulative Departure Curve
(Capacity) = C1 = C2 = C3= C4 = C5 = 600


2400


2700


3300


3600















Minimum Cumulative Arrival Curve
V1=600, V2=648, V3=702, V4=350
PHF 0.75


V4 = 350 vph


Uniform Cumulative Departure Curve
(Capacity) = C = C2 = C3= C4 = C5 = 600


0 300


600 900 1200 1500 1800 2100 2400


2700


3000


Time (sec)


Figure 5-19. Minimum reasonable cumulative arrival curve with period 1 visible


3300


3600


..0


I






















Maximum Cumulative Arrival Curve















(Capacity) = C = C2 3= C4=C5 = 600


0 300 600 900 1200 1500 1800 2100 2400 2700 3000 3300 3600 3900 4200 4500

Time (sec)


Figure 5-20. Maximum reasonable cumulative arrival curve with 5 periods


600



500

U)

. 400


"-
5 300
E


200



100



0



















Minimum Cumulative Arrival Curve

Vi=660, V2=600, V3=641.7, V4=350, V5=350
PHF 0.865






12 FOV





_Uniform Cumulative Departure Curve
(Capacity) = C = C2 = C3= C4 = C5 = 600








0 300 600 900 1200 1500 1800 2100 2400 2700 3000 3300 3600 3900 4200 4500

Time (sec)


Figure 5-21. Minimum reasonable cumulative arrival curve with 5 periods









CHAPTER 6
COMPARISONS WITH VEHICLE TRAJECTORY ANALYSIS

This chapter describes a process for reconciling cumulative curve delay and control delay

obtained from trajectory analysis. This reconciliation ensures that the delay limits associated

with the theoretical bounds properly represent delay and are consistent with trajectory analysis

(Objective 5).

The area between the cumulative arrival curve and the cumulative departure curve does not

represent either stopped delay or control delay, but rather a mixture of various delay and travel

time components. Trajectory analysis is used to demonstrate this fact and to establish the true

relationship between overflow delay and both stopped delay and control delay.

Appendix F of Chapter 16 of the 2000 HCM discusses the relationship between the initial

queue delay and deterministic queue delay using cumulative arrival curves. Five specific arrival

"cases" are discussed and the proper way to account for initial queue delay and deterministic

delay for each case is explained.

Unfortunately, the random portion of the control delay is not reflected in the cumulative

arrival and departure curves, nor is the portion of the control delay associated with deceleration

or post-stop bar acceleration. In addition, queue move-up delay and pre-stop bar acceleration

delay are over-represented when using a cumulative arrival/cumulative departure approach. To

prove these statements, an example has been prepared. The following technical terms are

referred to in the presentation of this example:

Delay Zone = Segment length over which control delay is measured. It includes a portion of the
approach link at the intersection and a portion of the departure link. For our examples, the delay
zone is 3900 feet long with 3600 feet on the approach link and 300 feet on the departure link.

Stopped Delay = Time that the vehicle is stopped in a queue (stationary wheels)

Acceleration Distance = Distance that the vehicle covers while accelerating from a complete
stop to its desired free flow speed










Pre-Stop Bar Free Speed Acceleration Time = Time that the vehicle would have taken to
travel from the front of the queue to the stop bar had it been able to move at its desired free flow
speed

Acceleration Delay = Time that the vehicle takes to accelerate form a complete stop to its
desired free flow speed minus the time that the vehicle would have taken to traverse the
acceleration distance had it been able to travel consistently at its desired free flow speed

Pre-Stop Bar Acceleration Delay = Time that the vehicle takes to travel from the front of the
queue to the stop bar minus the time that the vehicle would have taken to traverse this distance
had it been able to travel consistently at its desired free flow speed

Deceleration Distance = Distance that the vehicle covers while decelerating from its free flow
speed to a complete stop

Deceleration Delay = Time that the vehicle takes to decelerate form a its free flow speed to a
complete stop minus the time that the vehicle would have taken to traverse the deceleration
distance had it been able to travel consistently at its desired free flow speed

Move-Up Time = Time that it takes the vehicle to travel between queues

Free Speed Move-Up Time = Time that the vehicle would have taken to travel between queues
if it had been able to move at its desired free flow speed

Move-Up Delay = Time that the vehicle is delayed while traveling between queues = Move-Up
Time Free Speed Move-Up Time

Control Delay = Time that the vehicle is delayed due to intersection control = Time that the
vehicle takes to traverse the delay zone minus the time that the vehicle would have taken to
traverse the delay zone had it been able to travel consistently at its desired free flow speed =
Deceleration Delay + Stopped Delay + Move-Up Delay + Acceleration Delay

Interaction Delay = Delay resulting from travel speeds that are lower than the desired free flow
speed due to restrictions caused by other vehicles. It is not part of control delay.

Figures 50 through 53 document the differences between delay as represented by

cumulative arrival curves and true control delay as given by an analysis of vehicle trajectories.

As Dowling [3] has correctly noted:

Comparison of results between tools and methods is possible only if the analyst looks at
the lowest common denominator shared by all field data collection and analytical tools:
vehicle trajectories. At this microscopic level, the analyst can compare field data to
analysis tool outputs, whether the tool is microscopic or macroscopic. By computing









macroscopic MOEs from the vehicle trajectory data the analyst can compare the results of
macroscopic and microscopic tools to field data and to each other in a consistent manner.
This is the only appropriate method for comparing results between tools, validating the
model results against field data, or using the outputs of other tools to compute level of
service as defined by the Highway Capacity Manual.

Trajectory Example

Figure 6-1 provides an instructive example of how control delay accumulates for a single

vehicle traversing a signalized intersection using the true method for analyzing delay, trajectory

analysis. In this example, the vehicle initially travels at a free flow speed of 40 feet per second.

It enters the delay zone at distance 0 and travels at the free flow speed for 60 seconds until it

reaches a distance of 2400 feet. The vehicle then decelerates to a stop over a distance of 1000

feet, taking another 60 seconds to cover this distance. The 60 seconds of deceleration time can

be decomposed into 35 seconds of deceleration delay and 25 seconds of time traveling at the free

flow speed. The average speed during deceleration is 16.7 fps (1000 feet/60 seconds).

The vehicle then stops for 80 seconds, all of which is delay time. No progress forward is

made. The speed is zero during this period. The vehicle takes 50 seconds to move up from its

first stop to a second stop. The 50 seconds of move-up time can be decomposed in 40 seconds of

move-up delay and 10 seconds of time traveling at the free flow speed. The average speed

during move-up is 8 fps (400 feet/50 seconds). The vehicle then stops for another 90 seconds, all

of which is delay time. No progress forward is made and the speed is zero during this period.

The vehicle then accelerates back to the free flow speed. A portion of this acceleration

occurs prior to the stop bar. The vehicle travels 200 feet in 20 seconds to reach the stop bar.

This 20 seconds of pre-stop bar acceleration time can be decomposed into 15 seconds of

acceleration delay and 5 seconds of time traveling at the free flow speed. The remainder of the

acceleration occurs after the stop bar. The vehicle travels 300 feet in 10 seconds to reach the end

of the delay measurement zone. This 10 seconds of post-stop bar acceleration time can be









decomposed into 2.5 seconds of acceleration delay and 7.5 seconds of time traveling at the free

flow speed. The average speed during acceleration is 16.7 fps: (200 feet + 300 feet)/(20

seconds+10 seconds).

Summarizing, the vehicle experience 262.5 seconds of delay which is composed of 35

seconds of deceleration delay, 170 seconds of stop delay, 40 seconds of move-up delay, and 17.5

seconds of acceleration delay. The vehicle spends an additional 107.5 seconds of time traveling

at the free flow speed: 25 seconds of which occurs during the deceleration period, 10 seconds of

which occurs during move up, and 12.5 seconds of which occurs during acceleration (the

remaining 60 seconds occurs at the start of the period under free-flow conditions).

Trajectory analysis gives a true picture of vehicular delay. The only component of delay

that is not represented by this single-vehicle diagram is interaction delay, which is not a part of

control delay.

In setting up our trajectory analysis, we would like to minimize the amount of interaction

delay that is captured by making the delay zone as short as possible. The longer we make the

delay zone, the more unwanted interaction delay between vehicles will occur. However,

attempts to reduce interaction action delay by reducing the length of the delay zone can lead to a

situation where significant amounts of deceleration or acceleration delay go unmeasured because

they occur outside the delay zone. Free flow speeds may not be accurately obtained as well if

the delay zone is too short. Consequently, a certain unknown amount of interaction delay will

almost always be included in our control delay measurement. Fortunately, under most

conditions of interest, interaction delay is relatively small in comparison to control delay and can

be ignored.









Cumulative Arrival/Departure Curve Example

Figure 6-2 tracks the vehicle previously shown in Figure 6-1, but this time using a typical

set of cumulative arrival and cumulative departure curves. As in Figure 6-1, Vehicle X stops at

the back of the queue (thus "arriving") at time point 120 and vehicle X eventually crosses the

stop bar (thus "departing") at time point 360. In a traditional cumulative arrival/cumulative

departure analysis, the type of analysis discussed in Appendix F of Chapter 16 of the Highway

Capacity Manual, control delay is equated to the area between the two curves.

There are three principal problems with this approach. The first and most obvious is that

none of the deceleration delay is accounted for in the area between the curves since, by

definition, all of the deceleration delay occurs before the vehicle arrives at the back of the queue.

Analyzing the movement of vehicles between the two cumulative curves, we can see the

second problem with this view of delay; it includes two time components that are not delay at

all: Free Speed Move-Up Time and Free Speed Acceleration Time Prior to the Stop Bar. Upon

arriving at time point 120 there are 24 other vehicles situated between the stop bar and Vehicle

X. Contrary to popular belief, the vertical distance of 24 vehicles is not necessarily the length of

the queue at time 120 because some of the vehicles may be in motion, either moving-up between

queues or accelerating towards the stop bar just prior to departure. This is an important

distinction because, as we observed during the trajectory analysis, the time spent by vehicles in

motion can only partially be construed as delay time. This leads us to conclude that the

horizontal distance covered by vehicle X is not the delay experienced by Vehicle X since it

includes the free flow speed portion of the move-up time as well as the free flow speed portion of

the pre-stop bar acceleration delay.









This means that, once more contrary to popular belief, the time spent by Vehicle X

between the cumulative arrival and cumulative departure curves is not its control delay, or even

its stopped delay, but is rather made-up of the following 5 components:

1.) Stopped delay
2.) Move-up delay
3.) Free Speed Move-Up Time
4.) Pre-Stop Bar Acceleration Delay
5.) Pre-Stop Free Speed Acceleration Time

To convert this time to pure delay, we must subtract out the two free flow speed time

components.

The third problem is that none of the post-stop bar acceleration delay is accounted for

in the area between the curves.

Reconciling the Difference Between Cumulative Curves and Trajectories

Continuing our example, Figure 6-3 shows the simplified cumulative arrival/cumulative

departure curve view of the world converted into a trajectory analysis. This view ignores

deceleration delay, as well as the portion of acceleration delay that occurs downstream of the

stop bar. This can be represented graphically by having vehicles approach the queue at free-flow

speed (Line A on the figure) and depart the stop line at free flow speed (Line B in the figure). In

the naive world of cumulative arrival and departure curves, the vehicle is added to the

cumulative arrival curve, and delay time begins, when the vehicle arrives at the back of the first

queue. The vehicle is then added to the cumulative departure curve, and delay time ends, when

the vehicle departs the stop bar. In this example, delay time begins at T = 120 seconds and ends

at T = 360 seconds, for a total delay value of 240 seconds, 22.5 seconds less than the 262.5

seconds of delay obtained through proper trajectory analysis.

It is possible to reconcile the 262.5 seconds of delay produced through proper trajectory

analysis and the 240 seconds of delay given by the cumulative curves. First, the deceleration









delay (35 seconds) is added to the cumulative curve delay (240 seconds) to obtain an adjusted

delay of 275 seconds. The portion of the acceleration delay that occurs downstream from the

stop bar (2.5 seconds) is also added in to obtain a new adjusted delay of 277.5 seconds. Finally,

as previously discussed, it is necessary to subtract out the free flow speed portion of the move-up

time (10 seconds) and the free flow speed portion of the pre-stop bar acceleration time (5

seconds) to obtain a final adjusted delay of 262.5 seconds, which now matches the delay from

the trajectory analysis. This last adjustment is required because the cumulative procedure fails to

account for the fact that not all of the time spent between arrival at the back of the queue and

departure from the stop bar is delay time, some of the time is being productively used to cover

the distance (600 feet in this case) between the back of the queue and the stop bar (600 feet/40

fps = 15 seconds).

The Highway Capacity Manual delay formula also contains a random element of delay that

is not directly reflected in the cumulative arrival and departure curves. Consequently, the

delay calculated via these formulas would be somewhat higher than 240 seconds. Unfortunately,

this delay element is added in a macroscopic fashion, which makes it impossible to translate

into the microscopic situation shown here.

Since some of the errors in the cumulative arrival/cumulative departure procedure result in

the control delay being underestimated (failure to include deceleration delay or acceleration

delay past the stop bar) while others result in the delay being overestimated (inclusion of free

flow speed move-up time and free flow speed pre-stop bar acceleration time), the errors may, to

a large degree, cancel each other out. For example, initial simulation testing has shown that,

under a rather wide range of over-saturated conditions, the free flow speed move-up time and

free flow speed pre-stop bar acceleration time make up about 10% of the control delay.









Coincidentally, the acceleration delay and post-stop bar deceleration delay also sum to about this

10% value, producing overall delay results that look fairly good. However, it should be

recognized that this counter-balancing effect is not guaranteed, and conditions can arise wherein

the errors become significant.

Summarizing, a comparison of the control delay obtained from trajectory analysis and that

obtained from cumulative arrival/departure curves shows that the cumulative curves omit certain

valid portions of the control delay, while including other portions of time that are not delay at all.

To guarantee a true measure of control delay, the delay values obtained from these curves must

be adjusted by adding in the deceleration delay and the post-stop bar acceleration delay, and by

subtracting out the free flow speed portion of both the move-up time and the pre-stop bar

acceleration time.

Figure 6-4 illustrates the various delay-related travel time components in relation to a set

of cumulative arrival and departure curves. In Chapter 5 it was shown how a reasonable set of

cumulative arrival and cumulative delay curves associated with minimum and maximum delay

could be constructed using minimum peak hour factors. Recognizing the relationships depicted

in Figure 6-4, stopped delay (Ds) can be obtained from these cumulative curves by converting

the area between the curves (Acc = Dcc + Tcc) as follows:

Acc = Dec + Tcc = Ds + DMU + DA1 + TMU + TA1 (162)

Let Ecc=Non-Stopped Delay Portion of Cumulative Curve Area=DMu + DA1 + TMU + TA1

Therefore: Acc = Ds + Ecc

Acc= Ds + Ds (Ecc/ Ds)

Acc = Ds (1 + Ecc/ Ds)

Ds = Acc [1/(1 + Ecc/Ds)] (163)









So, if the non-stopped delay elements associated with the area between the cumulative

curves equals 35% of the stopped delay (Ecc / Ds = 0.35), then 74% of the area between the

cumulative curves is associated with stopped delay: Ds = Acc [1/(1 + 0.35)] = 0.74 Acc

Calculating Trajectory-Based Delay Components for the BuckQ Examples

Trajectory-based delay was calculated for the twelve BuckQ data sets (four examples, each

with three random number seeds) using the BuckTRAJ program. The results of the trajectory

analysis can be used to identify the non-stopped delay portion of the cumulative arrival area

based on equations 162 and 163. Tables 6-1 through 6-4 summarize the resulting percentages

that are used to convert the overflow delay, as reflected in the area between the cumulative

curves, into stopped delay. The conversion percentages are fairly stable regardless of the volume

levels or period, ranging from 74% to 79% and averaging about 77%. It is quite interesting to

note that using the area between the cumulative arrival curves as a measure of stopped delay can

be expected to produce results that are more than 20% too high. The results of our BuckTRAJ

runs were also used to investigate the relationship between the area between the cumulative

arrival curves and control delay. As a review of Tables 6-1 through 6-4 indicates, the conversion

percentages are also relatively stable for this case, ranging from 93% to 103% and averaging

about 98%. Given these results, it appears that using the area between the cumulative curves to

approximate control delay is not unreasonable since the various delay errors inherent in using the

cumulative curves almost exactly compensate for one another.

Changing cumulative curve delay to stopped delay requires the application of these delay

conversion factors. To obtain the true factors for the situation at hand, a complete trajectory

analysis is required. However, if we have enough information to conduct a complete trajectory

analysis then we can determine the control delay directly and our entire delay estimation

procedure is not needed. Since we don't have this information, typical conversion factors (such









as the 77% factor evident from our four examples) will need to be applied to the upper and lower

delay bounds and there will be some inherent error in this conversion process.

Calculating Cumulative Curve Delay for the BuckQ Examples

The formulas contained in sections B, D and E of Chapter 5 were applied to our four

BuckQ examples in order to calculate the area between the cumulative arrival curve and the

cumulative departure curve (which equals the overflow delay plus the uniform delay). Each

random number replicate for the four examples was examined separately, resulting in twelve sets

of delay calculations. The random number results were not aggregated since it is the explicit

intent of our delay estimation procedure to detect variations in delay due to these random

variations. Nine of the twelve data sets are representative of a "standard" analysis situation

wherein the true arrival rate can only be determined for period 4. The formulas contained in

Section B of Chapter 5 pertain in this case. However, two of the three data sets associated with

the lowest volume arrival pattern (625_700_650_350vph) have visible queues during period 1,

allowing an arrival rate to be calculated for this period as well. The formulas contained in

Section D of Chapter 5 pertain to this case. The remaining 625_700_650_350vph data set has

visible queues during all but one period and the formulas contained in Section E of Chapter 5

pertain to this case.

The cumulative curve delay for the 4-period Upper Bound assuming a minimum PHF of

0.80 was calculated and the resulting maximum cumulative overflow delay values for each 15-

minute period are provided on the right side of Table 6-5. The cumulative curve delay for the 4-

period Lower Bound was also calculated and the resulting minimum cumulative overflow delay

values for each 15-minute period are provided on the left side of Table 6-5. The middle of

Table 6-5 provides similar values for the "estimated actual" delay. This is the delay obtained

from the cumulative curve formulas when the actual arrival rates are used.









The cumulative curve delay was calculated for the Upper Bound when queues are visible

throughout period 1. The resulting maximum cumulative overflow delay values for each 15-

minute period are provided on the right side of Table 6-6. Cumulative curve delay was also

calculated for the cumulative Lower Bound curve for the case where period 1 queues are visible.

The resulting minimum cumulative overflow delay values for each 15-minute period are

provided on the left side of Table 6-6. The middle of Table 6-6 provides similar values for the

estimated actual delay.

Also included in Table 6-6 are the delay results for the case when all but one period is

visible. In this case, it is not necessary to establish upper and lower bounds since there is a

single, known delay solution

The cumulative curves do not address random delay. Random delay is an additional

source of delay that stems from headway variations in the arriving traffic stream. When volume

on an intersection approach exceeds the capacity of the approach then residual queues form and

the effect of random arrivals on delay is minimal. In effect, the residual queues "absorb" the

randomness. However, when no residual queues exist, then this variation in vehicle arrivals

leads to the under-utilization of some cycles, as the green time is "starved" due to episodes of

infrequent arrivals, and to the over-utilization of other cycles as the green time is "swamped" by

closely spaced arrivals. This random component of delay is recognized by the Highway

Capacity Manual [4] and is included as an element in the HCM's d2 term.

To account for the effect of random delay, the random component of the HCM's d2 term is

included as part of the cumulative curve delay for a given 15-minute period whenever a residual

queue does not exist at the beginning of that 15-minute period. The presence of a residual queue

is determined by comparing the cumulative number of arrivals at the beginning of the period to









the cumulative number of departures at the beginning of the period. If this value is greater than

the overall thruput for the approach, then a residual queue exists and the random delay

component is calculated and added to the other components of the cumulative curve delay

(overflow delay and uniform delay). Otherwise the random delay is given a value of zero.

Thruput is calculated for each 15-minute period by dividing the number of signals cycles

that occur during the period into the 15-minute capacity of that period. For example, if the

hourly capacity for the first 15-minute period is 600 vph and the average cycle length is 120

seconds, then the average thruput for the first 15-minute period is: (600/4)/(3600/120) = 150 / 7.5

= 20 vehicles. The maximum of the four period thruputs is used as the overall thruput.

This random delay adjustment is not applied to the lower bound since the lower bound

represents a minimum condition and a lack of variation in the traffic stream can lead to situations

where the random delay component is very close to zero even though no residual queue exists.

In general:

Cumulative Curve Delay (Acc) = Overflow Delay (OD) + Uniform Delay (Du)
+ Random Delay (DR) I (164)

Where I = 0 if a residual queue exists at the start of the period, and 1 otherwise

This can be reflected in the d2 term of the Highway Capacity Manual control delay

equation by modifying the 8kIX component to be 8kIX(T-min(t,T))/T. The modified d2 term

thus becomes:

d2 = 900T[(X-1)+sqrt[(X- 1)2 + 8kIX(T-min(t,T)/cT2] (165)

It is interesting to note that, during over-saturated conditions, variations in cycle-to-cycle

vehicle arrival patterns have much less of an effect on delay than variations in cycle-to-cycle

capacity stemming from relative driver aggressiveness. The amount of start-up lost time

experienced during a given cycle and the degree to which motorists utilize the yellow and all red









change intervals as green time are the important random variables when over-saturated

conditions exist.

Bracketing the Stopped Delay Prediction Results

As discussed in section D of this chapter, the cumulative curve delay must be multiplied by

the conversion factors provided in Tables 6-1 through 6-4 to obtain stopped delay. Once this is

done, the minimum and maximum reasonable delay curves (the curves associated with the

minimum PHF lower and upper bounds) can be used to bracket our prediction results and create

an envelope of reasonable delay. If the prediction results fall outside this envelope then

abandoning the prediction process would be a reasonable course of action. When this occurs, the

prediction results can either be replaced by the "minimum percent error" estimate obtained from

the minimum and maximum delay curves as was described in Chapter 5 (see equation. 89), or a

"hybrid" prediction curve can be constructed that makes uses of the theoretical boundary

whenever the prediction curve lies outside of it.

To illustrate how this theoretical bracketing is used, a series of tables with embedded

cumulative delay figures have been developed based on our four examples. Table 6-7 addresses

volume pattern 700_725_625_350vph with a separate analysis provided for each of the three

random number sets. The "corrected" delay values provided in this table are cumulative curve

values that have been multiplied by the required conversion factor. The reference value against

which all delay results are evaluated is actual stopped delay as identified through simulation.

Also provided in the table is the BuckQ prediction as well as the "minimum percent error"

estimate. A review of the embedded figures shows that predicted delay (delay estimated by our

limited-information second-by-second procedure based on a power function) falls well within the

theoretical envelope for all three runs. The heavy dashed lines delineate the theoretical

constraint on the solution space using a minimum PHF of 0.80 while the dotted PHF Min %









Error line is the theoretical "best estimate". Table 6-8 provides a comparison of the average

results for the three runs. The prediction continues to fall well within the theoretical envelope

and underestimates the final cumulative stopped delay by only 2% whereas the "minimum

percent error" estimate obtained from the theoretical curves underestimates delay by 10%. The

"estimated actual" delay obtained from using the true arrival rates to construct the cumulative

curves deviates from simulation by 13%.

Tables 6-9 and 6-10 address volume pattern 700_700_700_350vph. A review of the

embedded figures in Table 6-9 shows that the prediction falls just outside the theoretical

envelope for two of the three runs. (A review of the cumulative arrival and departure curves for

these two cases reveals that the cumulative arrival curve is curvilinear between the end of period

3 and the start of period 4, violating the linear assumption. It is this violation that causes the

resulting delay to be slightly less than the minimum.) Table 6-10 provides a comparison of the

average results for the three runs. The prediction falls well within the theoretical envelope for

the average, overestimating the final cumulative stopped delay by just 5 percent. The "minimum

percent error" estimate obtained from the theoretical curves overestimates the delay by 13%.

The "estimated actual" delay obtained from using the true arrival rates to construct the

cumulative curves deviates from the simulation by 9%.

Tables 6-11 and 6-12 address the highest volume pattern: 725_700_700_350vph. A

review of the embedded figures in Table 6-11 shows that the prediction continues to falls within

the theoretical envelope for all three runs. Table 6-12 provides a comparison of the average

results for the three runs. Once again, the prediction falls well within the theoretical envelope,

this time underestimating the final cumulative stopped delay by 10 percent. The "minimum

percent error" estimate obtained from the theoretical curves underestimates the delay by 14%.









The "estimated actual" delay obtained from using the true arrival rates to construct the

cumulative curves deviates from the simulation by 5%.

Tables 6-13 and 6-14 deal with the lowest volume pattern: 625_700_650_350vph. As

discussed previously, the theoretical delay envelope is not pertinent when the arrival rate can be

determined for three of the four periods, which happens with the second run. For this run, the

theoretical curves are omitted. A review of the remaining two embedded figures in Table 6-13

shows that the prediction falls outside the theoretical envelope for one of the two runs. When

this occurs, the theoretical upper bound overrides our predicted values, producing a hybrid

solution that is much more accurate. Table 6-14 provides a comparison of the average results for

the three runs. The prediction falls inside the theoretical envelope when the results are averaged

and the resulting delay overestimation is only 14%. The "minimum percent error" estimate

obtained from the theoretical curves overestimates the delay by 2% while the "estimated actual"

delay obtained from using the true arrival rates to construct the cumulative curves deviates from

the simulation by less than 1/ of a percent.

Table 6-15 summarizes the delay prediction results for all four volume pattern examples

presented in the dissertation. A review of this table indicates that the hybrid procedure does the

best job of estimating actual stopped delay with the average percent error being only 11% after

the fourth period. Even the intermediate periods are predicted with reasonable accuracy, having

an average percent error of 14% or less. If averages are considered instead of individual runs,

the average percent error falls to 8% for the fourth period with a percent error of 12% or less for

any period. These values compare very favorably to the 65% error that would occur if our

prediction procedure was not used and only visual delay were taken into account.









The final results are relatively satisfying. Using limited information, our analysis

procedure does a reasonable job of predicting stopped delay under a variety of over-saturated

volume patterns and the improvement over directly measurable delay is dramatic. In addition,

the predictions tend to fall within theoretically justifiable limits.











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Figure 6-2. Cumulative arrival-departure curve example


,E P
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1-rhr CONTROL DELAY w 24a
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Figure 6-3. Trajectory conversion of cumulative curve example














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c CiOHOCLAYeu
T FREE SPEED TIM

t DEC.LEAMTON DELAY

Toa FAZE. sSEo *cEr;;AtiloN tirW

D# CLuJLiATVVE CLaVoeAV

Tet a CLMUULAV CLfV FIEE SPEED NlIE

M TCPFW DELAY

Dw MCE-UP DELAY

Cma FE-STP AFB ACCEIRWTI DELAY

Du w PGST-STP BAR ACCERTION DELAY

Tw FREiE 8EED MOW-FI' TIME

To FPRESTPS B AHREE SEED SPEED ACCELERATION ME

Ta POSTTOP BAR FREE S EEDATCEA'NJ tEM


Figure 6-4. Delay and travel time components.


FREE
Ff.0#
I9L Hg


.::


_m


-F-


r"










Table 6-1. Calculation of cumulative curve delay conversion factors, volume pattern 625 700 650 350vph
Period


Random Number Set 1
Stopped Delay
Queue Move-Up Delay
Free Speed Queue Move-Up Time
Accel/Decel Delay
Pre-Stop Bar Accel Delay as % of Accel/Decel Delay
Pre-Stop Bar Acceleration (Al) Delay
Free Speed Pre-Stop Bar Accel Time as % ofA1Delay
Free Speed Pre-Stop Bar Acceleration Time
Non-Stopped Delay Portion of Cumulative Curve Area
Non-Stopped Delay Portion as % of Stopped Delay


V.


variable Source
Ds BuckQ
DMu BuckQ
TMu BuckQ
BuckQ
BuckTRAJ
DAI Calculated
BuckTRAJ
TAI Calculated
Ecc Calculated
0C0/Ds Calculated


Factor That Converts Overflow Delay to Stopped Delay Fs Calculated 76% 78% 77% 78%

Period
Random Number Set 2 Variable Source 1 2 3 4
Stopped Delay Ds BuckQ 6424 15372 32476 38385
Queue Move-Up Delay DMu BuckQ 35 533 2731 3105
Free Speed Queue Move-Up Time TMu BuckQ 28 398 1591 1870
Accel/Decel Delay BuckQ 1821 4479 6973 7924
Pre-Stop Bar Accel Delay as % of Accel/Decel Delay BuckTRAJ 46% 46% 46% 47%
Pre-Stop Bar Acceleration (Al) Delay DA1 Calculated 838 2060 3208 3724
Free Speed Pre-Stop Bar Accel Time as % of A1Delay BuckTRAJ 70% 73% 73% 72%
Free Speed Pre-Stop Bar Acceleration Time TAl Calculated 586 1504 2342 2681
Non-Stopped Delay Portion of Cumulative Curve Area Ecc Calculated 1487 4495 9871 11381
Non-Stopped Delay Portion as % of Stopped Delay Ecc/Ds Calculated 23% 29% 30% 30%


Factor That Converts Overflow Delay to Stopped Delay Fs Calculated 81% 77% 77% 77%


1
8223
399
268
2461
45%
1107
72%
797
2572
31%


2
19035
932
669
4824
40%
1930
88%
1698
5229
27%


3
33144
2538
1732
7268
42%
3053
86%
2625
9948
30%


4
38126
2725
1842
8285
43%
3563
82%
2921
11051
29%


V










Table 6-1. Continued
Period


Random Number Set 3
Stopped Delay
Queue Move-Up Delay
Free Speed Queue Move-Up Time
Accel/Decel Delay
Pre-Stop Bar Accel Delay as % of Accel/Decel Delay
Pre-Stop Bar Acceleration (Al) Delay
Free Speed Pre-Stop Bar Accel Time as % ofA1Delay
Free Speed Pre-Stop Bar Acceleration Time
Non-Stopped Delay Portion of Cumulative Curve Area
Non-Stopped Delay Portion as % of Stopped Delay


V.


variable Source
Ds BuckQ
DMu BuckQ
TMU BuckQ
BuckQ
BuckTRAJ
DAl Calculated
BuckTRAJ
TAl Calculated
Ecc Calculated
0C0/Ds Calculated


Factor That Converts Overflow Delay to Stopped Delay Fs Calculated 79% 78% 77% 77%


All Values are Cumulative


1
9858
435
331
2513
39%
980
84%
823
2569
26%


2
22344
1560
1076
4767
43%
2050
84%
1722
6408
29%


3
39242
3691
2300
7222
45%
3250
79%
2567
11808
30%


4
45610
4327
2617
8326
46%
3830
77%
2949
13723
30%


V










Table 6-2. Calculation of cumulative curve delay conversion factors, volume pattern 700 725 625 350vph
Period


Random Number Set 1
Stopped Delay
Queue Move-Up Delay
Free Speed Queue Move-Up Time
Accel/Decel Delay
Pre-Stop Bar Accel Delay as % of Accel/Decel Delay
Pre-Stop Bar Acceleration (Al) Delay
Free Speed Pre-Stop Bar Accel Time as % ofA1Delay
Free Speed Pre-Stop Bar Acceleration Time
Non-Stopped Delay Portion of Cumulative Curve Area
Non-Stopped Delay Portion as % of Stopped Delay


Factor That Converts Overflow Delay to Stopped Delay


Random Number Set 2
Stopped Delay
Queue Move-Up Delay
Free Speed Queue Move-Up Time
Accel/Decel Delay
Pre-Stop Bar Accel Delay as % of Accel/Decel Delay
Pre-Stop Bar Acceleration (Al) Delay
Free Speed Pre-Stop Bar Accel Time as % ofA1Delay
Free Speed Pre-Stop Bar Acceleration Time
Non-Stopped Delay Portion of Cumulative Curve Area
Non-Stopped Delay Portion as % of Stopped Delay


Factor That Converts Overflow Delay to Stopped Delay


V


E


variable Source
Ds BuckQ
DMu BuckQ
TMu BuckQ
BuckQ
BuckTRAJ
DAI Calculated
BuckTRAJ
TAI Calculated
Ecc Calculated
cc/Ds Calculated


1
12465
926
644
3394
38%
1290
71%
916
3775
30%


2
32341
3472
2405
6910
35%
2419
89%
2152
10448
32%


3
59334
7433
4888
9114
38%
3463
86%
2978
18763
32%


Fs Calculated 77% 76% 76% 76%


Period


V


E


variable Source
Ds BuckQ
DMu BuckQ
TMU BuckQ
BuckQ
BuckTRAJ
DAI Calculated
BuckTRAJ
TAI Calculated
Ecc Calculated
cc/Ds Calculated


1
15661
1187
900
3119
37%
1154
78%
900
4141
26%


2
38310
4184
2837
6490
37%
2401
80%
1921
11343
30%


3
69620
11605
6156
9438
38%
3586
75%
2690
24037
35%


Fs Calculated 79% 77% 74% 75%


4
68622
8807
5544
9934
39%
3874
84%
3254
21480
31%


4
83364
13341
7260
10307
40%
4123
75%
3092
27816
33%











Table 6-2. Continued


Period


Random Number Set 3
Stopped Delay
Queue Move-Up Delay
Free Speed Queue Move-Up Time
Accel/Decel Delay
Pre-Stop Bar Accel Delay as % of Accel/Decel Delay
Pre-Stop Bar Acceleration (Al) Delay
Free Speed Pre-Stop Bar Accel Time as % ofA1Delay
Free Speed Pre-Stop Bar Acceleration Time
Non-Stopped Delay Portion of Cumulative Curve Area
Non-Stopped Delay Portion as % of Stopped Delay


Factor That Converts Overflow Delay to Stopped Delay


V


E


variable Source
Ds BuckQ
DMu BuckQ
TMu BuckQ
BuckQ
BuckTRAJ
DAl Calculated
BuckTRAJ
TAl Calculated
Ecc Calculated
cc/Ds Calculated


1
13743
1089
775
3145
33%
1038
88%
913
3815
28%


Fs Calculated 78%


2
38723
4886
2997
6416
36%
2310
82%
1894
12087
31%


3
73079
10666
6415
8969
37%
3319
83%
2754
23154
32%


4
85601
12720
7458
10050
40%
4020
79%
3176
27374
32%


76% 76% 76%


All Values are Cumulative










Table 6-3. Calculation of cumulative curve delay conversion factors, volume pattern 700 700 700 350vph
Period


Random Number Set 1
Stopped Delay
Queue Move-Up Delay
Free Speed Queue Move-Up Time
Accel/Decel Delay
Pre-Stop Bar Accel Delay as % of Accel/Decel Delay
Pre-Stop Bar Acceleration (Al) Delay
Free Speed Pre-Stop Bar Accel Time as % ofA1Delay
Free Speed Pre-Stop Bar Acceleration Time
Non-Stopped Delay Portion of Cumulative Curve Area
Non-Stopped Delay Portion as % of Stopped Delay


V.


variable Source
Ds BuckQ
DMu BuckQ
TMu BuckQ
BuckQ
BuckTRAJ
DAl Calculated
BuckTRAJ
TAl Calculated
Ecc Calculated
0C0/Ds Calculated


Factor That Converts Overflow Delay to Stopped Delay Fs Calculated 77% 76% 76% 76%

Period
Random Number Set 2 Variable Source 1 2 3 4
Stopped Delay Ds BuckQ 15661 36773 69423 88873
Queue Move-Up Delay DMu BuckQ 1185 4090 11561 14343
Free Speed Queue Move-Up Time TMu BuckQ 902 2781 6196 7938
Accel/Decel Delay BuckQ 3092 6138 9307 10171
Pre-Stop Bar Accel Delay as % of Accel/Decel Delay BuckTRAJ 37% 39% 39% 41%
Pre-Stop Bar Acceleration (Al) Delay DA1 Calculated 1144 2394 3630 4170
Free Speed Pre-Stop Bar Accel Time as % ofA1Delay BuckTRAJ 78% 80% 76% 76%
Free Speed Pre-Stop Bar Acceleration Time TAl Calculated 892 1915 2759 3169
Non-Stopped Delay Portion of Cumulative Curve Area Ecc Calculated 4123 11180 24145 29620
Non-Stopped Delay Portion as % of Stopped Delay Ecc/Ds Calculated 26% 30% 35% 33%


Factor That Converts Overflow Delay to Stopped Delay Fs Calculated 79% 77% 74% 75%


1
12465
928
642
3420
38%
1300
71%
923
3792
30%


2
29923
3042
2132
6568
37%
2430
86%
2090
9694
32%


3
57995
7153
4699
9013
38%
3425
85%
2911
18188
31%


4
72168
9787
5925
10084
40%
4034
81%
3267
23013
32%


V










Table 6-3. Continued
Period


Random Number Set 3
Stopped Delay
Queue Move-Up Delay
Free Speed Queue Move-Up Time
Accel/Decel Delay
Pre-Stop Bar Accel Delay as % of Accel/Decel Delay
Pre-Stop Bar Acceleration (Al) Delay
Free Speed Pre-Stop Bar Accel Time as % ofA1Delay
Free Speed Pre-Stop Bar Acceleration Time
Non-Stopped Delay Portion of Cumulative Curve Area
Non-Stopped Delay Portion as % of Stopped Delay


V.


variable Source
Ds BuckQ
DMu BuckQ
TMu BuckQ
BuckQ
BuckTRAJ
DAl Calculated
BuckTRAJ
TAl Calculated
Ecc Calculated
0C0/Ds Calculated


Factor That Converts Overflow Delay to Stopped Delay Fs Calculated 78% 76% 76% 75%


All Values are Cumulative


1
13743
1085
779
3103
33%
1024
88%
901
3789
28%


2
36400
4644
2889
5920
37%
2190
81%
1774
11498
32%


3
71981
10854
6545
8611
38%
3272
82%
2683
23354
32%


4
89605
14263
8195
10066
42%
4228
78%
3298
29983
33%


V










Table 6-4. Calculation of cumulative curve delay conversion factors, volume pattern 725 700 700 350vph
Period


Random Number Set 1
Stopped Delay
Queue Move-Up Delay
Free Speed Queue Move-Up Time
Accel/Decel Delay
Pre-Stop Bar Accel Delay as % of Accel/Decel Delay
Pre-Stop Bar Acceleration (Al) Delay
Free Speed Pre-Stop Bar Accel Time as % ofA1Delay
Free Speed Pre-Stop Bar Acceleration Time
Non-Stopped Delay Portion of Cumulative Curve Area
Non-Stopped Delay Portion as % of Stopped Delay


V.


variable Source
Ds BuckQ
DMu BuckQ
TMu BuckQ
BuckQ
BuckTRAJ
DAl Calculated
BuckTRAJ
TAl Calculated
Ecc Calculated
0C0/Ds Calculated


Factor That Converts Overflow Delay to Stopped Delay Fs Calculated 77% 76% 75% 75%

Period
Random Number Set 2 Variable Source 1 2 3 4
Stopped Delay Ds BuckQ 19909 48528 89649 116151
Queue Move-Up Delay DMu BuckQ 2117 7524 17590 22139
Free Speed Queue Move-Up Time TMu BuckQ 1515 4341 8819 11428
Accel/Decel Delay BuckQ 2837 5827 8079 8568
Pre-Stop Bar Accel Delay as % of Accel/Decel Delay BuckTRAJ 38% 39% 40% 44%
Pre-Stop Bar Acceleration (Al) Delay DA1 Calculated 1078 2273 3232 3770
Free Speed Pre-Stop Bar Accel Time as % ofA1Delay BuckTRAJ 88% 80% 77% 77%
Free Speed Pre-Stop Bar Acceleration Time TAl Calculated 949 1818 2488 2903
Non-Stopped Delay Portion of Cumulative Curve Area Ecc Calculated 5659 15956 32129 40240
Non-Stopped Delay Portion as % of Stopped Delay Ecc/Ds Calculated 28% 33% 36% 35%


Factor That Converts Overflow Delay to Stopped Delay Fs Calculated 78% 75% 74% 74%


1
23943
3060
2083
3623
32%
1159
86%
997
7299
30%


2
55507
8310
5259
6277
35%
2197
92%
2021
17787
32%


3
96463
16446
9856
8591
37%
3179
88%
2797
32278
33%


4
121786
21141
12256
10068
41%
4128
83%
3426
40951
34%


V










Table 6-4. Continued


Random Number Set 3
Stopped Delay
Queue Move-Up Delay
Free Speed Queue Move-Up Time
Accel/Decel Delay
Pre-Stop Bar Accel Delay as % of Accel/Decel Delay
Pre-Stop Bar Acceleration (Al) Delay
Free Speed Pre-Stop Bar Accel Time as % ofA1Delay
Free Speed Pre-Stop Bar Acceleration Time
Non-Stopped Delay Portion of Cumulative Curve Area
Non-Stopped Delay Portion as % of Stopped Delay


V.


variable Source
Ds BuckQ
DMu BuckQ
TMu BuckQ
BuckQ
BuckTRAJ
DAl Calculated
BuckTRAJ
TAl Calculated
Ecc Calculated
0C0/Ds Calculated


Factor That Converts Overflow Delay to Stopped Delay Fs Calculated 76% 75% 75% 74%


All Values are Cumulative


1
20321
2470
1663
3272
36%
1178
92%
1084
6395
31%


2
5212
809
492
600
390/
234:
840/
196
1732
330/


Period
3
!4 96875
5 16789
3 9891
5 8309
S 41%
2 3407
S 83%
7 2828
27 32914
S 34%


4
122534
22567
12556
9584
44%
4217
79%
3331
42671
35%


V


Table 6-4. Continued











Table 6-5. Cumulative curve delay for standard 4-period case
0.80 PHF LOWER BOUND ESTIMATED ACTUAL 0.80 PHF UPPER BOUND
Random Period Period Period
Number
Volume Pattern Set 1 2 3 4 1 2 3 4 1 2 3 4


700 725 625_350vph









700_700_700_350vph









725 700 700_350vph


11753 29234 56730

12034 29541 61707

11935 29204 56654


11753 37021 83274

12034 38978 92930

11935 29204 60601




11829 39043 90783

12169 36563 86182

11935 32588 72744


68776 19463 58276

81296 19862 54020

70758 19807 57213




107397 19463 55002

125614 19862 49520

79204 19807 54963




121695 29297 81598

118668 21714 61722

100593 25848 73605


99671

96973

104156




94147

93373

105506




144363

120097

135397


112335 23701 70376

116386 22508 66509

124531 21796 66487




111382 25291 77028

119362 23276 69808

132882 23306 73060




178994 26505 83981

156275 23279 73553

171951 24969 80207


118010

117687

114469


138979

135023

128363




153946

140755

144540


130107

137276

128573


163129

167708

146966




184858

173241

172389










Table 6-6. Cumulative curve delay with multiple visible periods
0.80 PHF LOWER BOUND ESTIMATED ACTUAL 0.80 PHF UPPER BOUND
Random Period Period Period
Number
Volume Pattern Set 1 2 3 4 1 2 3 4 1 2 3 4

1 6227 18342 37939 44698 10649 24184 47491 53956 11793 28562 49797 55113

625_700_650_350vph 2 9718 24633 39689 49616

3 6583 18588 40481 48428 12078 25698 46675 54024 12288 31676 63073 71020









Table 6-7. Stopped delay prediction results for 700_725_625_350vph volume pattern
120 second cycle
Min PHF= 0.80


Random Number Set 1


120000 -Corrected Maximum

-Corrected Minimum


100000


- Simulation


Cumulative Stopped Delay


Period: 0 1


S*g/L Capacity:
Arrivals at BOQ:
Actual PHF:


629
716
0.82


Simulation 0 12465


2 3 4


659 655 639
740 628 348



32341 59334 68622


....... PHF Min % Error Pedicted
---. BuckQ Predicted
Corrected Actual

.


0 1 Period 2


to O


3 4


OD tp Ds Conver


Corrected


sion % 77%
Actual 19463
Actual 0 14987
/% Error 20.2%


Maximum 0
Corrected Maximum 0
Minimum 0
Corrected Minimum 0

PHF Min % Error Pedicted 0
% Error

BuckQ Predicted 0
% Error


23701
18250
11753
9050

12100
-3%

13796
11%


80000


Q 60000
C.
Q.
0
" 40000


20000


0


76%
58276
44290
36.9%

70376
53486
29234
22218

31394
-3%

36748
14%


76%
99671
75750
27.7%

118010
89687
56730
43115

58235
-2%

63544
7%


76%
112335
85374
24%

130107
98882
68776
52270

68389
0%

77325
13%








Table 6-7. Continued
Random Number Set 2


S*g/L Capacity:
Arrivals at BOQ:
Actual PHF:


648
720
0.84


Simulation 0 15661


660 609 604
728 624 368



38310 69620 83364


OD tp Ds Conversion %
Actual


Cor


Cor


Corrected Actual 0
% Error

Maximum 0
rected Maximum 0
Minimum 0
rrected Minimum 0


PHF Min % Error Pedicted 0
% Error

BuckQ Predicted 0
% Error


79%
19862
15691
0.2%

22508
17781
12034
9507

12390
-21%

17080
9%


77%
54020
41595
8.6%

66509
51212
29541
22747

31501
-18%

39625
3%


74%
96973
71760
3.1%

117687
87088
61707
45663

59912
-14%

61400
-12%


75%
116386
87290
5%

137276
102957
81296
60972

76588
-8%

77713
-7%








Table 6-7. Continued
Random Number Set 3


120000 -Corrected Maximum
-Corrected Minimum


100000


-Simulation


S*g/L Capacity:
Arrivals at BOQ:
Actual PHF:


Simulation 0 13743


635 645 561
732 632 364



38723 73079 85601


OD tp Ds Conver


Corrected


sion % 78%
Actual 19807
Actual 0 15449
% Error 12.4%


Maximum 0
Corrected Maximum 0
Minimum 0
Corrected Minimum 0


PHF Min % Error Pedicted 0
% Error


BuckQ Predicted 0
% Error


21796
17001
11935
9309


12031
-12%


16567
21%


80000


60000


40000


20000


0


76%
57213
43482
12.3%


66487
50530
29204
22195


30843
-20%


38141
-2%


76%
104156
79158
8.3%


114469
86996
56654
43057


57604
-21%


62987
-14%


76%
124531
94644
11%


128573
97715
70758
53776


69374
-19%


78925
-8%


0 1 Period 2 3 4









Table 6-8. Average stopped delay prediction results for 700_725_625_350vph volume pattern
120 second cycle
Min PHF = 0.80 P


- -Corrected Maximum
- -Corrected Minimum

Simulation
....... PHF Min % Error Pedicted


3 4


0 1 Pe 2
Period


Cumulative Stopped Delay


period: 0 1


S*g/L Capacity:
Arrivals at BOQ:
Actual PHF:


641
724
0.83


Simulation 0 13956


OD tp Ds Conversion %
Actual
Corrected Actual 0
% Error


Maximum 0
Corrected Maximum 0
Minimum 0
Corrected Minimum 0


PHF Min % Error Pedicted
% Error


BuckQ Predicted
% Error


2 3 4


651 636 601
733 628 360


36458


78% 76%
19711 56503
15374 43131
10.2% 18.3%


22669
17681
11907
9288


0 12173
-13%


0 15814
13%


67791
51747
29326
22386


31246
-14%


38171
5%


67344 79196


75%
100266
75534
12.2%


116722
87930
58364
43967


58584
-13%


62644
-7%


76%
117751
89098
13%


131985
99869
73610
55698


71450
-10%


77988
-2%


120000


100000



80000


60000



40000


20000



0









Table 6-9. Stopped delay prediction results for 700_700_700_350vph volume pattern
120 second cycle
Min PHF= 0.80


Cumulative Stopped Delay
Period: 0 1 2 3 4


Random Number Set 1


S*g/L Capacity:
Arrivals at BOQ:
Actual PHF:


Simulation 0 12465


659 655 596
704 680 380



29923 57995 72168


OD tp Ds Conver


Corrected


sion % 77%
Actual 19463
Actual 0 14987
% Error 20.2%


Maximum
Corrected Maximum
Minimum
Corrected Minimum


PHF Min% Error Pedicted 0
% Error


BuckQ Predicted 0
% Error


25291
19474
11753
9050


12357
-1%

13813
11%


76%
55002
41802
39.7%

77028
58541
37021
28136


38006
27%

39068
31%


76%
94147
71551
23.4%

138979
105624
83274
63288


79151
36%

69887
21%


76%
111382
84651
17%

163129
123978
107397
81622


98437
36%

91541
27%








Table 6-9. Continued
Random Number Set 2


S*g/L Capacity:
Arrivals at BOQ:
Actual PHF:


648
736
0.84


Simulation 0 15661


660 609 612
688 712 344



36773 69423 88873


OD tp Ds Conversion %
Actual


Cor

Cor


Corrected Actual 0
% Error

Maximum 0
rected Maximum 0
Minimum 0
rrected Minimum 0


PHF Min % Error Pedicted 0
% Error

BuckQ Predicted 0
% Error


79%
19862
15691
0.2%

23276
18388
12034
9507

12534
-20%

17698
13%


77%
49520
38130
3.7%

69808
53752
38978
30013

38519
5%

42183
15%


74%
93373
69096
-0.5%

135023
99917
92930
68768

81467
17%

66738
-4%


75%
119362
89522
1%

167708
125781
125614
94211

107730
21%

90042
1%








Table 6-9. Continued
Random Number Set 3


140000 -Corrected Maximum
-Corrected Minimum
120000 Simulation


S*g/L Capacity:
Arrivals at BOQ:
Actual PHF:


642
736
0.85


Simulation 0 13743


635 645 613
712 704 360



36400 71981 89605


....... PHF Min % Error Pedicted


100000


80000


60000


40000


20000


0


OD tp Ds Conver


Corrected


sion % 78%
Actual 19807
Actual 0 15449
% Error 12.4%


Maximum 0
Corrected Maximum 0
Minimum 0
Corrected Minimum 0


PHF Min % Error Pedicted 0
% Error


BuckQ Predicted 0
% Error


23306
18179
11935
9309


12313
-10%


16586
21%


76%
54963
41772
14.8%


73060
55525
29204
22195


31713
-13%


37329
3%


76%
105506
80184
11.4%


128363
97556
60601
46057


62573
-13%


62358
-13%


75%
132882
99662
11%


146966
110224
79204
59403


77201
-14%


82487
-8%


0 1 Period 2 3 4









Table 6-10. Average stopped delay prediction results for 700_700_700_350vph volume pattern
120 second cycle
Min PHF= 0.80 Pei


- -Corrected Maximum
- -Corrected Minimum
Simulation
....... PHF Min % Error Pedicted
- BuckQ Predicted


- Corrected Actual


Cumulative Stopped Delay


riod: 0 1


S*g/L Capacity:
Arrivals at BOQ:
Actual PHF:


640
729
0.85


Simulation 0 13956


OD tp Ds Conversion % 78%
Actual 19711
Corrected Actual 0 15374
% Error 10.2%

Maximum 0 23958
Corrected Maximum 0 18687
Minimum 0 11907
Corrected Minimum 0 9288


PHF Min % Error Pedicted 0 12401
% Error -11%


BuckQ Predicted 0 16032
% Error 15%


2 3 4

651 636 607
701 699 361


34365 66466 83549


76%
53162
40580
18.1%

73299
55951
35068
26768

36079
5%

39527
15%


75%
97675
73582
10.7%

134122
101038
78935
59464

74397
12%


66328
0%


75%
121209
91311
9%

159267
119982
104072
78401

94456
13%

88023
5%


A
A


140000


120000


100000


r 80000


- 60000
0
o
4-1

40000


20000


0


0 1 Period 2 3 4










Table 6-11 Stopped delay prediction results for 725_700_700_350vph volume pattern

120 second cycle
Min PHF = 0.80


Cumulative Stopped Delay
Period: 0 1 2 3 4


Random Number Set 1


S*g/L Capacity:
Arrivals at BOQ:
Actual PHF:


Simulation 0

OD tp Ds Conversion %
Actual
Corrected Actual 0
% Error


Maximum 0
Corrected Maximum 0
Minimum 0
Corrected Minimum 0


PHF Min % Error Pedicted 0
% Error


BuckQ Predicted 0
% Error


635
788
0.80


23943


77%
29297
22559
-5.8%


26505
20409
11829
9108


12595
-47%


19817
-17%


648 660 616
692 708 340


55507

76%
81598
62014
11.7%


83981
63826
39043
29673


40511
-27%


49338
-11%


96463

75%
144363
108272
12.2%


153946
115460
90783
68087


85660
-11%


85163
-12%


121786


75%
178994
134245
10%


184858
138644
121695
91271


110077
-10%


120764
-1%








Continued
Random Number Set 2


S*g/L Capacity:
Arrivals at BOQ:
Actual PHF:


662
768
0.83


638 621 644
724 700 356


Simulation 0 19909


OD tp Ds Conversion %
Actual
Corrected Actual 0
% Error

Maximum 0
Corrected Maximum 0
Minimum 0
Corrected Minimum 0

PHF Min % Error Pedicted 0
% Error

BuckQ Predicted 0
% Error


78%
21714
16937
-14.9%

23279
18158
12169
9492

12467
-37%

20074
1%


48528 89649 116151


75%
61722
46292
-4.6%

73553
55165
36563
27422

36634
-25%

45086
-7%


74%
120097
88872
-0.9%

140755
104159
86182
63775

79111
-12%

71435
-20%


74%
156275
115643
0%

173241
128198
118668
87814

104231
-10%

101389
-13%


Table 6-11.








Continued
Random Number Set 3


- -Corrected Maximum


1 2 3
Period


S*g/L Capacity:
Arrivals at BOQ:
Actual PHF:


Simulation 0 20321


140000


120000


100000


80000


60000


40000


20000


0


76%
25848
19645
-3.3%

24969
18977
11935
9071


12274
-40%


17366
-15%


635 645 610
704 700 364


52124 96875 122534


75%
73605
55203
5.9%


80207
60155
32588
24441


34759
-33%


41246
-21%


75%
135397
101548
4.8%


144540
108405
72744
54558


72585
-25%


70799
-27%


74%
171951
127243
4%


172389
127568
100593
74439


94017
-23%


101785
-17%


Table 6-11.


OD tp Ds Conversion %
Actual
Corrected Actual 0
% Error


Maximum 0
Corrected Maximum 0
Minimum 0
Corrected Minimum 0


PHF Min % Error Pedicted 0
% Error


BuckQ Predicted 0
% Error









Table 6-12. Average stopped delay prediction results for 725_700_700_350vph volume pattern
120 second cycle
Min PHF = 0.80 Per


- -Corrected Maximum
- -Corrected Minimum


3 4


0 1 Pe 2
Period


Cumulative Stopped Delay


iod: 0 1


S*g/L Capacity:
Arrivals at BOQ:
Actual PHF:


Simulation 0 21391


OD tp Ds Conversion %
Actual
Corrected Actual 0
% Error


77%
25620
19727
-7.8%


140000


120000


100000


80000


60000


40000


20000


0


2 3 4


640 642 623
707 703 353


52053


75%
72308
54472
4.6%


79247
59699
36065
27169


37302
-28%


45223
-13%


94329 120157


75%
133286
99520
5.5%


146414
109322
83236
62150


79119
-16%


75799
-20%


74%
169073
125678
5%


176829
131443
113652
84481


102775
-14%


107979
-10%


Maximum 0 24918
Corrected Maximum 0 19187
Minimum 0 11978
Corrected Minimum 0 9223


PHF Min % Error Pedicted 0 12445
% Error -42%


BuckQ Predicted 0 19086
% Error -11%









Table 6-13 Stopped delay prediction results for 625_700_650_350vph volume pattern
120 second cycle
Min PHF= 0.80


Cumulative Stopped Delay
Period: 0 1 2 3 4


Random Number Set 1


S*g/L Capacity:
Arrivals at BOQ:
Actual PHF:

Simulation 0


612
676
0.86

8223


669 655 516
688 652 360



19035 33144 38126


OD tp Ds Conver


Corrected


sion % 76%
Actual 10649
Actual 0 8093
% Error -1.6%


Maximum 0
Corrected Maximum 0
Minimum 0
Corrected Minimum 0

PHF Min % Error Pedicted 0
% Error

BuckQ Predicted 0
% Error


11793
8963
6227
4733

6194
-25%

10115
23%


78%
24184
18864
-0.9%

28562
22279
18342
14307

17424
-8%

23332
23%


77%
47491
36568
10.3%

49797
38343
37939
29213

33161
0%

42062
27%


78%
53956
42086
10%

55113
42988
44698
34865

38503
1%

48662
28%








Table 6-13. Continued
Random Number Set 2


S*g/L Capacity:
Arrivals at BOQ:
Actual PHF:

Simulation 0


665
628
0.87

6424


643 623 497
676 672 364



15372 32476 38385


OD tp Ds Conver


Corrected


sion % 81%
Actual 9718
Actual 0 7872
% Error 22.5%


BuckQ Predicted 0
% Error


6342
-1%


77%
24633
18967
23.4%

16427
7%


77%
39689
30561
-5.9%

33318
3%


77%
49616
38204
0%

39729
4%









Table 6-13. Continued
Random Number Set 3


60000


- -Corrected Maximum


-Corrected Minimum
50000 Simulation
....... PHF Min % Error Ped

40000 BuckQ Predicted


30000


20000


10000


0


0 w


S*g/L Capacity:
Arrivals at BOQ:
Actual PHF:

Simulation 0

OD tp Ds Conversion %
Actual
Corrected Actual 0
% Error

Maximum 0
Corrected Maximum 0
Minimum 0
Corrected Minimum 0

PHF Min % Error Pedicted 0
% Error

BuckQ Predicted 0
% Error


650 641 524
668 688 344



22344 39242 45610


643
672
0.86

9858

79%
12078
9542
-3.2%

12288
9708
6583
5200

6773
-31%

10197
3%


77%
46675
35940
-8.4%


63073
48566
40481
31170

37971
-3%

42741
9%


77%
54024
41598
-9%

71020
54685
48428
37289

44342
-3%

50672
11%


0 1 2 3 4
Period


78%
25698
20044
-10.3%

31676
24708
18588
14499

18274
-18%

24353
9%











Table 6-14. Average stopped delay prediction results for 625_700_650_350vph volume pattern
120 second cycle


Min PHF = 0.80


Period: 0


Cumulative Stopped Delay
1 2 3 4


60000 -Corrected Maximum


- -Corrected Minimum

- Simulation
....... PHF Min % Error Pedicted

- BuckQ Predicted


0 1 Period
Period


0 0,


2 3 4


S*g/L Capacity:
Arrivals at BOQ:
Actual PHF:


640
659
0.87


Simulation 0 8168


OD tp Ds Conve


Correcte


version % 79%
Actual 10815
dActual 0 8508
% Error 4.2%


Maximum
Corrected Maximum
Minimum
Corrected Minimum


12041
9472
6405
5039


PHF Min % Error Pedicted 0 6484
% Error -21%


50000



40000



30000
0
a.
2 20000



10000



0


8885
9%


654 640 512
677 671 356


18917 34954 40707


78%
24838
19291
2.0%


30119
23393
18465
14341


17849
-6%


21371
13%


77%
44618
34356
-1.7%


56435
43455
39210
30192


35566
2%


39374
13%


77%
52532
40625
0%


63066
48771
46563
36009


41422
2%


46354
14%


BuckQ Predicted 0
% Error












Table 6-15. Prediction comparison
Random uPe
NLrtrer
Volume Ptm Sdet


1



625_700_50_350vph 2



3




1



7CO_725_55_535vph 2



3




1



700_70_700_350vph 2



3


725_U_70_3v 1 2


725 700 ,OC35Dvph 2


-37% -25% -12% -10/O % -1/o -78%
1% -7% -_o -77/ -11% -13% -20o -10 78
-9% -7%/ -20% -13% -11% -13% -2 -10/o


PFF Mn % Error 22% 18%
Predicted 12% 12%
Hybrid 13% 12%


PHF Mn% Error 4
Predicted 5
Hybrid 6


14% 13/0 22%
14% 120%o 650/0 12%
14% 11 1 12/o

Total
6 4 18 2
5 6 23 3
6 6 m 3


13% 11% 100/o
12% 10PA0 % 8S I 65%
12% 10% L 8f% J


1 7
3 p70
3 10 J


Average
ofAbsolte
Percent Ofference


Frequercy
f Best
Prediction









CHAPTER 7
PERIOD ISSUES DURING OVER-SATURATED FLOW

This chapter describes deficiencies in the Highway Capacity Manual that can lead to

incorrect delay values during over-saturated conditions. These deficiencies must be corrected

before meaningful comparisons can be made to the predicted delay from our analysis procedure.

Methods for correcting these deficiencies are presented (Objective 6).

Correctly identifying the size of the residual queue is very important for accurately

calculating delay during over-saturated conditions. If the size of the residual queue is not

correctly identified at the start of each 15-minute period then the resulting delay calculations for

that period can be off by a substantial amount. The value of the d3 delay term is directly tied to

the length of the residual queue while the correct application of the random portion of the d2

delay term depends on whether or not a residual queue is present. As the following discussion

demonstrates, a cycle-by-cycle approach is required to accurately identify the residual queue, and

use of the period approach contained in the Highway Capacity Manual [4] would not provide the

desired result.

Simplified Example of Cycle-Period Issues in Calculating d3

Formula F16-6 in the Highway Capacity Manual is touted as yielding the residual queue.

The formula is:

Qb,i+1=max[0,Qb,i+ciT(Xi-1)]

Assuming that, at the start of the hour, there is no residual queue (Qb,i = 0) then, if the

volume is greater than the capacity for the first 15 minutes, this equation becomes:

Qb2= cT(X-1)

Recognizing that X = v/c, we can further simplify this equation to:

Qb2=cT(v/c-1)









Qb2= Tv -cT

Where v = volume in vph and c = capacity in vph. If we call V the volume for the first 15-

minute period then V = Tv (or v = V/T) and if we call C the capacity for the first 15-minute

period, then C = Tc (or c = C/T). The equation then simply reduces to:

Qb2= T(V/T) (C/T)T

Qb2= V C (166)

So, the residual queue for the start of period 2 equals the difference between the arriving

vehicles (15-minute volume) and the departing vehicles (15-minute capacity) for period 1.

However, this is not the correct procedure for determining the residual queue and Tables 7-

1 and 7-2 show why. The simple example illustrated in these tables has just two 15-minute

periods. The first period starts with no residual queue and the last period ends with little or no

residual queue. However, a sizeable residual queue does exist at the end of the first 15-minute

period and a value for d3 is calculated for the second 15-minute period based on this value.

Uniform arrivals are used to keep the example simple (and to avoid having to deal with the d2

term) but the results can be generalized to any arrival situation. Table 7-1 provides the second-

by-second cumulative arrival, cumulative departure and queue length information while the

resulting residual queue, thruput, and associated d3 values are provided in Table 7-2. (The full

data set associated with Table 7-1 is provided in Appendix C.)

This example will demonstrate that the HCM formula consistently over-predicts the length

of the residual queue and the associated value for d3. In other words, there is an upward bias in

the HCM formula and this bias can be substantial.

The uniform arrival rate for the first 15-minute period is V=180 vehicles (or v=720 vph)

and the uniform arrival rate for the second 15-minute period is V=130 vehicles (or v=520 vph).









Again, to keep the example simple, vehicles are assumed to depart at constant 2-second

headways whenever the indication is green and the effective green time is assumed to equal the

actual green time. Obviously, no departures occur when the signal is red.

Columns B through F of Table 7-1 pertain to a 60 second cycle length with 20 seconds of

green time and 40 seconds of red time. The start of the red for this cycle occurs at the start of the

first period, which results in the green time ending at the end of the first period. In other words,

the cycle is in complete synch with the period demarcation points. This is the best case situation

and, even in this case, the residual queue at the end of the first period is over-predicted by 4

vehicles and the resulting d3 is too high by about 15%.

There are 15 cycles during each 15-minute period (900/60 = 15) and 150 vehicles (C = 10

x 15 =150) will pass the stop bar during each 15-minute period for a capacity of c = 600 vph.

180/15 = 12 vehicles will arrive each cycle, with 4 arriving on the green and 8 arriving on the

red. A traffic technician counting from time 0 to time 900 seconds would count 180 arriving

vehicles (at either the stop bar or back of queue) and would count 150 vehicles departing. A

queue of 30 vehicles is present at time 900 and the approach is just beginning to receive the

green indication at this time. The modified HCM formula (166) would produce a residual queue

at the end of period 1 of 180 150 = 30 vehicles and a corresponding value for d3 in period 2 of

180.0 sec/veh. However, 4 of these 30 vehicles arrived during the previous green period and

are not part of the residual queue. The true residual queue is 26 and the correct value for d3 is

156.0 sec/vehicle.

The best way to obtain the residual queue is to look at the last end-of-red (start-of-green)

time point within period 1 which, for this example, is time point 880. At this time there are 176

cumulative arrivals and 140 cumulative departures, which result in 36 "queued" vehicles. (These









vehicles may not actually be stationary; they are simply vehicles situated between the back of

queue and the stop bar, whether moving or stationary.) However, this 36-vehicle "queue" at the

end of red is not the residual queue either. We must subtract out vehicles that will clear the stop

bar on the next green (the thruput), which for this example is 10. 36 10 = 26 is the true residual

queue, which matches our previous result.

Another source of error occurs if the start-of-red does not match up with the period

demarcation point and the period thruput is used, instead of the period capacity, to calculate the

HCM residual queue and associated d3. An example is provided in columns G through K. The

counted thruput for period 1 is only 144 vehicles (576 vph), considerably less than the 600 vph

capacity, which produces an (incorrect) residual queue of 36 vehicles and an associated d3 value

of 216.0 veh/sec. This type of error also occurs if the cycle length does not divide evenly into

900 seconds. If one is bent on using the erroneous HCM period-based approach, then one can at

least avoid this type of error by always using the calculated capacity, not the thruput. This is one

instance in queue accumulation and dissipation where the theoretical capacity is preferable to the

thruput.

The largest delay discrepancy is found with the 162-second cycle example analyzed in

columns AF through AJ. At time 900 there are 180 cumulative arrivals and 135 cumulative

departures, for an HCM residual queue of 45 vehicles. Consequently, the associated value for d3

is a whopping 250.0 sec/veh if one uses thruput instead of capacity. Using capacity continues to

produce a value of 30 for the residual queue and a corresponding value of 180.0 sec/veh for d3.

This is much closer to the truth but both values are still much too high.

The last end-of-red in period 1 occurs at time 756. There are 151 cumulative arrivals and

108 cumulative departures at this time, which produce 43 "queued" vehicles. Since we have 54









seconds of green time, the thruput is 27 vehicles (1 vehicle every 2 seconds), so the residual

queue is 43 27 = 16 and the associated value for d3 is only 97.2 seconds/vehicle, a hugh

difference of almost 100%.

Even for the best possible case, a cycle in synch with the period demarcation points, there

is an upward bias in both the residual queue and d3. The upward bias get worse if we have a

cycle length that does not divide evenly into 900 seconds, as shown in columns Q through AE.

And the bias gets even worse if the cycle length is a large one, as is shown in columns AF

through AT. The substantial difference in d3 that occurs between a cycle-based approach and a

period-based approach can be readily seen in Figure 7-1. The true cycle-based delay is always

less than the period-based delay.

A further demonstration of the loss of accuracy associated with period-based analysis can

be made by comparing the "actual control delay" (as obtained by summing up the actual queue

lengths on a second-by-second basis) with the control delay obtained by adding the di term to the

previously calculated d3 term. The results are presented in Figure 7-2. The cycle-based analysis

is much closer to the actual delay than the period-based analysis in every case. Also shown in

Figure 7-2 is the control delay value provided by CORSIM. As with the actual control delay, the

CORSIM delay results are much closer to the cycle-based results than the period-based results.

It should be noted that, because of CORSIM initialization issues and because this spreadsheet

example uses vertical queuing while CORSIM uses true horizontal queuing, an exact comparison

between the spreadsheet results and CORSIM cannot be made. However, in this particular case,

the differences caused by these items appear to be minor.

The solution to the period-cycle problem is to always start and stop the counts at the end-

of-red (start-of-green), and to keep track of how much time transpires for each count "period".









For the 162-second example just discussed, the traffic technician would, for the first period, start

counting at end-of-red time 108 and stop counting at end-of-red time 756. This makes period 1

of length 648 seconds (756-108). The cumulative arrivals during that period would be 130 and

the cumulative departures would be 108 (the two 108's are just a coincidence). The calculated

arrival rate would be 130 x 900/648 = 180 (or 720 vph) and the calculated departure rate

(capacity) would be 108 x 900/648 = 150 (or 600 vph). Both of these values check as the

stipulated arrival rate and capacity. The residual queue and associated d3 value would be

calculated as discussed in the previous paragraph. For period 2, we would start counting at time

756 and the process would be repeated. This type of counting discipline is needed to obtain

correct delay values when over-saturated conditions are present.

Residual Queue Discrepancy

It is very important to understand what is meant by the term "initial queue". It can be

argued that the term "residual queue" is preferable since it better represents the item of interest.

The residual queue for a particular lane is the number of "queued" vehicles that exist at

the start-of-green for that lane, minus the thruput of the lane. The thruput of the lane is the

number of vehicles that depart the stop bar during the subsequent green interval. To find the

residual queue at the end of each 15-minute period, one would evaluate the cycle that falls

closest to this time point.

The term "queue" is used loosely here to represent the number of vehicles situated between

the stop line and the back of queue. Under congestion, some of these vehicles may be moving

while some may be stopped, so there is no guarantee that they are all queued. The term "caged"

vehicles is coined in this research to describe these vehicles. (The word "trapped" was

considered for use, but this word has numerous other potential meanings in traffic engineering

whereas "caged" does not.)









It should be pointed out that the residual queue is not the number of vehicles between the

stop bar and the back of queue when the signal turns red (end-of-green) since these "caged"

vehicles include vehicles that arrived during the green interval. Only the portion of the caged

vehicles that arrived during the red interval represent the residual queue. In other words, only

those vehicles that experience a strict phase failure contribute to the residual queue; vehicles

experiencing a liberal phase failure do not.

Figure 7-3 uses another simple example to illustrate the difference between the true

residual queue and the residual queue calculated using the HCM approach. The HCM always

overestimates the residual queue, with the amount of the overestimation depending on the cycle

position that coincides with the period demarcation point. If the period demarcation point were

to coincide with the first end of red then the HCM approach would produce a residual queue of

17, which is 10 greater than the true residual queue of 7 since the thruput is not deducted in the

HCM approach. However, if the period demarcation point were to coincide with the start of red

then the HCM approach would produce a residual queue of 11, which is 4 greater than the true

residual queue of 7. The HCM approach mistakenly includes the 4 arrivals on green as part of

the residual queue. Furthermore, if the period demarcation point were to coincide with the end

of the second red then the HCM approach would produce a residual queue of 19, which is 10

greater than the true residual queue of 9. Consequently, depending on the exact location of the

demarcation point, the HCM approach produces a residual queue for this example that is too

large by a minimum value of 4 and a maximum value of 10. The residual queue bias is always

upward when the HCM approach is used with the maximum amount of the bias being

equal to the thruput and the minimum amount of the bias being equal to the number of

arrivals during the green indication.









As its name implies, the value of the initial queue delay term (d3) is heavily dependent on

the size of the initial (residual) queue. Consequently, an upward bias in the residual queue can

be expected to produce an upward bias in the initial queue delay, and a corresponding upward

bias in the control delay. This will only occur when volume exceeds capacity since the initial

queue delay is zero if volume is less than capacity. Also, since the amount of this upward bias

does not increase as the over-saturated volume-to-capacity ratio increases, but rather stays

"constant" at a value that fluctuates between the arrivals on green and the thruput, the relative

error will be greatest near a v/c ratio of 1.0 and will decrease as the v/c ratio increases. This

effect is clearly evident in Figure 4-23.

Detailed Example of Cycle-Period Issues in Calculating d3

In this research, rather easily obtainable departure information from stop line counts, along

with historical peak hour factors, are used to estimate both a minimum and a maximum

cumulative arrival curve. These curves are then used as a theoretical envelope to bracket the

real-time delay prediction results. Because cumulative curves are used in the theoretical

bracketing of the delay, it is very important to understand the difference in the "delay" produced

by cumulative arrival curves and the true delay associated with trajectory analysis. To do so, a

one-hour (3600 second) example has been developed that is summarized in Tables 7-3 through

7-6.

An important point needs to be made about capacity. Keeping things simple, capacity is

usually considered to be the number of vehicles that CAN pass the stop bar during a certain time

period given current operating conditions (including the most important operating condition, g/C

ratio). However, for the purposes of accurate queue accumulation, which is critical in

calculating the d3 term, capacity needs to be replaced by "thruput", the number of vehicles that

DO pass the stop bar during a certain time period given current operating conditions. Let's say









that, due to previous periods of over-saturated flow, we have accumulated a residual queue of 80

vehicles. The next 15-minute period has a flow rate of 400 vph and a capacity of 600 vph, with

capacity being calculated using the standard formula c = s(g/C). Using this definition of

capacity, the queue would shrink by 50 vehicles ([400-600]/4 = -50) during this period and the

initial (residual) queue for the next period would be 30 vehicles. However, because some wasted

green time occurs at the end of a few of the cycles, let's say that only 120 vehicles actually pass

the stop bar during this 15-minute period which is an effective capacity of only 480 vph (120 x

4 = 480). The end result is that the queue actually shrinks by only 20 vehicles ([400-480]/4 = -

20), producing a residual queue of 60 vehicles. The corresponding value for d3 will have

considerable error if thruput is not used instead of the standard definition of capacity.

Tables 7-3 through 7-6 summarize the comprehensive example. A 90-second cycle is used

in this example with the start of the green offset by about 15 seconds from the 15-minute period

demarcation points. At time zero, the signal is green and there is no queue. Since there is no

queue of any type (let alone a residual queue), the value of d3 for period 1 is simply zero.

The signal turns green at time 75 (seconds). This is the first start-of-green (or end-of-red).

23 vehicles have arrived at the stop bar or back of queue by time 75 and 7 vehicles have departed

from the stop bar. These 7 vehicles departed the stop bar during the green interval that was in

operation when period 1 began. At time 900, the demarcation point between periods 1 and 2,

178 vehicles have arrived and 156 vehicles have departed. So, on a period basis, we have 178

arrivals and 156 departures in period 1, with a resulting "queue" of 22 vehicles. In this example,

9 of the 22 caged vehicles are moving (between the stop bar and the front of the queue) and 13

vehicles are truly queued at time 900. In any event, 178 would be the volume counted by a

traffic technician who was instructed to begin counting at the top of the hour, and it is the









volume that would be used for period 1, the first 15-minute period, in an HCM multi-period

analysis. If we also allow capacity to vary on a 15-minute basis, as reflected by actual thruput,

then we would enter a value of 156 (which equals 624 vph) and, using equation F16-6 from the

HCM, the resulting residual queue at the end of the first period would be 22 vehicles. This

matches the number of caged vehicles at time 900. Using equation F 16-1 from the HCM with 22

for the initial (residual) queue (Qb), d3 for period 2 would then be calculated as 126 sec/veh by

the HCM. It should be noted that the HCS+ software does not allow capacity to vary by 15-

minute interval but instead requires a single capacity for the entire hour.

Unfortunately, these calculations are not correct because 22 is not the residual queue. The

period demarcation point does not occur at the start-of-green and the thruput has not been

deducted. The closest start-of-green time to the demarcation point between periods 1 and 2 (time

900) occurs at time 885. 176 vehicles have arrived by this time and 149 vehicles have departed.

The queue at this point is 27 vehicles in length (176-149) and, since it occurs at the start-of-

green, it is a true queue; all vehicles are stopped. However, although it is a true queue, it is not

the residual queue. To calculate the residual queue we must subtract out the number of vehicles

that depart the stop bar during the subsequent green period (the thruput). The subsequent end-of-

green occurs at time 915. 183 vehicles have arrived by this time and 165 vehicles have departed.

The "queue" at this point is 18 vehicles in length (183-165). However, it is neither a true queue

(15 of the 18 vehicles are moving) nor is it the residual queue. Subtracting the 149 cumulative

departing vehicles at the start-of-green (time 885) from the 165 cumulative departing vehicles

from the end-of-green (time 916) yields a thruput of 16 vehicles. Subtracting this thruput (16)

from the start-of-green queue (27) produces the true residual queue of 11 vehicles at the end of

period 1.









The capacity for the 900-second interval from time 885 to time 1785 is simply the thruput

for this period, which is obtained by subtracting the cumulative departures for these two times:

306-149 = 157. Since this thruput occurs over a 900-second interval, the equivalent hourly

capacity is calculated as: 157 x 3600/900 = 628 vph. Using equation F16-1 from the HCM with

11 for the residual queue (Qb) and 628 for the capacity, d3 for period 2 is correctly calculated as

only 69 sec/veh, not 126 sec/veh.

It should be noted that all yellow time is treated as green time in this example and that, for

our purposes, the end-of-green is actually the end-of-yellow. When the approach is operating

under capacity conditions, it is not uncommon for a CORSIM vehicle to cross the stop bar even

after the indication has turned red. Consequently, the accuracy of the departures at the end-of-

green is improved by using the departures that occur 1 second after the end-of green; time 916 in

this example.

At time 1800, the demarcation point between periods 2 and 3, 352 vehicles have arrived

and 313 vehicles have departed. On a period basis, we have 174 arrivals (352-178) and 157

departures (313-156) in period 2, with a resulting "queue" of 39 vehicles (352-313). In this

example, 7 of the 39 caged vehicles are moving (between the stop bar and the front of the queue)

and 32 vehicles are truly queued at time 1800. 174 would be the volume counted by a traffic

technician who was instructed to count at 15-minute intervals, and it is the value that would be

entered into an HCM multi-period analysis for period 2, the second 15-minute period. If we also

enter capacity, as reflected by actual thruput, into the HCM analysis, then we would enter a

value of 157 (which equals 628 vph) and, using equation F16-6 from the HCM, the resulting

residual queue at the end of the second period calculated by HCS+ would be 39 vehicles. This

matches the number of caged vehicles at time 1800. Using equation F16-1 from the HCM with









39 for the initial (residual) queue (Qb), d3 for period 3 would then be calculated as 225 sec/veh

by the HCM.

These calculations are once again wrong because 39 is not the residual queue. As before,

the problem is twofold; the period demarcation point does not occur at the start-of-green and the

thruput has not been deducted. The closest start-of-green time to the demarcation point between

periods 2 and 3 (1800) occurs at time 1785. 351 vehicles have arrived by this time and 306

vehicles have departed. The queue at this point is 45 vehicles in length (351-306) and, since it

occurs at the start-of-green, it is a true queue; all vehicles are stopped. However, although it is a

true queue, it is not the residual queue. The number of vehicles that depart the stop bar during

the subsequent green period (the thruput) must be subtracted to calculate the residual queue. The

subsequent end-of-green occurs at time 1815. 353 vehicles have arrived by this time and, 1

second later, 320 vehicles have departed. The "queue" at this point is 33 vehicles in length (353-

320). However, it is neither a true queue (15 of the 33 vehicles are moving) nor is it the residual

queue. Subtracting the 306 cumulative departing vehicles at the start-of-green (time 1785) from

the 320 cumulative departing vehicles from the end-of-green (time 1816) yields a thruput of 14

vehicles. Subtracting this thruput (14) from the start-of-green queue (45) produces the true

residual queue of 31 vehicles at the end of period 2.

The capacity for the 900-second interval from time 1785 to time 2685 is the thruput for

this period, which is obtained by subtracting the cumulative departures for these two times: 459-

306 = 153. Since this thruput occurs over a 900-second interval, the equivalent hourly capacity

is calculated as: 153 x 3600/900 = 612 vph. Using equation F16-1 from the HCM with 31 for

the initial (residual) queue (Qb) and 612 for the capacity, d3 for period 3 is correctly calculated as

only 173 sec/veh, not 225 sec/veh.









At time 2700, the demarcation point between periods 3 and 4, 502 vehicles have arrived

and 465 vehicles have departed. On a period basis, we have 150 arrivals (502-352) and 152

departures (465-313) in period 3, with a resulting "queue" of 37 vehicles (502-465). In this

example, 6 of the 37 caged vehicles are moving (between the stop bar and the front of the queue)

and 31 vehicles are truly queued at time 2700. 150 would be the volume counted by a traffic

technician who was instructed to count at 15-minute intervals, and it is the value that would be

entered into the multi-period HCM analysis for period 3, the third 15-minute period. If we also

enter capacity, as reflected by actual thruput, into the HCM analysis, then we would enter a

value of 152 (which equals 608 vph) and, using equation F16-6 from the HCM, the resulting

residual queue at the end of the third period calculated by the HCM would be 37 vehicles. This

matches the number of caged vehicles at time 2700. Using equation F16-1 from the HCM with

37 for the initial (residual) queue (Qb), d3 for period 4 would then be calculated as 155 sec/veh

by the HCM.

As before, these calculations are incorrect because 37 is not the residual queue. The period

demarcation point does not occur at the start-of-green and the thruput has not been deducted.

The closest start-of-green time to the demarcation point between periods 3 and 4 (2700) occurs at

time 2685. 502 vehicles have arrived by this time and 459 vehicles have departed. The queue at

this point is 43 vehicles in length (502-459) and, since it occurs at the start-of-green, it is a true

queue; all vehicles are stopped. However, although it is a true queue, it is not the residual queue.

To calculate the residual queue we must subtract out the number of vehicles that depart the stop

bar (the thruput) during the subsequent green period. The subsequent end-of-green occurs at

time 2715. 506 vehicles have arrived by this time and, 1 second later, 474 vehicles have

departed. The "queue" at this point is 32 vehicles in length (506-474). However, it is neither a









true queue (12 of the 33 vehicles are moving) nor is it the residual queue. Subtracting the 459

cumulative departing vehicles at the start-of-green (time 2685) from the 474 cumulative

departing vehicles from the end-of-green (time 2716) yields a thruput of 15 vehicles.

Subtracting this thruput (15) from the start-of-green queue (43) produces the true residual queue

of 28 vehicles at the end of period 3.

The capacity for the 900-second interval from time 2685 to time 3554, the start of the last

full green in period 4, is the thruput for this period, which is obtained by subtracting the

cumulative departures for these two times: 588-459 = 129. Since this thruput occurs over an

869-second interval, the equivalent hourly capacity is calculated as: 129 x 3600/869 = 534 vph.

Using equation F16-1 from the HCM with 28 for the initial (residual) queue (Qb) and 534 for the

capacity, d3 for period 4 is correctly calculated as only 64 sec/veh, not 155 sec/veh.

This example clearly demonstrates that the delay error caused by using period-based

arrivals and capacities instead of cycle-based arrivals and capacities can be quite large.

The period-based method simply does not produce the correct residual queue.

The situation becomes even worse when we use the period-based method with a constant

capacity value as is required by the HCS+ software. The thruput for the entire hour in this

example is 590 vehicles. If we calculate the hourly capacity using the standard c = s(g/C)

formula, the result is 594 vehicle per hour, which is very close. Using a single value of 590 for

the hourly capacity produces residual queue lengths of 31 at the start of period 2, 57 at the start

of period 3 and 60 at the start of period 4, with a final queue of 6 at the end of period 4. The

associated values of d3 are 186 sec/veh for period 2, 348 sec/veh for period 3, and 200 sec/veh

for period 4. All of these values are much higher than they should be. The situation can be

remedied somewhat by using a single capacity value that is calculated using information taken









only from over-saturated periods. If this is done for our example, the d3 delay results are close to

those obtained for the variable capacity period-based scenario: 134 sec/veh for period 2, 229

sec/veh for period 3, and 65 sec/veh for period 4.

Table 7-3 documents the calculation of the arrivals and departures on both a period basis

and a cycle basis and also provides the associated queue length calculations. The wide disparity

in the calculation of the residual queue is clearly evident from a review of the values contained in

the last three columns.

Table 7-4 provides the dl, d2 and d3 results for both a period-based and cycle-based

approach. Summing these values, the total control delay for each period and the cumulative

control delay are also calculated and a comparison made to CORSIM control delay. A review of

this table shows that the largest deviation from the CORSIM results occurs when a period-based

analysis with a single (fixed) hourly capacity is used (the HCS+ software approach). The period-

based analysis can be improved significantly by allowing capacity to vary over the four 15-

minute periods; the period-based variable capacity approach. However, to approach the

CORSIM results, a cycle-based analysis must be used to calculate d3, an approach which uses the

correct definition of the residual queue. A tremendous improvement is made in the calculation

of control delay when the correct approach is taken.

As was discussed previously in Chapter 6, d2 requires adjustment when the residual queue

is not zero to eliminate the delay effects associated with random arrivals. As is shown in Table

7-5, when this additional correction is made, a further improvement in the delay results occurs,

especially for intermediate periods. However, the effect of the d2 adjustment in this particular

case is minor in comparison to the cycle-based correction.









As was also discussed in Chapter 6, the d3 term is based on a cumulative curve approach.

Consequently, a further adjustment is warranted to convert cumulative curve delay to control

delay. This adjustment to d3 is provided in Table 7-6 and, in this particular case, has minimal

effect. (The adjustment percentages were obtained from a BuckTraj analysis of CORSIM-

generated vehicle trajectories.)

This example clearly demonstrates that the accuracy of the delay calculations is

greatly increased under the preferred option, a cycle-based approach with proper

definition of residual queue, varying capacity by period, and a d2 term that eliminates the

effect of randomness during over-saturated conditions.

Previously, in Tables 6-7 through 6-15, a comparison was made between stopped delay

obtained from simulation and "actual" stopped delay obtained from cumulative curves for our

four examples. Although the delay obtained from these two sources were generally in close

agreement, there were instances where discrepancies arose. The use of a period-based approach

instead of a cycle-based approach may be the cause of these discrepancies.


















Table 7-1. Generalized example of cycle-period delay discrepancies data


Cycle Length
Red/Green Pattern
Period 1
Actual Arrivals & Departures/Thruput (vph)
Actual Arrivals & Departures/Thruput (@15min)
Period 2
Actual Arrivals & Departures/Thruput (vph)
Actual Arrivals, Dep/Thruput (veh per 15min)
Actual Arrivals & Depf/hruput (@30min)

A=Arrivals, D=Departures, Q=Queue
Time (sec)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56


Start on Half Green

vph 720
180 140

vph 520
130 158
310 298


Start on Full Red

720 600
180

520 600
130 150
310 300

A D Q

0 0
0 0 0
0 0 0
S00
1 0
1 0
1 0
1 0
2 0 2
2 0 2
2 0 2
2 0 2
2 0 2
3 0 3
3 0 3
3 0 3
3 0 3
3 0 3
4 0 4
4 0 4
4 0 4
4 0 4
4 0 4
5 0 5
5 0 5
5 0 5
5 0 5
5 0 5
6 0 6
6 0 6
6 0 6
6 0 6
6 0 6
7 0 7
7 0 7
7 0 7
7 0 7
7 0 7
8 0 8

8 0 8
G 8 1 8
G 8 1 7
G 9 2 7
G 9 2 7
G 9 3 7
G 9 3 6
G 9 4 6
G 10 4 6
G 10 5 5
G 10 5 5
G 10 6 5
G 10 6 4
G 11 7 4
G 11 7 4
G 11 8 3
G 11 8 3


60 seconds
Start on Full Green

720
180 "144

520 600
130 150
310 294

A D Q

0 0
G 0 0 0
G 0 0 0
G 1 1 0
G 1 1 0
G 1 1 0
G 1 1 0
G 1 1 0
G 2 2 0
G 2 2 0
G 2 2 0
G 2 2 0
G 2 2 0
G 3 3 0
G 3 3 0
G 3 3 0
G 3 3 0
G 3 3 0
G 4 4 0
G 4 4 0
G 4 4 0
4 4 0
4 4 0
5 4 1
5 4 1
5 4 1
5 4 1
5 4 1
6 4 2
6 4 2
6 4 2
6 4 2
6 4 2
7 4 3
7 4 3
7 4 3
7 4 3
7 4 3
8 4 4
8 4 4
8 4 4
8 4 4
8 4 4
9 4 5
9 4 5
9 4 5
9 4 5
9 4 5
10 4 6
10 4 6
10 4 6
10 4 6
10 4 6
11 4 7
11 4 7
11 4 7
11 4 7


Start on Half Green

720
180 147

520 600
130 150
310 297

A D Q
o o

G 0 0 0
G 0 0 0
G I 1 0
G 1 1 0
G 1 1 0
G 1 1 0
G 1 1 0
G 2 2 0
G 2 2 0
G 2 2 0
2 2 0
2 2 0
3 2 1
3 2 1
3 2 1
3 2 1
3 2 1
4 2 2
4 2 2
4 2 2
4 2 2
4 2 2
5 2 3
5 2 3
5 2 3
5 2 3
5 2 3
6 2 4
6 2 4
6 2 4
6 2 4
6 2 4
7 2 5
7 2 5
7 2 5
7 2 5
7 2 5
8 2 6
8 2 6
8 2 6
826
8 2 6
8 2 6
9 2 7
9 2 7
9 2 7
9 2 7
9 2 7
10 2 8
10 2 8
10 8
G 10 3 8
G 10 3 7
G 11 4 7
G 11 4 7
G 11 5 6
G 11 5 6


Start on Full Red

720
180 143

520 -
130 156
310 299

A D Q

0 0
0 0 0
0 0 0
1 0 1
1 0 1
1 0 1
1 0 1
1 0 1
2 0 2
2 0 2
2 0 2
2 0 2
2 0 2
3 0 3
3 0 3
3 0 3
3 0 3
3 0 3
4 0 4
4 0 4
4 0 4
4 0 4
4 0 4
5 0 5
5 0 5
5 0 5
5 0 5
5 0 5
6 0 6
6 0 6
6 0 6
6 0 6
6 0 6
7 0 7
7 0 7
7 0 7
7 0 7
7 0 7
8 0 8
8 0 8
8 0 8
8 0 8
8 0 8
9 0 9
9 0 9
9 0 9
9 0 9
9 0 9
10 0 10
10 0 10
10 0 10
10 0 104

G 11 1 10
G 10
G 11 2 9
G 11 2 9


78 seconds
Start on Full Green

vph 720 "
180 148

vph 520
130 146
310 294

A D Q

0 0
G 0 0 0
G 0 0 0
G 1 1 0
G 1 1 0
G 1 1 0
G 1 1 0
G 1 1 0
G 2 2 0
G 2 2 0
G 2 2 0
G 2 2 0
G 2 2 0
G 3 3 0
G 3 3 0
G 3 3 0
G 3 3 0
G 3 3 0
G 4 4 0
G 4 4 0
G 4 4 0
G 4 4 0
G 4 4 0
G 5 5 0
G 5 5 0
G 5 5 0
G 5 5 0
5 5 0
6 5 0
6 5 1
6 5 1
6 5 1
6 5 1
7 5 1
7 5 2
7 5 2
7 5 2
7 5 2
8 5 2
8 5 3
8 5 3
8 5 3
8 5 3
9 5 3
9 5 4
9 5 4
9 5 4
9 5 4
10 5 4
10 5 5
10 5 5
10 5 5
10 5 5
11 5 5
11 5 6
11 5 6
11 5 6


Start on Half Green

vph 720
180 146

vph 520 M
130 153
310 298

A D Q

0 0
G 0 0 0
G 0 0 0
G 1 1 0
G 1 1 0
G 1 1 0
G1 1 00
G 1 1 0
G 2 2 0
G 2 2 0
G 2 2 0
G 2 2 0
G 2 2 0
G 3 3 0
3 3 0
3 3 0
3 3 1
3 3 1
4 3 1
4 3 1
4 3 1
4 3 2
4 3 2
5 3 2
5 3 2
5 3 2
5 3 3
5 3 3
6 3 3
6 3 3
6 3 3
6 3 4
6 3 4
7 3 4
7 3 4
7 3 4
7 3 5
7 3 5
8 3 5
8 3 5
8 3 5
8 3 6
8 3 6
9 3 6
9 3 6
9 3 6
9 3 7
9 3 7
10 3 7
10 3 7
10 3 7
10 3 8
10 3 8
11 3 8
11 3 8
11 3 8
11 3 9


Start on Full Red

vph 720
180 135

vph 520 M
130 162
310 297

A D Q

0 0
0 0 0
0 0 0
1 0
1 0 1
1 0 1
1 0 1
1 0 1
2 0 2
2 0 2
2 0 2
2 0 2
2 0 2
3 0 3
3 0 3
3 0 3
3 0 3
3 0 3
4 0 4
4 0 4
4 0 4
4 0 4

5 0 5
5 0 5
5 0 5
5 0 5
5 0 5
6 0 6
6 0 6
6 0 6
6 0 6
6 0 6
7 0 7
7 0 7
7 0 7
7 0 7
7 0 7
8 0 8
8 0 8
8 0 8
8 0 8
8 0 8
9 0 9
9 0 9
9 0 9
9 0 9
9 0 9
10 0 10
10 0 10
10 0 10

10 0 10
11 0 11
11 0 11
11 0 11
11 0 11


162 seconds
Start on Full Green

vph 720
180

vph 520 M
130 144
310 290

A D Q

0 0
G 0 0 0
G 0 0 0
G 1 1 0
G 1 1 0
G 1 1 0
G 1 1 0
G 1 1 0
G 2 2 0
G 2 2 0
G 2 2 0
G 2 2 0
G 2 2 0
G 3 3 0
G 3 3 0
G 3 3 0
G 3 3 0
G 3 3 0
G 4 4 0
G 4 4 0
G 4 4 0
G 4 4 0
G 4 4 0
G 5 5 0
G 5 5 0
G 5 5 0
G 5 5 0
G 5 5 0
G 6 6 0
G 6 6 0
G 6 6 0
G 6 6 0
G 6 6 0
G 7 7 0
G 7 7 0
G 7 7 0
G 7 7 0
G 7 7 0
G 8 8 0
G 8 8 0
G 8 8 0
G 8 8 0
G 8 8 0
G 9 9 0
G 9 9 0
G 9 9 0
G 9 9 0
G 9 9 0
G 10 10 0
G 10 10 0
G 10 10 0
G 10 10 0
G 10 10 0
G 11 11 0
G 11 11 0
S 11 11 0
11 11 0

















Table 7-2. Generalized example of cycle-period delay discrepancies summary
A B C D E F G H I J K L M N P Q R S T U V W X Y Z AA AB AC AD AE
Cycle Length: 60 seconds 78 seconds
Red/Green Pattern Start on Full Red Start on Full Green Start on Half Green Start on Full Red Start on Full Green Start on Half Green
Cycle Length (sec) C 60 Thruput 60 60 78 78 78
Green Time (sec) g=G 20 10 20 10 20 10 26 13 26 13 26 13
Red Time (sec) 40 40 40 52 52 52
g/C=G/C 033 033 033 033 033 033
A=Arnvals, D=Departures, Q=Queue Cumulative A D Q d A D Q d A D Q d A D Q d A D Q d A D Q d
Theoretical Arnvals & Dep/Capacity (vph) X= vl/cl v,= 720 600 = c 720 600 720 600 720 600 720 600 720 600
PERIOD-BASED ANALYSIS T = 0 25
Actual Arnvals & Departures/Thruput (vph) 720 600 720 576 720 588 720 572 vph 720 593 vph 720 5824
Actual Arnvals, Dep/Thruput & Qb (@15mm) Period 1 180 150 180 144 180 147 180 143 180 148 180 146
ActualArrlvals & Departures/Thruput (vph) 520 520 520 520 Ivph 520 vph 520
Actual Arnvals, Dep/Thruput & Q (15mm) 130 150 130 150 130 150 130 156 130 146 130 153
Actual Arnvals, Dep/Thruput & Q (@30mm) 310 300 10 310 294 16 310 297 13 310 299 11 310 294 16 310 298 12
u&t Period 2 100 025 False Qb 100 025 100 025 100 025 100 025 100 025
Calc d, & Q,=cT(X-1) based on theoretical capacity 30 r180 30 "180 30 "180 30 F180 30 1 80 30 1 80
Calc d3=1800Qb(l+u)t/CT & Qb based on actual thruput
CYCLE-BASED ANALYSIS Time
Actual Arnvals, Departures/Thruput & Q at end of 1st red 40 8 0 True Q 60 12 4 50 10 2 52 104 0 78 156 52 65 130 26
Actual Arrivals, Departures/Thruput & Qb at end of last red 880 176 140 26 900 180 144 26 890 178 142 26 832 1664 1300 234 858 171 6 1352 234 845 1690 1326 234
Actual Arval & Dep/Thruput ForTime Shown Period 1 840 168 140 (176-140-10) 840 168 140 840 168 140 780 156 130 780 156 130 780 156 130
Adjusted Arrivals & Dep/Thruput (15mn) 900 180 150 900 180 150 900 180 150 900 180 150 900 180 150 900 180 150
Adjusted Arrvals & Departures/Thruput (vph) 720 600 720 600 720 600 720 600 720 600 720 600
Actual Arr, Dep/Thruput & Q0 ForTime Shown at end of last red 1780 307 290 7 1800 310 294 6 1790 309 292 7 1768 305 286 6 1794 309 291 5 1781 307 289 6
Actual Arrval & Dep/Thruput For Time Shown 900 131 150 900 130 150 900 131 150 936 139 156 936 138 156 936 138 156
AdjArnval&Dep/Thruput(15min) 900 131 150 900 130 150 900 131 150 900 134 150 900 132 150 900 133 150
Adj Arrival & Dep/Thruput (vph) 524 600 c 520 600 522 600 535 600 529 600 532 600
u&t Period 100 025 1 00 025 100 025 1 00 025 100 025 1 00 025
Calculated d3= 1800Qb(1+u)t/cT based on actual thruput 156 156 156 140 140 140
Calculated dj=0 5C[1-g/Cf/[1-g/C] 20 20 20 26 26 26
Penod 2 Actual "Control" Delay (d + d3) In sec 20990 20390 20690 21899 21180 21576
Penod 2 Actual "Control" Delay (di + d3) in sec/veh 140 136 138 140 145 141
A AF AG AH Al AJ AK AL AM AN AO AP AQ AR AS AT
Cycle Length: 162 seconds
Red/Green Pattern Start on Full Red Start on Full Green Start on Half Green
Cycle Length (sec) C 162 162 162
Green Time (sec) g =G 54 27 54 27 54 27
Red Time (sec) 108 108 108
g/C=G/C 033 033 033
Cumulative A=Arnvals, D=Departures, Q=Queue A D Q d A D Q d A D Q d
Theoretical Arnvals & Dep/Capacity (vph) X= vl/cl 720 600 720 600 720 600
PERIOD-BASED ANALYSIS T = 0 25
Actual Arnvals & Departures/Thruput (vph) vph 720 540 vph 720 583 2 vph 720 561 6
Actual Arnvals, Dep/Thruput & Q (@15m) Period 1 180 135 180 146 180 140
Actual Arrivals & Departures/Thruput (vph) vph 520 vph 520 vph 520
Actual Arnvals, Dep/Thruput & Qb (15mm) 130 162 130 144 130 158
Actual Arnvals, Dep/Thruput & Q (@30mm) 310 297 13 310 290 20 310 298 12
u&t Period2 100 025 1 00 025 100 025
Calc d3 & Qb=ciT(X-1) based on theoretical capacity 30 180 30 180 30 180
Calc d3=1800Qb(1+u)t/CT & Qb based on actual thruput
CYCLE-BASED ANALYSIS
Actual Arnvals, Departures/Thruput & Qb at end of 1st red 108 21 6 00 162 324 108 135 270 54
Actual Arnvals, Departures/Thruput & Q at end of last red 756 151 2 1080 162 810 1620 1188 162 783 1566 1134 162
Actual Arval & Dep/Thruput For Time Shown Period 1 648 129 6 108 648 1296 108 648 1296 108
Adjusted Arrivals & Dep/Thruput (15mm) 900 180 150 900 180 150 900 180 150
Adjusted Arvals & Departures/Thruput (vph) 720 600 720 600 720 600
Actual Arr, Dep/Thruput & Q0 For Time Shown at end of last red 1728 300 270 3 1782 307 281 0 1755 303 275 1
Actual Arrival & Dep/Thruput For Time Shown 972 148 162 972 145 162 972 147 162
Adj Arnval & Dep/Thruput (15 mi) 900 137 150 900 135 150 900 136 150
Adj Arrival & Dep/Thruput (vph) 550 600 539 600 544 600
u&t Period 2 100 025 1 00 025 100 025
Calculated d3= 1800Qb(1+u)t/cT based on actual thruput 97 9 97
Calculated dj=0 5C[1-g/C l[1-g/C] 54 54 54
Penod 2 Actual "Control" Delay (d, + d3) in sec 26084 24797 25580
Penod 2 Actual "Control" Delay (di + d3) in sec/veh 161 172 162















300




250




200


10 -




1 00


0 -


SP e rio d B asked U s ing C a p a city

Period B asked, U sing Thruput

O C ycle-B a sed (Thruput= C capacity)


---7


78
Cycle L e n g th


Figure 7-1. Cycle v. period initial queue delay analysis


m


I


I


. I















* Period-Based Using Capacity

Perio d-Based, Using Thruput

O Cycle-Based (Thruput=C apacty)

O AA ctua ue ue -B ase d

o CORSIM Actual


S t rt of G reen


Start of G reen i



Middle of Green I


M middle of Green


m Stirtof Green _


78


78

Cycle L e n g th


Figure 7-2. Cycle v. period "control delay" analysis


3i0 r


300 -


200


1 0





1 0





c n


60


162


i


::::
::::
::::
::::
::::
::::
::::
::::
::::
::::
::::
::::
::::
::::
::::
::::
::::
::::
::::
::::
::::
::::


:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::


:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::












Table 7-3. Detailed example of cycle-period delay discrepancies, residual queue determination


90 sec cycle length Cumulative
Volume Pattern: 690_690_590_345 Time Arrivals Departures


Random Number Set 3

Start of Period 1

Start of First Full Green


Start of Last Green in Period 1

Start of Period 2

End of Last Green (+ls) in Period 1


Start of Last Green in Period 2

Start of Period 3


End of Last Green (+ls) in Period 2


Start of Last Green in Period 3

Start of Period 4


End of Last Green (+ls) in Period 3


Start of Last Full Green

End of Last Green

End of Period 4


Point at BOQ From SB

0 0 0


Arrivals (Volume) Departures (Capacity)


P Total


Residual Queue


Time vehicles Arriving Cvle Period Time Vehicles.I Dparling Cycle Period v.:le Caged Cycle Basis Period Basis


Period


75 23


885 176 149

900 178 156

916 | 183 I


1785

1800 352 313


1816


2685 502 459

2700 502 465


2716 5061 4


3554

3567

3600 596 590


al BOO Basis Basis Period


Slop Bar Basis Basis Thrupul atEOR VarCap VarCap Fixed Cap

0 0 0
7
16
142 142 156 11
27


22 31


18
ll:, 45


39 57


-2
14 43


37 60


-41
15 2



-2
2 0


6 6


TOTALS:
Initial Vehicles Ignored
Final Vehicles Ignored


596 567 596
23
6
596


590 583 590
7
0
590









Table 7-4. Detailed example of cycle-period delay discrepancies, delay comparison
d_ d2 dl Total Delay (seclveh) Cumulative Total Delay (seclveh)
PERIOD Cycle Basis Period Basis Cycle Basis Period Basis Cycle Basis Period Basis Cycle Basis Period Basis Cycle Basis Period Basis
VarCap Var Cap Fixed Cap VarCap Var Cap Fixed Cap VarCap Var Cap Fixed Cap VarCap Var Cap Fixed Cap Corsim VarCap Var Cap Fixed Cap Corsim
1 0 0 0 54 82 108 31 31 31 86 113 140 82 86 113 140 82
2 69 126 186 72 69 97 31 31 31 172 227 315 132 132 169 226 120
3 173 225 348 33 33 41 31 31 31 237 289 420 168 157 205 284 151
4 64 155 200 4 6 2 30 31 31 98 192 233 89 139 203 276 150



Table 7-5. Detailed example of cycle-period delay discrepancies, delay comparison with modified d2 term
d3 d2 dl Total Delay (seclveh) Cumulative Total Delay (sec/veh)
PERIOD Cycle Basis Cycle Basis Cycle Basis Cycle Basis Cycle Basis
Original d2 Modified d2 Original d2 Modified d2 Original d2 Modified d2 Original d2 Modified d2 Corsim Original d2 Modified d2 Corsim
1 0 0 54 54 31 31 86 86 82 86 86 82
2 69 69 72 52 31 31 172 152 132 132 121 120
3 173 173 33 0 31 31 237 204 168 157 147 151
4 64 64 4 1 30 30 98 96 89 139 139 150



Table 7-6. Detailed example of cycle-period delay discrepancies, delay comparison with d3 adjustment
d d, d, Total Delay (sec/veh) Acc to Dc Cumulative Total Delay (seclveh)
PERIOD Cycle Basis Cycle Basis Cycle Basis Cycle Basis Conversion Cycle Basis
Modified d2 Modified d2 Modified d2 Modified d2 Corsim Percentage Modified d2 With d3 Adjustment Corsim
1 0 54 31 86 82 108% 86 92 82
2 69 52 31 152 132 102% 121 123 120
3 173 0 31 204 168 100% 147 147 151
4 64 1 30 96 89 99% 139 138 150




















11 b

V I


L~L:








-


{..*I*I -, *


-. 'r ED


Tr',


Figure 7-3. Upward bias in HCM residual T"l:queue calcul :' ation

Figure 7-3. Upward bias in HCM residual queue calculation


. r-


*uI i ~ 'I- ..r~e-


'1


'.III *I~iI


i.. i~ .'" i."


I 6


-aI- r- Rcr









CHAPTER 8
CONCLUSIONS, APPLICATIONS, AND RECOMMENDATIONS

Conclusions drawn from the research effort, along with potential applications of the real-time
delay estimation procedure that was developed as the core element of this research, are presented
in this chapter.
Research Findings

The following findings have resulted from the research at hand:

1. The research has demonstrated that it is possible to develop a reasonably accurate real-
time procedure for estimating actual stopped delay under conditions of limited
information. The procedure developed, which utilizes a series of adjustments to the
measured arrival rate entering the field of view to estimate the true arrival rate at the back
of the queue, is capable of predicting the unseen component of delay for both under-
saturated conditions and over-saturated conditions.

2. The research indicates that there are two important variables for predicted non-visible
delay: the number of consecutive cycles during which the end of the queue remains
outside the field of view, and the speed at which the queue propagates towards its source.

3. The importance of the queue propagation effect increases as the over-saturated volume-
to-capacity ratio increases. At lower over-saturated v/c ratios queue propagation has little
effect on the predicted delay but at higher v/c ratios it has a substantial effect.

4. This research demonstrates that it is possible to identify both minimum and maximum
cumulative arrival curves for the entire analysis time frame. These curves are established
through the use of arrival information obtained at the end of the analysis period when all
queues are visible, along with historical minimum peak hour factors.

5. A series of equations are presented which allow these minimum and maximum curves to
be calculated for any set of arrival conditions, and for any number of 15-minute analysis
periods. The concept of a peak period factor is introduced to handle analysis time frames
greater than one hour.

6. Given these minimum and maximum cumulative arrival curves, the research
demonstrates that it is possible to calculate a set of theoretical upper and lower bounds on
the solution space for overflow delay. A series of equations are presented which allow
these bounds to be calculated.

7. The research demonstrates that these theoretical bounds can be used, in an ex post facto
manner, to bracket the real-time stopped delay estimation procedure. They can also be
used to identify an independent "most probable" arrival pattern by selecting an
intermediate curve between the upper and lower bounds that minimizes the maximum
percent error between the estimate and the actual delay.









8. This research demonstrates that, contrary to popular belief, the area between the arrival
and departure curves is not the delay (either stopped or control) incurred by approaching
vehicles. It is rather a mixture of delay and non-delay time elements. Consequently, the
Highway Capacity Manual (HCM) assertion that the area between the cumulative arrival
curve and the uniform departure curve can be added to the di term and the random
portion of the d2 term to obtain control delay is not quite right.

9. An evaluation of trajectory analysis during over-saturated conditions is used to reconcile
the difference between stopped delay and the area between the cumulative arrival and
cumulative departure curves. Typical factors for converting cumulative curve delay
(overflow delay) into stopped delay are presented.

10. It is demonstrated that the operational definition of an initial (residual) queue as
presented in the HCM is not correct. The research shows that, in order to identify the
true residual queue on an approach, the situation must be evaluated at the end of the red
period for the approach, and the expected thruput during the subsequent green period
must be subtracted from the observed "queue". Failure to do so leads to an
overestimation of the initial queue and a corresponding overestimation of the initial
queue delay.

11. It is shown that, all other things being equal, the degree of delay overestimation
stemming from the HCM's improper definition of the residual queue tends to increase as
the cycle length increases.

12. It is also shown that, all other things being equal, the degree of overestimation by the
HCM is highest during over-saturated periods having v/c ratios slightly above 1.0

13. The 2000 Highway Capacity Manual's multi-period signalized intersection analysis
procedure uses a period-based technique for queue accumulation. This technique has
certain drawbacks that can produce substantial errors when calculating control delay
during over-saturated conditions. The degree of error increases with increasing cycle
length. A cycle-based counting technique is proposed to remedy this deficiency.

14. As presented in the 2000 Highway Capacity Manual, the random delay component of the
incremental delay term incorrectly contributes to control delay even during over-saturated
periods that are preceded by an initial queue. The result is an artificial increase in control
delay. The amount of the increase is highest when the random delay component is
greatest, which once again occurs at over-saturated v/c ratios close to 1.0 A modification
to the d2 delay term is proposed to remedy this situation.

15. During over-saturated conditions, variability in capacity due to cycle-to-cycle changes in
driver aggressiveness is more important with respect to delay than variations in the
arrival pattern at the back of the queue. The single hour-long capacity value found in the
HCS+ software represents an artificial restriction on capacity variation that contributes to
incorrect delay results during congested conditions.









Application of the Research


The major accomplishment of this research was the development of a theoretically

constrained delay estimation technique that is based on limited information. Use of the

technique to estimate control delay on an over-saturated intersection approach for a one-hour

analysis time frame would proceed as follows:

10. Using the vehicle detection equipment for the approach of interest, real-time second-
by-second data were collected on the number of vehicles crossing the stop bar, the
number of vehicles entering the field of view, the length of the visible queue, and the
presence or absence of a stationary vehicle in the last queue position of the field of
view.

11. This data set is entered into the BuckQ module of the BuckGO delay estimation
software, which measures the length of the visible queue and estimates the length of
the non-visible queue at every second of the one-hour analysis time frame. Second-
by-second cumulative stopped delay is then calculated using this queue information.

12. The stopped delay prediction is converted by BuckQ to control delay using a series of
conversion ratios that vary by cycle length and v/c ratio. The conversion ratio varies
between 1.2 and 1.4 with 1.3 being a typical value. The BuckQ predicted control
delay is considered the final control delay for use in real-time traffic control.

13. The time during the last 15-minute period at which the end of the queue becomes
visible is recorded, as is the cumulative number of vehicles that have crossed the stop
bar at that time. At the end of the one-hour analysis time frame, the cumulative
number of vehicles that have crossed the stop bar is also recorded. This information
is used to calculate the arrival rate during the last 15-minute period.

14. The minimum reasonable Peak Hour Factor (PHF) for the approach and time period
in question is obtained from historical traffic counts. The BuckBOUND module of
the BuckGO delay estimation software constructs a theoretical set of upper and lower
bounds using this minimum PHF and the calculated arrival rate during the last 15-
minute period.

15. The BuckCURVE module of the BuckGO delay estimation software then calculates
the cumulative curve delay associated with both the upper and lower bounds.

16. The cumulative curve delay is then converted to stopped delay by the application of a
correction factor (approximately 0.75).

17. The corrected maximum theoretical stopped delay associated with the upper bound,
and the minimum theoretical stopped delay associated with the lower bound, are
compared to the predicted stopped delay. If the predicted stopped delay falls outside









of the theoretical bounds during any of the four 15-minute periods, then the predicted
delay is appropriately adjusted to remain within the bounds. The resulting "hybrid"
stopped delay is considered the final stopped delay prediction. Note that the
theoretical bracketing of the predicted stopped delay is carried-out in an ex post facto
manner, after the analysis time frame has expired.

18. The hybrid stopped delay results are converted to control delay using a series of
conversion ratios that vary by cycle length and v/c ratio. The conversion ratio varies
between 1.2 and 1.4 with 1.3 being a typical value. The hybrid control delay is
considered the final control delay prediction for project evaluation purposes.

Using this process, the proposed delay estimation technique proves useful for both real-time

traffic control and project evaluation.

Three examples are provided to demonstrate how the delay estimation procedure

developed in this research might be used in real world applications to improve the results

obtained.

Example 1: Signal System Retiming Evaluation

A consultant has been hired to retime 10 traffic signals that are part of a closed-loop

system along a busy arterial (SR 4) in Pahokee, Florida. Initial (before) travel time runs are

conducted along SR 4 during all analysis periods of interest; including the weekday AM and PM

peak periods. New signal timings are developed, implemented, and fine-tuned. Final (after)

travel time runs are then conducted along SR 4 during the same periods as the initial runs. All

periods show a significant reduction in travel time along SR 4 with implementation of the new

timings. Unfortunately, the citizens of Pahokee are not happy with the retiming project so a new

consultant is hired to check the work.

It quickly becomes evident to the new consultant that, although traffic flow along SR 4

seems pretty good, side street delay is excessive at many critical locations with repeated phase

failures and extensive recurring queues. The new consultant repeats the "after" SR 4 travel time

runs, but this time with the BuckGO suite of delay estimation software installed as an add-on to









the closed-loop software. BuckGO is used to measure actual side street stopped delay when

queue lengths do not exceed the field of view, and to estimate stopped delay when they do.

Appropriate ratios, based on cycle length and v/c ratio, are then applied to the stopped delay

values to obtain control delay.

To the pleasure of the citizens of Pahokee, the new consultant reinstalls the initial timings.

The "before" travel time runs are then repeated, this time with BuckGO in operation. The new

consultant obtains almost exactly the same "before" and "after" travel time results along SR 4 as

the original consultant; the improvement along SR 4 was indeed real. However, after reviewing

the BuckGO results, the new consultant realizes that side street delay skyrocketed when the new

timings were installed.

Example 2: Real-Time Traffic Signal Control

A real-time adaptive traffic signal control program has been installed at the "T"

intersection of Main Street and Elm Street in Clewiston, Florida. The adaptive software uses

volume information obtained from upstream inductance loop detectors to optimize signal timings

at this isolated signal. Unfortunately, the local traffic engineer has received numerous citizen

complaints that not enough green time is being provided to the side street approach during peak

periods. The engineer investigates and determines that multiple cycle failures and recurring

queues occur on the side street during the weekday PM peak hour. The engineer calls in a

consultant for assistance.

The consultant reviews the situation and determines that, during the PM peak hour, side

street queues extend well past the upstream inductance loop, which is located a healthy 400 feet

from the stop bar. Because of this, the loop is not counting the true demand on the approach and

is, therefore, not allocating green time based on a delay determination that is derived from the

true arrival rate.









The consultant addresses the problem by adding the BuckGO module to the adaptive

software. When queues extend beyond the upstream loop, BuckGO estimates the true (higher)

arrival rate and associated (greater) delay based on the number of adjacent blind periods that

occur. This improved estimate of delay results in a reallocation of green time that greatly

reduces the extent and duration of the recurring side street queues. Complaints from the citizens

of Clewiston concerning the signal timing at this intersection disappear.

Example 3: Signalized Intersection Capacity Analysis

A traffic engineer would like to determine the delay on various approaches to an existing

signalized intersection that is experiencing severe congestion. However, the queues are so long

on some of the approaches (over 1/2 mile), and build so rapidly, that counting arriving vehicles is

virtually impossible. The engineer is smart enough to know that merely counting vehicles

crossing the stop bar will not produce a true picture of delay since these counts do not measure

the true demand on the approach, only the supply. The engineer, once again, calls in a consultant

for assistance.

The consultant reviews the situation and determines that BuckQ could be used to estimate

the arriving volume on the over-saturated approaches. The appropriate 15-minute count data are

collected and, for the under-saturated, near-saturated, and slightly over-saturated movements, a

multi-period analysis based on HCM principals is conducted to determine approach control

delay. However, for the grossly over-saturated movements, BuckGO is used.

Potential Areas of Extended Research

The following areas of additional research have been identified:

1. Extension of the procedure to examine other cycle lengths and other fields of view -
Although preliminary analyses were made that involved three cycle lengths (80, 120 and
160 seconds) and two fields of view (8 and 12), the final detailed analysis included only
one cycle length (120 seconds) and one field of view (12). It would be of interest to
expand the range for both of these important items to quantify their effect. In addition,









the g/C ratio was held constant at about 0.30 Varying this value and examining the
results would also be of interest.

2. Investigation of other methods for adjusting the arrival rate during adjacent blind
periods Although it appears to work rather well over the range of simulation conditions
investigated in this dissertation, the use of a power function based on the number of
adjacent blind periods is only one of many possible methods for adjusting the arrival rate
at the back of the queue. Other options could be investigated, including the use of a
logarithmic function instead of a power function, or the use of the length of the adjacent
blind period in seconds as the input to the function rather than the number of adjacent
blind periods. We could also develop logic that would ignore any isolated break in the
number of vehicles entering the field of view when determining whether or not the non-
visible queue had dissipated. This would eliminate false termination of the blind period
due to "sleepers", queued motorists who failed to advance into the field of view in a
timely manner due to some distraction.

3. Accounting for the effect of trucks in the traffic stream Trucks have a twofold effect
on queue formation and discharge: 1.) They have a discharge headway that is greater than
that of passenger cars (CORSIM assumes 120%), and 2.) They are longer than passenger
cars, which causes fewer vehicles to be observable within a given field of view. Also,
as both Tarko [29] and Cohen [31] discovered, in the real world trucks have a third effect
on queue formation and discharge, their presence lengthens the headways of passenger
cars in the traffic stream. All of these effects could be examined in future work,
especially as they relate to our choice of a 5 second headway for re-setting the non-
visible queue to zero.

4. Accounting for the effect of arrival type Vehicles typically arrive at an approach in
one of three basic ways: 1. They arrive randomly, 2. They arrive in platoons that reach
the approach at the same time every cycle (since the approach is "fed" by an upstream
signal with an equivalent cycle length), or 3. They arrive in platoons that reach the
approach at different times during the cycle (since the approach is "fed" by an upstream
signal with a different cycle length). The sensitivity of our delay estimation procedure to
arrival type is another fertile area for future simulation-based research.

5. Accounting for the effect of multiple approach lanes The queue arrival and
discharge situation is complicated when lane-changes can occur, as is the case for side
streets with multiple approach lanes. Modifying the delay estimation procedure to handle
queue accumulation under such conditions would be of practical benefit.

6. Accounting for the effect of queue mixing Vehicle queues often mix together on an
approach under high-volume or over-saturated conditions. For example, it is not
uncommon for queues from a left turn lane to spillback into the adjacent thru lane during
peak periods. This mixing of queues offers a particularly challenging problem if one
desires to apportion delay by movement, rather than by approach.









7. Field application of the delay estimation procedure Application of the delay
estimation procedure described in this document to an intersection approach would
probably be the most enlightening extension to the research. The logical place to start
would be with the simplest possible case, a single lane approach where left turns are not
permitted (such as in a downtown area) having very few trucks and random arrivals.
During our research efforts, some time was spent experimenting with video taken on a
multi-lane main street approach in Gainesville, Florida and on a multi-lane side street
approach in Jacksonville, Florida. Work was begun on a software program for producing
queue length and count information from the video. This is a challenging endeavor in
and of itself given the peculiarities of video detection.

8. Development of additional examples In order to illustrate the calculation of the
theoretical upper and lower bounds, and associated overflow delay, for the 5-period and
n-period cases, additional examples could be developed.

9. Use of other measures of effectiveness (MOE) It may be that other MOEs besides
delay are of interest when evaluating intersection performance. Such MOEs might
include variables mentioned in this document, including: predicted queue length,
predicted back-of-queue position, number of phase failures, number of vehicle re-queues,
number of adjacent blind periods, or percent of time that the queue is not visible. Or they
might involve totally new variables. A 2004 paper by Zhang and Prevedouros [51],
which was based on a web-survey with 2017 responses, suggests that "waiting time"
(a.k.a. "delay") is not as important to motorists as other factors. These factors include:
traffic signal responsiveness (related to the delay when no vehicles are present on
conflicting movements), extent of phase failures, availability of left turn lanes and
phasing, and pavement quality. In addition, as discussed by Tarnoff and Ordonez [1], the
use of alternative MOEs may be particularly appropriate when over-saturated conditions
prevail:

"When saturation exists, different measures of effectiveness should be usedfor evaluating
system performance. During under-saturated conditions, stops and delays are the MOEs
typically used. When saturated conditions exist, the objective is to minimize the time
period during which these conditions exist, and the MOEs in use include queue lengths,
number of cycle failures and the percent of time that intersections are congested. This is
accomplished by controlling the direction of queue build-up to avoid spillback and
minimize cycle failures. "

10. Use of other simulation programs If may prove beneficial to make use of a second
simulation program (such as SYNCHRO/SIMTRAFFIC, VISSIM, PARAMICS or
AIMSUN) to check the results obtained with CORSIM and to make use of features
inherent in these programs that may be superior to those in CORSIM. For example,
when dealing with a multiple lane approach, CORSIM often has the first few vehicles in
the queue starting simultaneously, which is quite a deviation from reality.









11. Closely spaced intersections The usefulness of any procedure that is developed would
be enhanced if it could be applied to closely spaced intersections, including those that
occur in the vicinity of freeway ramp terminals.

12. Arterial evaluation The procedures developed in this research should prove useful for
arterial evaluation, at least as far as side street approach delay and delay in the main street
left turn lane are concerned. The value of the procedures would be maximized where
significant periods of over-saturated operation occur. Consequently, integrating the
results of this research into a larger arterial evaluation tool would be of interest.

13. Development of an automated delay estimation module Finally, the ultimate
extension of this research would be the development of a closed-loop or traffic signal
controller module that would automate the delay estimation procedure. The module
would provide real-time delay estimation, even during over-saturated conditions, and
would apply ex post facto delay adjustments once queues have cleared. The module
night be patented and marketed to both traffic signal controller manufacturers and traffic
signal software development firms.








APPENDIX A
DATA SETS FOR BUCKQ TESTING

CORSIM 5.1





216 Total Runs

12 volume levels

3 random number sets per volume level

3 cycle lengths

2 fields of view










~-,,


Mean = 1.80


C.
w!
wL


I


1.70 1.71 1.72 1.73 1.74 1.75 1.76 1.77 1.78


IN.


1.79 1.80 1.81 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89 1.90


Headway, 1 Hour Average (seconds)


Figure A-1. Queue discharge headway histogram


10

5

0













2U

18

16

14
Mean = 2.58
12

S10

o 8

6

4

2

0
1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00
Start-Up Lost Time, 1 Hour Average (seconds)


Figure A-2. Start-up lost time histogram













~2flfl ______ ______ ______ ______ ______ -


* 80 second cycle (55s red)

* 120 second cycle (82s red)

A 160 second cycle (11Os red)


y = 0.839x 4.9248
R2 = 0.9987 0
y =- 0.8108x 4.3722
R2 = 0.9996








A linear relationship exists between control delay and stopped
delay that is independent of cycle length


0 20 40 60 80 100 120 140 160
Control Delay (sec/veh)


180 200 220 240 260 280 300


Figure A-3. Comparison of control delay and stopped delay by cycle length (g/C =0.30)


y = 0.8676x 5.4561
R2 = 0.9989


inni














* All 3 cycles combined


y = 0.8587x 7.2045
R2 = 0.9948


0 20 40 60 80 100 120 140 160
Control Delay (sec/veh)

Figure A-4. Comparison of control delay and stopped delay (g/C =0.30)


180 200 220 240 260 280 300













(g/C = 0.30)

* 80 second cycle (55s red)

* 120 second cycle (82s red) y= 1.0123x -14.998
y = 1.0271x 13.275 R2 = 0.9994
A 160 second cycle (110s red) R2 = 0.9997







y 0.839x 4.9248
R2 = 0.9987






A linear relationship exists between control
delay and "stopped plus queue move-up"
delay that is independent of cycle length


40 60 80 100 120 140 160
Control Delay (sec/veh)


180 200 220 240


260 280 300


Figure A-5. Comparison of control delay and stopped plus queue move-up delay by cycle length (g/C = 0.30)


" 240
U
I 220

$ 200

? 180

| 160

140

S 120

g 100
0.
80

60

40

20

0


0 20













* All 3 cycles combined


y = 1.008x 13.062
R2 = 0.9986






fA


40 60 80 100 120 140 160
Control Delay (sec/veh)


180 200 220 240 260 280 300


Figure A-6. Comparison of control delay and stopped delay plus queue move-up delay (g/C =0.30)


0 20













300
290
2 9 0 ----------------------------------------------------------_ _
280 All 3 Cycles Combined
270
260
250 1 57x
240 y = 0.0021e10s7X
230 R2 = 0.7879
220 --
210
200
1 9 0 ------------- _ _
180
180 y = 1044.9x -1007.1
170
170 FR2 =0.8088
160
150 -
140 --
130 ---
120 -
110
100 --*-
90
80
70
60
5 0 --------

30
20
40 _____________________------__-----------------------
3 0 _ __-----------------------------------------------
2 0 _-------------------------------------------------
1 0 _-------------------
0


0.95


1.00


1.05 1.10
v/c Ratio During 1st 45 Minutes


1.15


Figure A-7. Relationship between v/c ratio and stopped delay












300
290
280
270
260
250
240
230
220
210
200
190
180
170
160
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0
0


1.05


1.15


v/c Ratio During 1st 45 Minutes


Figure A-8. Relationship between v/c ratio and stopped delay by cycle length


* 80 second cycle

* 120 second cycle

A 160 second cycle y = 0.0002e12.732x
R2= 0.8521 y = 0.0712e6.9348x
t__ R2 = 0.8318








____________ y = 0.0073e8.8577x
__i____ __ R2 = 0.791
A __
: -


.95














* All 3 Cycles Combined
y = 0.0015e10.488x
R2 = 0.7837





y______= 1217.2x -1176.8
R2 = 0.7995



1W
---~? 4r

= Jl
......................


0.95


1.00


1.05


1.10


1.15


1.20


v/c Ratio During 1st 45 Minutes


Figure A-9. Relationship between v/c ratio and stopped plus queue move-up delay












300
290
280
270
260
250
240
230
220
210
200
190
180
170
160
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0
0.95


1.20


v/c Ratio During 1st 45 Minutes


Figure A-10. Relationship between v/c ratio and stopped plus queue move-up delay by cycle length


1.00 1.05 1.10 1.15













300
290
280 All 3 Cycles Combined Y=0.0058e93406x
270 R 2 0.7939
260 ___________
250
240
230 'k 'I
220 IT
210
210r y 1213.9x -1161.3
2002
190 zoo", *0= 0.8091
180
170
160
150 __
140 _
130
120
110
100
90
80 jag
70
60 _
50
40
30 000100
20
100000
0


0.95


1.00


1.05


1.10


1.15


v/c Ratio During 1st 45 Minutes


Figure A-11. Relationship between v/c ratio and control delay


1.20












290
280 U 80 seconds 13
270 y = 0.0002e12.732x 0576.6575x
260 120 seconds R2 = 0.8521 = 0.1157e
250 R2 = 0.837
240 A 160 seconds
230
220 A
210
S200 4 A
190
S180
170 IA
S160 -A00__ -
Q 150 = 0.0141e8.4536x


S110
2 130--------- R2^^--- =^ 0.7914-------


S110

< 70
60
50 E
40
30 -
20
10
2 0 ---------------------------------------------------------
1 0 -------------------------------------------------------
0
0.95 1.00 1.05 1.10 1.15 1.20
v/c Ratio During 1st 45 Minutes


Figure A-12. Relationship between v/c ratio and control delay by cycle length













30C
29C
28C
27C
26C
25C
24C
23C
22C
21C
20C
* 19C
> 18
S17
S16
c 15c
S14C
13C
. 12(
112
S10
9(
8(
7(


)
0


-AI 1


7 T7


%J V4 -T_


Figure A-13. Relationship between vehicle re-queues and control delay


0 y = 0.1953x + 73.131
0 2
R = 0.998

S_ = 0.1472x + 56.361 /_
S2 R2 = 0.9965 A --



Sl y = 0.0946x + 41.357
S-- R2 =0.9983
) "-"


A strong linear relationship exists between
vehicle re-queues and control delay


A 160 second cycle

* 120 second cycle

* 80 second cycle


0 100 200 300 400 500 600 700 800 900 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Vehicle Re-Queues


i
I


31' -~ '


0I E I











2500 ..
2400 A 160 second cycle
2300 120 second cycle y= 15327x -15313

R2200 1 = 0.8198 A
2100 A 80 second cycle
2000
1900 -/
1800 -A A
1700
1600 The number of vehicle re-queues
increases as the cycle length decreases
n 1500 If cycle length is accounted for, the -/ a ch
Q 1400 number of vehicle re-queues is A
0 1300 linearly related to the v/c ratio A / = 7213.5x 7244.3
S1200 A R2 = 0.7858
1100 -
S 1000 -. --
900 or A -
800
700 A___" A____--I-I
600 y = 5560.2x 5561.7
500 R R2 = 0.8507 -
400 A
300 AL 0.98 09 ----
200
100 -- ---
0.95 0.96 0.97 0.98 0.99 1.001.01 1.02 1.031.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.141.15 1.16 1.17 1.18 1.19 1.20
0.95 0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20


v/c Ratio During 1st 45 Minutes


Figure A-14. Relationship between v/c ratio and vehicle re-queues












2500
2400
2300
2200
2100
2000
1900
1800A A 1.8
1700 y= 0.0002e13787x
1600 If cycle length is not accounted for, the number of 2 0.4
1500 vehicle re-queues is only weakly related to the v/c ratio
S1400 --A -
S1300 A
S1200 A y= 6341x 6186
S1100
| 1000_ R2 = 0.329
S91000

800 A,
700 -----
600 A- AA ,-
500 ---_ A_ ,--
400
300 A____,_
200 AA_ _L
100 A __A
0 1 1
0.95 0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20


v/c Ratio During 1st 45 Minutes


Figure A-15. Relationship between v/c ratio and vehicle re-queues by cycle length









































01
0.95 1.00 1.05 1.10 1.15


v/c Ratio During 1st 45 Minutes


Figure A-16. Relationship between v/c ratio and cycles with phase failure


1.20

















If cycle length is not accounted for,
the number of cycles with phase
failures is unrelated to v/c ratio


60




50




40




i 30
0.


S 20




10


v/c Ratio During 1st 45 Minutes


Figure A-17. Relationship between v/c ratio and cycles with phase failure by cycle length


y = 3.765x + 20.481
R2 = 0.0003


0 -
0.95


1.15












100%
y = 3.2432x 2.6335 A
90%- R2 = 0.6417 A '
A y= 2.4516x -1.9204
8%-------A A R2 0.5826











o If cycle length is accounted for, the
S26021x -085 percentage of cycleswith phase
30% R2 07536






3 = failures is linearly related to vlc ratio A 160 second cycle

20% 120 second cycle
70%















0.95 1.00 1.05 1.10 1.15 1.20
vic Ratio During 1st 45 Minutes
30 R2 0.76 failures is linearly related to v/c ratio A 160 second cycle

20%-- 120 second cycle

A 80 second cycle
10% -



0.95 1.00 1.05 1.10 1.15 1.20

v/c Ratio During 1st 45 Minutes


Figure A-18. Percentage of cycles in 1 hour with phase failure by cycle length












100%


90% AA


80% _-----------___-A_--_--____-----_---------


70% y = 2.2224x -1.6343 _____A
R2 = 0.4634




S50%A AA
-I If-----A-A-A
If cycle length is not accounted for, the percentage
o 40% of cycles with phase failures is still linearly related
"o to the v/c ratio, but not as strongly




2 0 % --------------------------------------------------------------------
30%


20%


10% --

0%
0.95 1.00 1.05 1.10 1.15 1.20

v/c Ratio During 1st 45 Minutes


Figure A-19. Percentage of cycles in 1 hour with phase failure




























C)




C0
0

C)
00


300.0

280.0

260.0

240.0

220.0

200.0

180.0

160.0

140.0

120.0

100.0

80.0

60.0

40.0

20.0

0.0


0 10000 20000 30000 40000 50000 60000 70000
Sum of Adjacent Blind Period Counter(ABPC)


Figure A-20. Linear relationship between ABPC and stopped delay


80000











300.0

280.0

260.0

240.0

220.0

200.0

180.0

160.0

140.0

120.0

100.0

80.0

60.0

40.0

20.0

0.0


0 10000 20000 30000 40000 50000 60000 70000
Sum of Adjacent Blind Period Counter (ABPC)


Figure A-21. Exponential relationship between ABPC and stopped delay


80000











280 --Y = yo.oo-.e R2 = 0.7979
R2 = 0.8486
260 A/ y = 72.435eE y = 55.233e2E
240 R = 0.8117 R2 = 0.9244


18240
220
1 40 / A=8 .19 :


~ 180--I-( ^ Ir--y = =51.186e2E-05<





S1200 A0

100 /" I" -- 80 sec/12 FOV
80 ________ 120 sec/12 FOV

40
A 2 160 sec/1 2FOV
kA 160 sec/8 FOV

40 120 sec/8 FOV

20 80 sec/8 FOV

0
0 10000 20000 30000 40000 50000 60000 70000 80000
Sum of Adjacent Blind Period Counter (ABPC)
Figure A-22. Relationship between ABPC and control delay









APPENDIX B
TYPICAL PEAK HOUR FACTORS












1.000


I



0.800
C 9 0 - - - - - - - - --- ----- -- ---- ---- --




0.800 ------------------------------------------------- ------- -----------------


-4-Tues, 3/28/06
---Wed. 3/29/06
--- Thur, 3/30/06



0.700
4:00-5:00 4:15-5:15 4:30-5:30 4:45-5:45 5:00-6:00 5:15-6:15 5:30-6:30 5:45-6:45 6:00-7:00

Time Period


Figure B-1. US 1 S. PM peak hour factor, southbound (outbound) flow














1.000


0.900


0.800









0.700


4:00-6:00


4:15-6:15


Figure B-2. US 1 S. PM peak period factor, southbound (outbound) flow


Peak Period Factor




------------^---- ---------------------------------------------











---Tues, 3/28/06
----Wed. 3/29/06
--$--Thur, 3/30/06


4:30-6:30

Time Period


4:45-6:45


5:00-7:00










Table B 1. US 1 machine counts (Southern St. Johns County)
North of SR 206
Tuesday, 3/28/06
End


Start Time Time
4:00 4:15
4:15 4:30
4:30 4:45
4:45 5:00
5:00 5:15
5:15 5:30
5:30 5:45
5:45 6:00
6:00 6:15
6:15 6:30
6:30 6:45
6:45 7:00
Peak Hour:


Southbound
303
250
249
282
282
316
299
234
209
196
165
158
1179


Northbound
176
188
197
167
207
207
212
159
136
132
152
98
793


1-Hour


PHF


Both Period SB NB
479 1 4:00-5:00 0.894 0.924
438 2 4:15-5:15 0.942 0.917
446 3 4:30-5:30 0.893 0.940
449 4 4:45-5:45 0.933 0.935
489 5 5:00-6:00 0.895 0.926
523 6 5:15-6:15 0.837 0.842
511 7 5:30-6:30 0.784 0.754
393 8 5:45-6:45 0.859 0.910
345 9 6:00-7:00 0.871 0.852
328
317
256
1972


2-Hour

Period
4:00-6:00
4:15-6:15
4:30-6:30
4:45-6:45
5:00-7:00


PPF

SB NB
0.876 0.892
0.839 0.869
0.818 0.835
0.784 0.809
0.735 0.768










Table B-1. Continued.
Wednesday, 3/29/06


End
Start Time Time
4:00 4:15
4:15 4:30
4:30 4:45
4:45 5:00
5:00 5:15
5:15 5:30
5:30 5:45
5:45 6:00
6:00 6:15
6:15 6:30
6:30 6:45
6:45 7:00
Peak Hour:


SB
252
262
246
260
301
320
279
235
211
184
173
135
1160


1-Hour


PHF


Both Period SB NB
403 1 4:00-5:00 0.973 0.836
477 2 4:15-5:15 0.888 0.877
415 3 4:30-5:30 0.880 0.874
444 4 4:45-5:45 0.906 0.890
487 5 5:00-6:00 0.887 0.896
536 6 5:15-6:15 0.816 0.894
508 7 5:30-6:30 0.815 0.810
425 8 5:45-6:45 0.854 0.853
395 9 6:00-7:00 0.833 0.800
323
308
266
1975


2-Hour

Period
4:00-6:00
4:15-6:15
4:30-6:30
4:45-6:45
5:00-7:00


PPF

SB NB
0.842 0.841
0.826 0.859
0.795 0.817
0.767 0.799
0.718 0.770










Table B-1. Continued.
Thursday, 3/30/06


End
Start Time Time
4:00 4:15
4:15 4:30
4:30 4:45
4:45 5:00
5:00 5:15
5:15 5:30
5:30 5:45
5:45 6:00
6:00 6:15
6:15 6:30
6:30 6:45
6:45 7:00
Peak Hour:


SB
254
246
271
276
297
297
256
246
179
179
177
168
1126


1-Hour


PHF


Both Period SB
459 1 4:00-5:00 0.948
433 2 4:15-5:15 0.918
456 3 4:30-5:30 0.960
438 4 4:45-5:45 0.948
483 5 5:00-6:00 0.923
493 6 5:15-6:15 0.823
461 7 5:30-6:30 0.840
423 8 5:45-6:45 0.794
336 9 6:00-7:00 0.982
383
320
292
1875


2-Hour


NB
0.901
0.963
0.930
0.913
0.932
0.896
0.906
0.835
0.770


PPF


Period SB NB
4:00-6:00 0.902 0.916
4:15-6:15 0.870 0.887
4:30-6:30 0.842 0.898
4:45-6:45 0.803 0.872
5:00-7:00 0.757 0.849


0.718 0.768
0.902 0.916
0.812 0.845


Lowest
Highest
Average


0.784
0.982
0.885


0.754
0.963
0.881















1.000


0.900 ___A__o


U-l

I
a-\


0 -Tues, 3/28/06, Estrella,
0.800 Wed, 3/29/06, Estrella- -

SThur, 3/30/06, Estrella-- -
A- Tues, 3/28/06, Lewsi Speedway
-- Wed, 3/29/06, Lewis Speedway
J- Thur, 3/30/06, Lewis Speedway

----------------------- -------------------------------------------------------
0.700
4:00-5:00 4:15-5:15 4:30-5:30 4:45-5:45 5:00-6:00 5:15-6:15 5:30-6:30 5:45-6:45 6:00-7:00


Time Period


Figure B-3. US 1 N. PM peak hour factor, northbound (outbound) flow













1.000


0.900 -i__ vv U, 0JIL.-JI'..J, LYvvI 4jp
S, l- Thur, 3/30/06, Lewis Speec






00~------ 0.80------------------------^^**^^^^<----------- ---------------
---------------------
a--




00 0.800





0.0 --------------------------------------------AL------------------^---



0.700
4:00-6:00 4:15-6:15 4:30-6:30 4:45-6:45 5:00-7:00

Time Period


Figure B-4. US 1 N. PM peak period factor, northbound (outbound) flow















Table B-2. US1 Machine counts (northern St. Johns County)


North of Estrella Avenue
Tuesday, 31/28/0
Start Time End Time Northbound Southbound
4 00 4:15 214 251
4:15 4:30 220 216
4:30 4:45 245 320
4:45 5:00 235 231
5:00 5:15 274 261
5:15 5:30 240 249
5:30 5:45 261 258
5:45 6:00 226 202
6:00 6:15 155 167
6:15 6:30 175 176
630 6:45 145 158
6:45 7:00 106 110
Peak Hour. 994 1061


Wednesday, 312956
Start Time End Time Northbound Southbound
4 00 4:15 211 238
4:15 4:30 227 242
4:30 4:45 234 276
4:45 5:00 234 250
5:00 5:15 281 286
5:15 5:30 254 237
5:30 5:45 269 241
5:45 6:00 181 179
6-00 6:15 176 192
0 6:15 6:30 173 177
630 6:45 143 164
645 7:00 150 132
Peak Hour. 1038 1014


1-Hour PHF
Period NB SB
4:00-500 0933 0.795
4:15-615 0889 0.803
4:30-530 0907 0.829
4:45-5.45 0.922 0.957
5:00-6.00 0.913 0.929
515-6'15 0845 0.849
5 30-6'30 0783 0.778
545-6'45 0 775 0.870
6 00-700 0830 0.868


4:000-5 0 96
4:15-515 0868
430-530 0892
4.45-5.45 0.923
5.00-6.00 0.876
5:15-615 0810
5:30-630 0 743
5:45-645 0930
6:00-700 0912


2-Hour PPF
Period NB SB
4:00-600 0.874 0777
4:15-615 0.847 0744
4:30-6:30 0.826 0728
4:45-6:45 0.781 0.815
5:00-7:00 0.722 0.757


4:00-6:00 0.841 0852
4:15-6:15 0.826 0832
4:30-6:30 0.802 0803
4:45-6:45 0.761 0.754
5.00-7:00 0.724 0.703


South of Lewis Speedway
Tuesday, 31/2806 1-Hour PHF 2-Hour PPF
Start Time End Time Northbound Southbound Both Period NB SB Period NB SB
400 415 251 291 542 1 4.00-5:00 0.960 0.798 4:00-6:0D 0901 0.754
4:15 4:30 264 288 552 2 4:15-5:15 0.967 0.829 4:15-6:15 0 859 0.721
4:30 4:45 252 378 630 3 4:30-5:30 0.959 0.813 4:30-6:30 0 823 0.689
4:45 5:00 270 250 520 4 4:45-5:45 0.931 0.823 4;45-6:45 0.705 0.692
5:00 5:15 258 338 596 5 5.00-8:00 0.902 0.794 5:00-7:00 0.724 0.652
5'15 5:30 256 263 519 6 5:15-6:15 0.813 0.BB1
5 30 5:45 2BB 262 550 7 5:30-:30 0.747 0.B16
5'45 6:00 237 211 44B B 5:45-:45 0.777 D.B9B
600 6:15 155 191 346 9 600-7:00 0.870 0.901
615 6:30 181 191 372
630 6:45 164 165 329
6.45 7:00 130 141 271
PeakHour: 1300 1517 2817

Wednesday, 31/2B 9
StartTime EndTime Northbound Southbound Both
400 415 245 321 566 1 4.00-5:00 0.947 0.B4B 4:00-6:00 0872 0.802
415 430 266 273 539 2 4:115-5:1 0.964 0.864 4:15-6:15 843 0.756
4:30 4:45 258 364 622 3 430-5:30 0.911 0.861 4:30-6:30 0 13 0.726
4:45 5:00 239 276 515 4 445-5:45 0.913 0.841 4:45-6:45 0.776 0.699
5:00 5:15 267 345 612 5 5.00-0:00 0.871 0.798 5:00-7:00 0.741 0.651
5:15 5:30 289 269 558 6 5:15-6:15 0.796 0.871
5'30 5,45 261 271 532 7 5:30-6:30 0.792 0.794
545 600 190 216 406 B 5:45-6:45 0.941 0.B90
600 615 160 188 368 9 6.00-700 0.902 0.926
615 630 196 186 382
630 645 172 179 351
6.45 7.00 159 143 302
PeakHour. 1053 1254 2307


Thursday, 3/30/06
Start Time EndTime Northbound Southbound
4:00 4:15 224 233
4:15 4:30 241 226
4:30 4:45 242 277
4:45 5:00 240 254
5:00 5:15 295 252
5:15 5:30 279 231
5:30 545 239 241
5:45 600 210 220
6:00 6.15 182 198
6:15 6,30 179 157
6:30 6:45 133 171
6:45 7:00 131 153
Peak Hour: 1056 1014


4 00-5 00
415-515
4.30-5.30
4:45-5:45
5:00-6:00
5:15-6:15
5:30-6:30
5:45-645
6:00-7:00


4:00-6:00 0.835 0873
4:15-6:15 0.817 0857
4.30-0:30 0.791 0.026
4:45-6.45 0.744 0.848
5:00-7'00 0.698 0.05


Thursday, 3130 6
Start Time End Time Northbound Southbound Both


400 4:15
4:15 4:30
4:30 4:45
4:45 5:00
5-00 5:15
5:15 5130
5:30 545
5:45 600
6:00 6.15
6:15 6.30
6:30 6:45
6:45 7:00
PeakHour:


584 1 400-5:00 0.958 0.833 4:00-6:00 0904 0.769
613 2 4:15-5:15 0.972 0.880 4:156:1 0878 0.736
709 3 430-5:30 0.939 0.044 4:30-6:30 0.841 0.691
599 4 4.45-5:45 0.912 0.792 4:45-6.45 0.787 0.642
678 5 510-6:00 0.894 0731 5:00-7'00 0737 0.597
567 6 5:15-6:15 0.31 0871
542 7 530-6:30 0.877 0791
476 8 5:45-6:45 0.798 0879
412 9 6.00-7:00 0.889 0.865
372
334
333
2599


Lowest 0.747 0.731
Highest 0.972 0.926
Average 0890 0842


Lowest
Highest
Average

Lowest
Highest
Average


0.743 0.778
0.978 0.963
0.873 0.879

0743 0.731
0.978 0.963
0.882 0.860


0.698 0.703
0.874 0.873
0.792 0.798

0.698 0.597
0.904 0.873
0.306 0.752


0.724 0.597
0.904 0.802
0.1 0.705












1.000


0.900


0.800 ,
0.800 --------------------------------
-- E of Univ, Tues, 10/5/04 -
---@ E of Univ, Wed. 10/6/04 I
---E of Univ, Thur, 10/7/04 -
4 W of Arl, Tues, 10/5/04 l
W of Arl, Wed, 10/6/04
-- W of Arl, Thur, 10/7/04 eak Hour Factor
-A-W of SSide, Tues, 10/5/04 Hur c
-W of SSide, Wed, 10/6/04
0.7C ---W of SSide, Thur, 10/7/04 -
4:00-5:00 4:15-5:15 4:30-5:30 4:45-5:45 5:00-6:00 5:15-6:15 5:30-6:30 5:45-6:45 6:00-7:00
Time Period


Figure B-5. Atlantic Boulevard PM peak hour factor, eastbound (outbound) flow












1.000


0.900


0.800









0.700


Figure B-6. Atlantic Boulevard PM peak period factor, eastbound (outbound) flow


4:00-6:00 4:15-6:15 4:30-6:30 4:45-6:45 5:00-7:00

Time Period















Table B-3. Atlantic Boulevard machine counts
East f Unversit Boulevard
Tuesday, 10/504 1-Hour PHF 2-Hour PPF
StartTime End Time Eastbound Westbound Both Period EB WB Period EB W E
400 415 406 290 606 1 400-500 0935 065 400-600 0932 0856
415 430 363 317 680 2 415-515 0967 0967 415-615 0956 0864
430 445 366 307 673 3 43305 0964 0866 430-630 0944 0851
4:45 5:00 384 009 693 4 445-545 0976 0893 445-645 0914 0825
5:00 5:15 388 293 681 5 500-600 0971 0883 500-700 0877 0810
5:15 5:30 358 369 727 6 515-615 0954 0898
5:30 5:45 384 347 731 7 530-630 0934 0889
545 600 377 295 672 8 545645 0877 0889
600 615 347 314 661 9 600-700 0875 0866
615 630 326 278 604
630 645 273 229 502
645 700 269 267 536
PeakHour 1514 1318 2832

Wednesday, 106/04
StartTime End Time Eastbound Westbound Both
400 415 359 285 644 1 400-500 0960 0922 400-6 0940 0 97
415 430 382 259 641 2 415-515 0977 0894 415-615 0948 0896
4:30 4:45 364 300 664 3 430-530 0971 0939 430630 0937 0892
4:45 5:00 389 314 703 4 445-545 0962 0946 445-645 0924 0857
5:00 5:15 385 339 724 5 500-600 0929 0940 500-700 0889 0825
5:15 5:30 373 320 693 6 515-615 0933 0951
530 545 350 310 660 7 530-630 0916 0923
545 600 323 306 629 8 545-645 0899 0850
600 615 383 281 664 9 600-70 0052 0857
6 15 630 348 248 596
630 645 323 206 529
645 700 252 228 480
Peak Hour 1511 1273 2784


Thursday, 10/704
StatTime End Time Eastbound Westbound Both
4 0 415 387 288 675
4 15 430 375 257 632
4:30 4:45 399 287 686
4:45 5:00 384 305 689
5:00 6:15 387 358 745
5:15 5:30 334 327 661
5 30 545 294 314 608
545 600 247 336 583
6 00 615 383 282 665
615 630 392 277 669
630 645 364 272 636
645 700 283 257 540
PeakHour 1504 1277 2781


1 400-500 0968 0932 400-600 0879 0863
2 415-515 0968 0843 415-615 0878 0861
3 430-530 0942 0692 430-630 0883 0 86
4 445-545 0904 0911 445-645 0888 0863
5 500600 0815 0932 500-700 0856 0846
6 5 15-6 15 0821 0937
7 530-630 0839 0900
8 545645 0884 0868
9 600-700 0907 0965


West ofArlington Road
Tuesday, 10/504 1-Hour PHF 2-Hour PPF
:artTime EndTime Eastbound Westbound Both Period EB WB Period EB WB StartTime
400 4 15 482 298 780 1 400-500 0 846 0 957 4 00-6 00 0 761 938 4:00
415 430 441 296 737 2 415-515 0815 0 945 415-615 0 750 0 934 4:15
430 445 407 311 718 3 430-5 30 0 849 0 945 4 30-6 30 0 739 0 928 4:30
445 500 302 286 588 4 445-545 0 708 0 937 445645 729 0 896 4:45
500 515 287 282 569 5 500-6 0 778 0 934 500-7 00 0734 0894 500
5:15 5:30 416 316 732 6 515-615 0 845 0 938 515
5:30 5:45 549 300 849 7 530-630 0835 0958 530
5:45 6:00 457 283 740 8 545-645 902 0 945 545
6:00 6:15 434 286 720 9 600-700 0 873 0 943 600
615 630 394 281 675 615
630 645 364 231 595 630
645 700 323 281 604 645
PeakHour 1856 1185 3041 Peak

Wednesday, 10/04
artTime EndTime Eastbound Westbound Both Start Time
400 4 15 459 272 731 1 400-5 00 0956 0921 400-600 0945 0 912 4:00
415 430 428 242 670 2 4 15-5 15 971 0 945 4 15-6 15 0 890 0 915 4:15
4:30 4:45 428 304 732 3 4 30 0 0 955 0 978 4 3060 894 0 915 4:30
4:45 5:00 441 302 743 4 445-545 0 967 0 943 445-645 0 888 0 879 4:45
5:00 5:15 450 305 755 5 500-6 O 951 0 929 500-700 857 845 500
5:15 5:30 468 313 781 6 515-615 0 912 0 910 515
530 545 452 261 713 7 530-6 30 0899 0939 530
545 600 411 284 695 8 545-645 877 0898 545
600 6 15 503 281 784 9 600-7 00 0 829 0 847 600
615 630 443 241 684 615
630 645 407 214 621 630
645 700 315 216 531 645
PeakHour 1787 1224 3011 Peak

Thursday, 10/704
:artTime EndTime Eastbound Westbound Both Start Time
400 4 15 446 293 739 1 4 00-500 0 949 0 958 4 00-6 00 0 B48 0 876 4:00
415 430 460 262 722 2 4 15-5 15 927 0 900 415-6 15 0 803 0 859 4:15
4:30 4:45 484 287 771 3 430-530 0 893 0 907 4 30-6 0 799 0 857 4:30
4:45 5:00 448 281 729 4 445-545 917 0 900 445-645 0 789 0 844 4:45
5:00 5:15 403 319 722 5 500-600 896 0 872 5 00700 0 768 0 829 500
5:15 5:30 393 270 663 6 515-615 0 748 938 515
530 545 399 278 677 7 5 3-6 30 772 0 925 530
545 600 250 246 496 8 545-645 0 793 0982 545
600 6 15 523 249 772 9 600-7 00 0 46 0979 600
615 630 444 256 700 615
630 645 443 255 698 630
645 700 360 243 603 645
PeakHour 1728 1157 2885 Peak


West of Southsde Boulevard
Tuesday, 10/504 1-Hour PHF 2-Hour PPF
EndTime Eastound Westbound Boh Penod EB WB Penod EB WE
4:15 375 325 700 1 400-500 0869 0937 400-600 0 67 0889
4:30 378 320 698 2 415-515 0825 0919 4 15-615 840 865
4:45 267 341 608 3 4 30-5 30 832 0905 4 30-630 837 0848
5:00 294 292 586 4 445-545 0864 0957 445-645 0859 0924
515 308 300 608 5 500600 0877 0949 5 00-700 865 906
530 373 302 675 6 515-615 0867 0917
545 314 262 576 7 530-630 0937 0956
600 313 282 595 8 545645 0951 0955
615 293 262 555 9 600-7 00 0951 095
630 335 272 607
645 333 261 594
700 313 247 560
Hour 1314 1278 2592

Wednesday, 106104
EndTime Eastbound Westbound Boh
4:15 383 302 685 1 400-500 0891 1 1400-600 04 0901
4:30 329 287 616 24 15-515 0960 0915 4 15-6 15 934889
4:45 338 346 684 3 430530 0938 092 430630 0929 0 78
5:00 315 326 641 4 445-545 0986 0939 445-645 0911 0876
515 316 307 623 5 50-600 0 975 0922 5 00-7 O 0 904 856
530 299 305 604 6 515-615 0920 0894
545 316 286 602 7 530-630 0932 0859
600 301 334 635 8 545645 0911 0840
615 342 270 612 9 600-700 0906 0978
630 316 257 573
645 287 261 548
700 295 268 563
Hour 1365 1261 2626

Thursday, 10/704
EndTime Eastbound Westbound Both
4:15 373 329 702 1 4 00-500 954 0958 4 00-6 00 0907 0927
4:30 355 312 667 2 4 15-55 0 983 0976 4 15-6 15 937 0939
4:45 343 321 664 3 4 30-5 30 0 957 0956 4 30-6 30 0 928 0 929
5:00 352 299 651 4 445-545 0952 0942 445-645 0922 0904
515 358 321 679 5 5 00 6 00 0896 0919 5 007 00 0 911 891
530 317 287 604 6 5 15-615 0 909 0955
545 336 303 639 7 5 30 0917 00955
600 272 269 541 8 545645 0911 0929
615 351 299 650 9 6007 00 0945 0926
630 328 287 615
645 328 256 584
700 320 266 586
Hour 1423 1261 2684


Lowest 0815 0843 0856 0810
Highest 0977 0967 0956 0897
Average 0922 0908 0910 0858

Lowest 0708 0840 0729 0810
Highest 0986 0982 0956 0939
Average 0.904 0924 0.872 0880


Lowest 0 708 0 847
Highest 0 971 0 982
Average O 871 0 932


0 729 0 829
8 945 0 938
0 813 0 888


Lowest 0825 0840 837 0848
Highest 0986 0978 0937 939
Average 0919 0932 0 893 095












1.000


0.900


0.800









0.700


4:00-5:00 4:15-5:15 4:30-5:30 4:45-5:45 5:00-6:00 5:15-6:15 5:30-6:30 5:45-6:45 6:00-7:00

Time Period


Figure B-7. University Blvd. PM peak hour factor, northbound (outbound) flow












1.000


--- --------------------------------------------------------------
Peak Period Factor












0.8( ---E of Univ, Wed. 10/6/04
-E-E of Univ, Thur, 10/7/04
W of Arl, Tues, 10/5/04

*- W of Arl, Wed, 10/6/04
W of Arl, Thur, 10/7/04
- -- - - - - - -







-A-W of SSide, Tues, 10/5/04
-- E of U niv, T hur, 10/7/04 -- - -- - -- - -- - -- % - -- - -
W of Arl, Tues, 10/5/04 1%%-4%%
-0 W of rl, W ed, 10/6/04 -- - - - - - - -A
M W of A rl, T hur, 10/7/04 -- - - - - - - - - -
-W of SSide, Wed, 10/6/04
W of SSide, W ed, 10/6/04 -- - - - - - - - - -
0.7( -W of SSide, Thur, 10/7/04
4:00-6:00 4:15-6:15 4:30-6:30 4:45-6:45 5:00-7:00

Time Period


Figure B-8. University Blvd. PM peak period factor, northbound (outbound) flow















Table B-4. University Boulevard machine counts (Jacksonville).


South of Los Santos Dnv
Tuesday, 8/19i03
ne EndTime Northbound Soutl
415 403
430 440
445 486
5:00 481
5:15 487
5:30 495
5:45 525 2
600 457
615 351
630 383
645 313
700 312
akHour 1988 1

Wednesday, 8/2003
ne EndTime Northbound Soutl
415 422
430 423
4:45 480 2
5:00 488 2
5:15 495 2
5:30 513 2
545 507
600 468
615 374
630 347
645 343
700 282
akHour 1976 1

Thursday, B821l3
ne End Time Northbound Soutl
415 414
430 443
4:45 479
5:00 455
5:15 496 2
5:30 476 2
545 498
600 434
615 407
630 350
645 327
700 303
ak Hour 1906 1


1-Hour PHF
Period NB SE
1 400500 0931 0900
2 415-515 0972 0957
3 430-530 0984 0986
4 445-545 0947 0959
5 500-600 0935 0941
6 515-615 0870 0909
7 530-30 0817 0956
8 545-645 0823 0922
9 60 -700 0887 0883


1 400-500 0929
2 415515 0953
3 430-530 0963
4 445-545 0976
5 500-600 0966
6 515-615 0907
7 530-630 0836
8 545-645 0818
9 600-700 0900








1 400-500 0935
2 415-515 0944
3 430-530 0961
4 445-545 0966
5 5 000- 0956
6 515-615 0911
7 530-630 0804
8 545-645 0874
9 600-700 0852


2-Hour PPF
Period NB SE
400600 0899 0917
415-615 0886 0924
430-630 0873 0925
445-645 0031 0096
500-700 0791 0858


400-600 0925 0913
4 156 0913 0913
430-630 0895 0918
445-645 0861 0929
500-700 0 11 0909













400-600 0927 0908
415-615 0926 0923
430-630 0902 0919
445-645 0864 0919
500-700 0826 0914


North of Arington Road
Tuesday, 8/19/03
StatTime EndTime Norbound Sou
400 415 316
4 15 430 393
430 445 398
4:45 5:00 451
5:00 5:15 444
5:15 5:30 456
5:30 5:45 486
545 600 410
6 0o 615 334
6 15 630 347
630 645 303
645 700 291
Peak Hour 1837


Wednesday, 8/20,03
StartTime End Time Norbround Southbound Both


4 00 415
415 430
430 445
4:45 5:00
5:00 5:15
5:15 5:30
5:30 5:45
545 600
6 00 615
615 630
630 645
645 700
Peak Hour


1-Hour PHF 2-Hour PPF
thbound Both Period NB SB Period NB E


316 1 400500 0864
393 2 415-515 0935
398 3 4 30-530 0959
451 4 445-545 0945
444 5 5 0-600 0924
456 6 515-615 0867
486 7 530-630 0811
410 8 545-645 0850
334 9 6 00-7 00 0919
347
303
291
1837


327 1 4 00-500 076
363 2 455515 0926
370 3 4 30-530 0929
423 4 445-545 0960
411 5 00-600 0960
443 6 515-615 0903
425 7 530-630 0862
422 8 545-645 0825
310 9 6 00-700 0863
309
352
244
0 1702


Thursday, 8/2103
StartTme End Time Norbrtound Southbound Both


400 415
415 430
430 445
445 500
5:00 5:15
5:15 5:30
5:30 5:45
5:45 6:00
6 00 6 15
615 630
630 645
645 700
Peak Hour


Lowest 0017 0803 0791 0858
Highest 094 0906 0927 0929
Average 0913 0937 0875 0913

Lowest 0781 0864 0750 0841
Highest 094 0906 0927 0939
Average 0.902 0 936 0.861 0 90


331 1 400-500 0925
366 2 415-515 0939
406 3 4 3 30-530 0962
400 4445-545 0964
425 5 5 00-600 0970
432 6 515-615 0930
440 7 5 30-630 0865
410 8 545-645 0851
3559 6 00-700 0865
317
314
242
0 1707

Lowest 0811
Highest 0970
Average 096


4 00600 063
4 15-615 0867
430-30 0855
445-645 0031
5 00-7 0 00790


400-600 0 89
4 156 0894
4 30-6 30 0878
445-645 0873
5 00-7 00 0023













400-600 0912
4 15-615 0919
4 30-6 30 0905
45-645 0879
5 00-7 00 0834


aouin oTiaywoo0 HOao
Tuesday, 8119/03
StartTime End Time Northbound Southbound Both


467 14 0[
629 2 4 1
574 3 4 30
650 4 4 45
652 5 500
675 6 5 1
711 7 53
570 8 54'
547 9 6 0
539
489
440
2688


Wednesday, 820.03
StartTime End Time Northbound Southbound Both


295 200
305 188
349 222
410 216
381 222
410 235
427 225
398 216
303 231
274 247
324 245
230 219
1628 898


495 1 4 00
493 2 4 1
571 3 4 3
626 4 4 4
603 5 5 0
645 6 5 1
652 7 5 3
614 8 545
534 9 6 00
521
569
449
2526


Thursday,8/21/3
StartTime End Time Northbound Southbound Both
400 415 293 209 502 1 4 0
415 430 314 214 528 2415
430 445 370 206 576 3 4 3
4:45 5:00 413 216 629 4 445
5:00 5:15 413 236 649 5 5 00
5:15 5:30 382 207 589 6 5 1
5:30 5:45 427 227 654 7 5 30
545 600 355 223 578 8 545
6 00 615 348 239 587 9 6 0
615 630 312 236 548
630 645 293 212 505
cr 1nn 1 "1 0


1-Hour PHF 2-Hour PPF
Penod NB SB Period NB S


k500 0853 0923 400-600
-5 15 0912 0960 4 15-6 15
-530 0922 0886 430-630 C
-545 0920 0094 45-6 45
-60 0895 0864 5 00-700
-6 15 0833 0874
-630 0781 0931
5645 0860 0944
-700 0896 0956


-5 00 0829 0930 4 00-600 871 0 917
-515 0 881 0955 415-615 0873 0934
-5 30 0945 0952 4306 30 0864 0918
-545 0953 0955 4 45-6 45 857 0930
-600 0946 09555 00-700 0 804 0931
-6 15 0900 0965
-630 0821 0930
-645 0816 0950
70 0873 0953








-500 0841 0978 4 00-600 0869 0921
5 15 0914 0924 4 15-6 15 885 0925
-530 0955 0916 4 30630 0884 0936
-545 0957 0939 445-6 45 0862 0939
-60 00 923 0946 5 00-7 00 0805 0930
-615 0885 0937
-630 0644 0960
-645 0921 0952
-7 00 0843 0926


Lowest 0701 0064 0750 0041
Highest 0957 0970 0805 0939
Average 0886 0936 840 0903












1.000


0.900 If --- -





080-------------- ------- ---------------------- -----------------



0.800

-*--Tues, 4/4/06
-- Wed, 4/5/06 ------
--Thur, 4/6/06 Peak Hour Factor




0.700
4:00-5:00 4:15-5:15 4:30-5:30 4:45-5:45 5:00-6:00 5:15-6:15 5:30-6:30 5:45-6:45 6:00-7:00

Time Period


Figure B-9. SR A1A S. PM peak hour factor, southbound (outbound) flow































- --- - -

---Tues, 4/4/06
-&-Wed, 4/5/06
----Thur, 4/6/06


4:00-6:00


4:15-6:15


Peak Period Factor


4:30-6:30


Time Period


Figure B-10. SR A1A S. PM peak period factor, southbound (outbound) flow


1.000


0.900


0.800








0.700


4:45-6:45


5:00-7:00











Table B-5. SR A1A machine counts (Crescent Beach)


North of Riverside Boulevard
Tuesday, 4/4/06
Start Time End Time Southbound Northbound
4.00 4.15 230 149
4.15 4.30 187 161
4:30 4:45 201 151
4:45 5:00 180 166
5:00 5:15 225 158
5:15 5:30 198 154
530 5645 195 154
545 600 154 146
6.00 6.15 148 122
615 630 152 151
630 645 123 134
6.45 7.00 125 118
Peak Hour: 804 629

Wednesday, 4/5/06
Start Time End Time Southbound Northbound
400 4'15 181 195
415 430 175 177
430 4'45 194 171
4:45 5:00 198 159
5:00 5:15 223 171
5:15 5:30 217 167
5:30 5:45 210 163
5.45 6.00 169 161
6 00 615 176 149
6.15 6.30 153 132
630 6'45 162 151
6.45 7.00 140 128
Peak Hour: 848 660


Start Time
400
4:15
4:30
4:45
5:00
5'15
5.30
545
6.00
6'15


Thursday, 4/6/06
End Time Southbound Northbound
415 181 200
4:30 181 188
4:45 192 162
5:00 190 191
5:15 227 171
530 198 153
5.45 187 174
600 182 158
6.15 190 150
6'30 163 137


6.30 6.45
645 700
Peak Hour:


Both
379
348
352
346
383
352
349
300
270
303
257
243
1433


Both
376
352
365
357
394
384
373
330
325
285
313
268
1508


Both
381
369
354
381
398
351
361
340
340
300
286
247
1502


1-Hour
Period
1 4:00-5:00
2 4:15-5:15
3 4:30-5:30
4 4:45-5:45
5 5:00-6:00
6 5:15-6:15
7 5:30-6:30
8 5:45-6:45
9 6:00-7:00


1 4:00-5:00
2 4:15-5:15
3 4:30-5:30
4 4:45-5:45
5 5:00-6:00
6 5:15-6:15
7 5:30-6:30
8 5:45-6:45
9 6:00-7:00








1 4:00-5:00
2 4:155:15
3 4:30-5:30
4 4:45-5:45
5 5:00-6:00
6 5:15-6:15
7 5:30-6:30
8 5:45-6:45
9 6:00-7:00


SB
0.867
0.881
0.893
0.887
0.858
0.878
0.832
0.937
0.901


0.944
0.886
0.933
0.951
0.918
0.889
0.843
0.938
0.896








0.969
0.870
0.889
0.883
0.874
0.956
0.950
0.895
0.820


NB
0.944
0.958
0.947
0.952
0.968
0.935
0.930
0.916
0.869


0.900
0.958
0.977
0.965
0.968
0.958
0.928
0.921
0.927








0.926
0.932
0.886
0.902
0.943
0.912
0.889
0.927
0.917


Lowest 0.820 0.869
Highest 0.969 0.977
Average 0.898 0.932


2-Hour PPF
Period SB NB
4.00-6:00 0.853 0.933
4:15-6:15 0.827 0.913
4,30-6:30 0 807 0.905
4:45-6:45 0.764 0.892
5 00-7:00 0 733 0.900


4'00-6:00 0 878 0.874
4.15-6:15 0876 0.931
4'30-6:30 0 863 0.931
4:45-6:45 0845 0.916
5.00-7:00 0813 0.893













400-6:00 0847 0.873
4:15-6:15 0852 0.882
4:30-6:30 0 942 0.848
4'45-6:45 0816 0.834
5:00-7:00 0.780 0.866


0.733 0.834
0 878 0.933
0 826 0893












1.000
1.000 -----------------------------------------------------------
















---Tues, 4/24/01, Marlin
0.800 Wed, 4/25/01, Marlin Peak Hour Factor------
-- Thur, 4/26/01, Marlin -
4- Tues, 4/24/01, Corona
**- Wed, 4/25/01, Corona
U Thur, 4/26/01, Corona
-A-Tues, 4/24/01, PGA
--Wed, 4/25/01, PGA
-0-Thur, 4/26/01, PGA
0.700~. --------------------------------------------------











4:00-5:00 4:15-5:15 4:30-5:30 4:45-5:45 5:00-6:00 5:15-6:15 5:30-6:30 5:45-6:45 6:00-7:00
Time Period
---- ---- ---- ---- ---- ---








Tues, 4/24/01, C orona - - - - - - -




T hu r, 4/2 6/0 1 P G A- - - - - - - - -




Time Period


Figure B-11 SR A1A N. PM peak hour factor, southbound (outbound) flow












1.000


0.900


-Tues, 4/24/01, Marlin
--Wed, 4/25/01, Marlin
0.800 --Thur, 4/26/01, Marlin
4- Tues, 4/24/01, Corona
Wed, 4/25/01, Corona
*- Thur, 4/26/01, Corona
-A-Tues, 4/24/01, PGA
--Wed, 4/25/01, PGA
-0-Thur, 4/26/01, PGA


0.700


Figure B-12. SR A1A N. PM peak period factor, southbound (outbound) flow


4:00-6:00 4:15-6:15 4:30-6:30 4:45-6:45 5:00-7:00

Time Period























Table B-6. SR A1A machine counts (Ponte Vedra) PDF 17 KB


North of Marlin Avenue
Tuesday, 42401
ne End Time Southbound Northbound Both
415 497 533 1030
430 573 481 1054
445 600 450 1050
500 590 502 1092
5:15 594 600 1194
5:30 598 582 1180
5:45 592 539 1131
6:00 593 504 1097
615 551 446 997
630 509 423 932
645 562 384 946
700 485 387 872
ak Hour 2377 2225 4602

Wednesday, 4/2501
ne EndTime Southbound Northbound Both
415 542 469 1011
430 567 410 905
445 513 456 969
500 553 458 1011
5:15 563 532 1095
5:30 567 505 1072
5:45 553 507 1060
6:00 531 507 1038
615 575 434 1009
630 542 459 1001
645 577 358 935
700 445 292 737
ak Hour 2214 2051 4265

Thursday, 4/2601
ne End Time Southbound Northbound Both
415 475 551 1026
430 600 488 1088
445 543 497 1040
500 563 492 1055
5:15 559 561 1120
5:30 590 603 1193
5:45 617 528 1145
6:00 586 533 1119
615 557 474 1031
630 518 487 1005
645 521 424 945
700 451 387 838
a Hour 2352 2225 4577


1-Hour PHF
Period SB NB
1 400500 0933 0922
2 415-515 0972 0847
3 430-530 0903 0089
4 445545 0992 0926
5 500-600 0994 0927
6 515-615 0976 0 90
7 530-630 0946 0B7
8 545645 0934 0872
9 600-700 0937 0919


1 400-500 0959
2 415-515 0968
3 430530 0968
4 445545 0986
5 500-600 0976
6 515-615 0960
7 30-630 0957
8 545-645 0964
9 600-700 0927


1 400-500 0909
2 415-515 0944
3 430530 0956
4 445-545 0944
5 500-600 0953
6 515-615 0952
7 530-630 0923
8 545-645 0931
9 600-700 0919


2-Hour PPF
Period SB NB
400-600 0955 0873
415-615 0966 0855
430-630 0953 0543
445645 0959 0829
500700 0937 0805


400-600 0968 0905
415-615 0961 0097
430630 0956 0906
44564 50966 0883
500-700 0943 0644


400-600 0910 0002
415-615 0935 0866
430630 0918 0865
445-645 0914 0050
500-700 0091 0029


Northof Corona Road
Tuesday, 4,2401
tartTime EndTime Soubound Nco
400 415 381
415 430 436
430 445 415
4:45 5:00 445
5:00 5:15 386
5:15 5:30 419
5:30 5:45 405
545 600 446
6 00 6 15 368
615 630 341
630 645 347
645 700 289
Peak Hour 1655


1-Hour PHF
rthbound Both Period SR N
362 743 1 400-500 0942 08
340 776 2 415-515 0945 09
369 704 3 4 30-530 0935 9
424 869 4 4455 45 0930 9
408 794 5 500-600 0928 0 9
379 798 6 515-615 0918 08
398 803 7 5 30-6 30 074 00
340 786 8 545-645 0842 08
314 682 9 6 00-700 0914 09
294 635
273 620
256 545
1609 3264


Wednesday, 42501
rTime EndTime Southbound Northbound Both
100 415 398 298 696
115 430 397 323 720
130 4 45 387 344 731
:45 5:00 388 395 783
:00 5:15 422 356 778
i:15 5:30 409 377 786
i:30 5:45 364 407 771
545 600 428 324 752
00 6 15 388 351 739
i15 630 370 269 639
i30 645 332 249 581
i45 700 303 249 552
PeakHour 1583 1535 3118

Thursday, 4/2601
rtTime EndTime Soutbound Northbound Both
100 415 394 371 765
15 4 30 390 383 773
130 445 433 335 768
1:45 5:00 365 460 825
i:00 5:15 424 368 792
i:15 5:30 401 408 80
i:30 5:45 428 420 848
445 600 417 355 772
00 6 15 367 358 725
15 630 348 317 665
i30 645 335 287 622
i45 700 316 262 578
Peak Hour 1618 1656 3274


1 400-500 0986 08
S4 15-515 0944 08
3 430l530 0951 9
1 445545 0938 0 9
5 500-600 0940 03
6 515-615 0928 0
7 5 30-6 30 0905 8
8 545-645 0887 08
9 6 00-7 00 089 07








1 400-500 0913 068
2 415-515 0931 08
3 430-530 0937 08
4 445-545 0945 09
5 500-600 0975 09
6 515-615 0942 09
7 530-630 0911 0 8
8 545-645 0879 09
9 600-700 0931 08


2-Hour PPF
3 Period SB NB
81 400-600 0934 0890
09 415-615 0930 0 876
32 430-630 0904 0063
49 4456 45 0885 834
34 500-700 0841 0816
99
45
98


61 400-600 0 933 03867
97 415-615 030 0 84
32 4 30-630 0 922 867
43 4 456 45 0906 0 838
99 500-700 0801 0793
96
30
50
96








42 400-600 0939 0042
40 415-615 0931 0839
54 4 30-6 30 0 919 0821
00 445-6 45 0901 0 08
23 500-700 0807 0826
17
63
20
55


North of PA Tour Bouleard
Tuesday, 424/01
rTime End Time Soutound NorthcboL
100 415 342 327
415 430 351 263
130 445 326 335
1:45 5:00 334 350
5:00 5:15 377 347
5:15 5:30 357 326
5:30 5:45 335 366
545 600 340 298
300 615 353 281
3 5 630 302 281
330 645 284 245
345 700 250 229
PeakHour 1403 1389


Wednesday, 4/2501
StarTime End Time Souobound Northbound Both


1-Hour PHF 2-Hour
id Both Period SB NB Penod
669 1 400-500 0964 0911 400-600 1
614 2 4 15-5 15 0920 0925 4 15-6 15 [
661 3 430-530 0924 0970 430-630 [
684 4 445-545 0930 0949 4 45-6 45 1
724 5 500-6 00 0934 0913 500-700 1
683 6 5 15-615 0970 0868
701 7 5 30-630 0942 0037
63 8 545645 0906 0927
634 9 6007 00 0842 0922
583
529
479
2792


400 415 326 250 576 1 4 0
415 430 333 322 655 2 4
430 445 300 278 578 3 4 3
445 500 338 304 642 4 44E
5:00 5:15 325 301 626 5 5 0C
5:15 5:30 337 299 636 6 5 15
5:30 5:45 306 330 636 7 5 3
5:45 6:00 340 315 655 8 545
600 615 315 266 581 96 0
615 630 272 232 504
630 645 282 241 523
645 700 234 223 457
PeakHour 1308 1245 2553

Thursday, 4/26/01
StartTime End Time Souotbound Northbound Both


400 415
415 430
4 30 4 45
445 500
5:00 5:15
5:15 5:30
5:30 5:45
5:45 6:00
600 615
615 630
630 645
645 700


677 1 4 OC
657 2 415
691 3 4 3
651 4 4 4
710 5 5 0
682 6 515
677 7 5 30
678 8 5 4
637 9 6 00
561
532
535


HeakHour 1329 1418 2747


Lowest 0909 0840 0891 0805
Highest 0994 0964 0960 0906
Average 0954 0909 0943 0862

Lowest 0842 0 796 0841 0793
Highest 0994 0972 0968 0915
Average 0.937 0901 0.924 0856


Lowest 0842 0796 0841 0793
Highest 0986 0 949 0939 0 390
Average 0925 0 888 0909 0 844


Lowest 0842 0816 0861 0803
Highest 0974 0972 0950 0915
Average 0933 0905 0 920 0862


PPF
SB NB
916 0 892
919 0876
903 0003
889 0852
861 0810


S958 0 909
0954 0915
1931 0881
S925 0 867
6 06 0 836


1943 0001
1 51 0862
1 930 0 850
0924 0021
1 903 0 003


-500 0959 0896 4 00-61
3-5 15 0959 0936 4 15-6
-5 30 0962 0972 4306:
-545 0966 0935 4456.
-600 0962 0943 500-7
-6 15 0954 0917
-630 0907 0866
-645 0889 0837
-7 00 0875 0904


-500 0954 0941 4 00-6
-5 15 0 950 0874 4 15-6
-5 30 0963 0878 430-6
-545 0974 0076 445-6.
5-600 0952 0914 500-7
-6 15 0971 0906
-630 0918 0876
-645 0876 0816
-7 00 0056 0929


Sta
4
4
4
4
5
5
5


Sta
4
4
4
4
5

5
5
6
6









Table B-7. Appendix B data summary.


Peak
Direction


Atlantic Boulevard
> 0.70 US 1 North
US 1 North
> .University Boulevard
US 1 South
University Boulevard
Atlantic Boulevard
University Boulevard
> 0.80 SR A1A
Atlantic Boulevard
SR A1A
SR A1A
SR A1A


West of Arlington Road
North of Estrella Avenue
South of Lewis Speedway
South of Baywood Road
North of SR 206
North of Arlington Road
East of University Boulevard
South of Los Santos Drive
North of Riverside Boulevard
West of Southside Boulevard
North of Corona Road
North of PGA Tour Boulevard
North of Marlin Avenue


Eastbound
Northbound
Northbound
Northbound
Southbound
Northbound
Eastbound
Northbound
Southbound
Eastbound
Southbound
Southbound
Southbound


3-Day
Minimum
PHF


U. 81
0.784
0.811
0.815
0.817
0.820
0.825
0.842
0.842
0.909


3-Day 3-Day
Average Maximum
PHF PHF

0.871 0.971
0.873 0.978
0.890 0.972
0.886 0.957
0.885 0.982
0.906 0.970
0.922 0.977
0.913 0.984
0.898 0.969
0.919 0.986
0.925 0.986
0.933 0.974
0.954 0.994


Overall Minimum
Overall Average
Overall Maximum


0.71
0.80


0.87
0.91


0.96
0.98


0.91 0.95 0.99









APPENDIX C
GENERALIZED CYCLE-PERIOD DELAY EXAMPLE:



















Table C-1. Generalized example of cycle-period delay discrepancies data.


A
rCv1 Lin.gl


]' C D E F 0 H I J l
-------------- iefondi --


Pva1r6rwen P4nrn

AcrualArrimal & 6 paplurerThIrupih f (vph)
ActualArrivals & DXparren-hrupuF l(@15rln]
P.Iodd 2


,, 1. ,- p, 9 r 2,,,,

ifArrivil. O uOpaeturi.'s. .Qaueui



T-
4


7
2




11
12

13



14


18
12

19


26






21


91
34

28




7q


40
41
42
431
4
42
4n
4






95


12
32

64
















26
41


























88


92art1 n P1ll R




5I0 s 60



&22 862
130 160






0




2 0 2
1 0


I 0 1
2 6 7
2 0 2



2 0 2







S 0 6
2 0 2















4 7 4













40 4 4

10 6
10 6 5













10 6 41








12 68 2



13 10 4
132 16


K L M N 0


P* R S T U V W X Y Z AA AB AC AD
r---d-T-- ll


120
20 144 2
S1 MI
01 I II
130 1












0 0
0& D
D D







G 2 D

G U
0 D



3 0
G 4 4



0 4 2
043 A 2
04 4 4
S440





S541
a 4 6
2 A 4






642
S4 2
1 1 7





C I







I 6
4 7




















6 A 2
C 4 9
I 6

6













. ** 7


iw Holf Gr,"n


1wo 14D

572 B ll

310 ~







1242 12
A 6 26

4 0


5 a
7 I I 6
7 1 8 I



B I I 4
5 1 I 6



710




11







7 2
3 2 1
I 1 1





4, 2
2 7 1





2 12



513
5 2


































,o 5
4 4




















S 2 3
8 7 4














11

















C. .




c I 3
8 2 6


AE


-- -


SiWn OnFLA Rd

IBB
1I5 143 I


13D 15e





a o
I u I


2 0 2
S u0
10
101
1 o
I 0
20

2 0
20

30


303
3 0







1 n
ii U
5 0 4
4 0
S U 5
S5 0







































9 1 1 6
8 U1
60

6 0
07




a o




2 0 U







12 6



8 11 2 8




813 6 6
G 6 8
1 2 1
128
1124
12s
71 1 1
71 11 2





71 1 1
7 12 6 10


AJ rK AL AM AN Ao AP AO AR AS AT
id2 seond-


S-r 0. FUt C we

rph 272 -




10




S 0 D
a 1 1 u
0 1 1 2
7 1 D 9
1 1
S2 2
7 2 2 2



B 3 3 2




3 4 4 2
0 4 4
01 4 4
6 4 4


7 2 2



7 6 6 9
II

6 5
6I
76t


$Wan- HalfCiTaen

uph "90 P
10 146

Wh 570
130 J53
310 a f I

& D n




0 1 1

0 1 I
G 1 I












4 a- 1

4
1 3
li 3
0 i 4









S 3 1

3 1 4
7 2 2













G 3 D4
3 3 2
3 3
33

3 3















7 3
















10 1 8
4 I















41 3
41 3
4 3 2















1 3
12 3 3
I 3
S .3

63 12
S 4 4
6 74













1 2 12

10 3











Il 24 8


AF AG AN Al

Sort on FI Red

wh nO ?No
1i0 135

w 1h 2 I
310 1








& D 2


0 0 3
1 D
I I
1
1
10 1








3 2 3
2 2D 3
2





3 3
3 3
S3

04
4 2 4





6 6
5 20




6 2 6














0 9 9
6 2D




















12 1 12
7 2D











1 0 13







19 U 13
13 2 12

13 2 12
13 2 12
1? n
20 T
130 1:
1 0 13
13 03^
13 13
14 U1^


S. t On Full O err

vph ?T-H
170 54B

vph





S1 1 9


0 1 1 2
u 0





2 2 2
o0110




G 2 2 2
3 2 2 2
S61 1 0

0 4 2 U
1 3 3 2





7 4 4
7 4 4








G 2


G 7
0 a


(G G 6
1770















S 1 11 2
71 8 6 2























13 11 2

12 1 2
14 11
2 1 11 I
HB 1 3


ph 272M
17n 11 2


130 158
310

A 6 6




1 1 1
0 1

0 1 0
1 0

G 1 2
S 2 2 0

G 2 2 0
G 2
G 2 2 0


G 3 0




6440
G 4 4 0
S 4 4
G 4 4 0




6 46 0










0 5 4
G 5 D







1 5 3
15 5

















1 5 7
10 0
61
6 6 1

















125
1 15
6 23

6 3
9 3

9 5 4
91 4
9 1 4


10 6 A

12 5 5









12 6 7


13 5 6


















Table C-1. Continued

A=Arrivals, D=Departures, Q=Queue: A
Time (sec)


14








90
7-? 14
73 15


77 I5
78 16

1 016

bli 7s
1 16
















Il IQ
P6 17
















Z4



50 3
e8 27
18


























:42 40
04 1


R4? 2
Ig


S20







II8 ,. 221
12 20

104 C'21
05 231


21 ^ 21






22l 24

25 25
15 23









37 26
V2 214
34: 27,
128 **











325 27
15 25



31 26





432 286
43 28


35 27
47 25
35 30
32 28

t43 03


3S7 29

43a 30

451 30


D Q A D Q


AD Q AD Q


ADQ ADQ A DQ ADQ A DQ


4 5 3
14 S q
14 0 l
,4 5 j





I-FL Ia
*13
15 1
15 5 1O







2 18




4 19
o





3
















I4 18
,


















14
a













9U 4

27 10 ~1
?i


?3 8







22 8
22 8

23 0 4
23 10
23 18 5












33 88
fl I8 4
'4 10 4
Y4 IS 4
?4 18 5
24 81
24 18

?5 16
25 18 7
25 8I
26 5 1

26 18
26 18
?7 a
2'1 I13

27 18

28 10 10
28 18 10
28 I8 10
28 18 10
2H 10 10
28 I8 i

29 18 11
29 18 11
30 IV 11
00l 10 17
8U 10 1'1
30 18 12


14 0 14
14 14
14 0 14
14 0 14
1 0 15
16 1 11
1 0 1"
15 1 B,



1 11
16 IB

l7 0 7B












2' 1 5






25,1 17
17 0 I

1?

17 17
lB lB






E8 17
1 0 11 B
2D 0 lD
1G 0 1



2D 0 2B
19 10 3





7 0 7
2D 0 2
1 0 2D
?D U 21
20 U 41



2t U 21
2t Cl t,
2t 8

22 1 20
22 2 2
2 2 213

2~3 20


5 IH

24 1 2
fl4 2D
24 5 HI

24 6 IB
25 5 Il
25 9 1?
25 90 lB

2B 10 lB
?B 1 1 I
2B &

2B 2 14
2U J 14
fl I0 14
37 4 13
27 4 13


23 6 12
2B l@ 2
'B 7 11

38 7 II
28 98 1
2)9 I I t
29 t0 ID
29 3Q 1B
30 10
3. 21
3D 2'2 9


14 5 8
14 a
14 5 9
14 5
15 5 9

1 5
15 10

12 5
16 0 1
5 10






















2 0
I 1 21
1 6 S1


17 5 2
11 5 21
1, 11


























IS I
1 12



18 5 3
18 5 i2
19 5 32
18 4
18 5 13
20 5 3



20 5 S3
0 5 S3
9 5 S3



40 $ 4
21 5 1 S
29 5 14
20 5 S4



21 5 15
21 5 16

21 5 S6
*31 5 ^


23 5 16
22 5 17
22 5 S7
32 S ^
22 1,I
23 5 17
73 5 S7
23 5 S8

23 5 HS
234 5 S8
234 5 19
24 5 19
24 $ 10
25 5
23 5 20
25 5 20
25 20

235 5 20

236 5 ?1
26 5 21
26 5 21
21 $ 21

21
21 6 21
U8 1 21
28 8 20
28 8 20

28 9 9
28 9 8
33 gQ S3

29 1 16
29 1 S8
30 12 I8
130 13 7

30 E3 S7


















Table C-1. Continued

A=Arrivals, D=Departures, Q=Queue: A
Time (sec)

152 30
163 31


1661
166 31



166 36


157 1 61



1733
1 14






17[1 39





189 30
186 G 2
166 0 6

167 G 3


















6@8 39
162 5 3










121 g 3
17 0 15


17 3






156 41
168 37




116 16



917 41
19B 64



131 41







14 6 47
217 43
661 5 0





223 0 4

265 5 41


66B 5 46




229 5 46
266 5 46
46
5 46





666 5 41
234 40



23G 47


D Q A D Q


AD Q AD Q


ADQ ADQ A DQ ADQ A DQ


30 I1 20
21 11 ?0
31 11 20
31 11 o0
31 I1 20


32 6t 21



33 12 ?1

33 13 20
33 t3 20
34 64 60
34 14 19
34 6 19
32 15 19
35 IB 18

35 17 17
3S6 IB 17
35 IB 17
36 19 16
36 29 16
36 20 16
36 720 16
36 61 16


37 26 15
37 27 16
37 23 4
38 24 14
36 64 14
3B 25 13
3B 26 13
38 26 13
39 25 12
39 27 12
39 ?? 16
39 ?B 11
39 26 11
40 29 11
40 29 11
410 3B 1

40 31 10


41 67
41 33

42 34 B
42 56 6
42 35 ?

42 SB ?
43 36

43 37
43 5B 6
44 B56
44 3B
44 3B
44 3B


45 SB 7
46 SB ?
46 SB ?

46 38 6
46 36 6
46 3B6
46 36
41 36 6

47 36 6


















Table C-1. Continued

A=Arrivals, D=Departures, Q=Queue:
Time (sec)

235
23?

239
24?
241

243
244
245
24?
24t
24-
254B
249



214
25?
25?
27


269
251


264
239
255



273
271

2753
27T
2??









281
22,


285
?B?
2?S




27B

291
292


293

297


302


310
3?7
301


314
315
31B


317
291
2S234

239
23?E5
23?




230
















2??^


AD Q AD Q


AD Q


AD Q AD Q AD Q


A DQ ADQ A DQ

















Table C-1. Continued

A=Arrivals, D=Departures, Q=Queue:
Time (sec)
31B
81,
321
322

325
32E
32?
3 E

33?

333


33?

334
347
34B
344
34,
34T
34
3 4

34B

36B
354




36M


364

36?


3713

374
37m


37&
379
381

304

385
360
3sI


397



395
394


407


AD Q AD Q


AD Q


AD Q AD Q


AD Q A D Q AD Q A D Q


















Table C-1. Continued

A=Arrivals, D=Departures, Q=Queue:
Time (sec)

401
402?
402
404
405
400
405
400
410
411
410

414
410
4125
417
418
420
421
420
420
494
411
425




43B



440
441
42

443






4M0
445


441
447
443

445


440
447
44?

410
451
450
46
454

407
45B


481
462


464
48;
40



471

47 '
47'1


477"
47B
479



402
483
48


AD Q AD Q AD Q


AD Q AD Q


AD Q A D Q AD Q A D Q


G 00






B2
GR9





R3
B3
B3
01






B3
01

02

E3



0M
B3
03



Bfi
06
104



R5

B5
Be
00


k6
00
00
00
E



g2
B/

07
B7
B7

R1

BS
BS
09
00


BS

B0
00


BO
s0
90
90






G 91
G 92
00
0 0&




G 12
G 92

G 53
S00






0 13
G 01
G &o
0 4'




0 0
0 03
Q 5S
0094




G @6
G 04
0 00



PB
97



















Table C-1. Continued

A=Arrivals, D=Departures, Q=Queue:
Time (sec)

484

400
407
48B
491

492
494
493

498

497
409
491
400
801
501
503
204
05

5OO
507



$17
018
514

810


510



520
524



5B

530
$ 1


534
$S8


358

540
541
82
064




547


83
361
3^2

$86
B65


AD Q AD Q


EC 07
C 07
C 97
C 09
C SB
C 88

C 98
: SB0
C. yo
C 88


C 181
C 188
C 100

C 100
100
100
100

100
ltil

100
107

102
107
003
10$

100
10B
















1807






C 100

4. 000
100
100
C 110
S110



C 100
C 10t



C 100
C 100
C 110



C 1100
C 110
C 110
C 110
107
112
116
113
113
117
113


AD Q AD Q


ADQ ADQ A DQ ADQ A DQ


S97

097
90
97
B0
918

B0



00
00

500
10




0 10
010

01o
012


207

02
007


103

t73
037
002
003

074
084
004
034


057
001




070

~05
075
070
007


t01


107






007
012

07
103






11


12


12


07
97
97

6 00
ID I?
0 90
6 800



0 88
0 8

I88
o Do
6 18
G 10

0 10t


G 1BD


e 107
6 100

6 100
G 100





G 13
0 100
G 101
6 101



S 192
S 18
e 102






S 1IN



6 103


e 10
G 180
6 180



6 100
G 108
0 108




G 100




0 188
6 190

100






















103



















Table C-1. Continued

A=Arrivals, D=Departures, Q=Queue:
Time (sec)

567
558
575
571
571

574
575

578
585
lB

581
582
SUE
584
5850
5B?
58B


591
5982
591
595

597
69T
a1-
eUD
601
BO?
603

B55
607
609
810
811
Bl ?
613
6514
515
51B
157
511
Bl0
,18

512
62B
521

57
632
52



B 7
62


87
6S I


524



551
64:t
2 5


635
8 1
e3 ^
54 i
e4 @

582 0
845 0
644
@4 i

585
647
84 e
e4 T
58 0
541 0


AD Q AD Q


AD Q AD Q


ADQ ADQ A DQ ADQ A DQ


113

114
114
141


116
11
115
15
115






118
115



115



Is
417
1f
o 17

l?











C, Ill
118

G 51


121

0 1l
Oi 171
C, I1ll

0 22
G I23



G 2
G t17
0 417
OA 1717
O 1521
O 2$1

OA 1217
C, 122
,S 421
G n??
Q5 hn







O 74
O t74
0 *24



G 23
OL 174
0 175



C, 125

O 25
S 1754
G ?4
G 524
Q5 74


G 125
G 125
G 175





0 t7
G 72
O 178
G I 2
O 17
OE 121


G 27
G I2I
G 497

O5 178
O t 5




O 50
0 I?

O 173
(S US


C, 125
G 28
G 199
G 25

O||| 175
G 125


113

114
114
114



114

11l
11l

115
11l
116



lIB
118




11B

117
117
117
117
117


112
11B
11B
11B
119
119
119
125
150
171
17I
120



1257
12$


177
127
1253
174
124
174
15
155





125
1;2
13?







123
1574





157
175
178
174
125
121






1751
155
175
126
178
15B







128
178
157
6 315
177
177

177
12S
1?8

1JB
1?9
129
129
179

1SD



















Table C-1. Continued

A=Arrivals, D=Departures, Q=Queue:
Time (sec)

5B
B6 ?
67 3
8562 1
654
65
67 5
5B
6B
556

6,
666
G S

B4 B
665


G U
566

Eil
B671
677
87 5
674
675
eG

67
87
681
6807
fGi
B84


8ee
591
G9A




G94
695

B55


69s
7GG
7UI


194

7oS
7o7




?14
7But
717
700
721











724
072









72B
7129



713
7141
715
71B
17
715
71B

721
23
776 6






725


727
72B
728
701
732


AD Q AD Q AD Q AD Q AD Q


130


131



132
31
121
SO

133
317


134
135

133
13




135
1 8


134
340




166
1357
357




136



141
197
179
146
146

1616
136

163
137
137
143
SA




141


lAG
14
141



145
139





13






S141

140
140
4 11
1 41



- 411
6 142

2 1432



2 143





G 16






1* W


109 2"
1M 24
196 22
1D9 22

1I4 22
199 13
1D9 23
169 73
166 23


109 24
1I9 24

D9 ?4
19 24
169 25


109 35
166 26
196 76





1D9 21


166 21
139

109
169 26
106 23
1D 29

166 29
1B9 29
169 29
166 26
139 79
11D9 30
119 30
1D3 30

1DO 31

118 31
1D9 31
1D9 31
ill 31

11 30
11 3Q
172 30


114 29
114 28
115 28
116 71
176 71
17 27

118 26
116 78
166 26

171 26

121 24
162 24
122 23
121 24
172 74

122 24


ADQ A DQ ADQ A DQ



















Table C-1. Continued

A=Arrivals, D=Departures, Q=Queue:

Time (sec)

733





740
741
7473


7145
747
747
746

7U51
797






77 ?

Z;l3
795







79?











767







79 3
84


















02


714
37










74 7
74






766
79?












671
67
67?
61


AD Q AD Q


AD Q AD Q AD Q


147

147
147


142
143


149
149

119
719







Ill
150







761
150

















154
71



151
172





152
513
$16











1562





I2
156
154








3 7e9

3 761
2 171














3; 76


AD Q A D Q AD Q A D Q


147
14
147

147
147
146

746


149
44

149
49


154










S 157













167
0 1
191















S 151

151
167















151
158
182
152
152
734
153

753


3 154

o 54
0 169
G 155












G 186

G 116



7 19









7 61
7 61







162
163

163


47 7 108

747 117 3?

4g3 177 46
747 178 34
749 1 18 46
546 179 43


745 174 47





7.4 109 417
796 108 41
058 1 08 0




94In 12inn
l8 108 0
756 108 43






0 51 178 47
'57 114 4M



G 955 111 47

S 72 113 2
O a 3 1 4
7 793 113 4
7 73 114 4?
G 4 119 34


0 94 116 2?
G 9 117 3
S795 118 27
17 119 37
2 11 I



7 92 21 36
532 131 21
G 757 122 36
2 7 123 36
73 1 23 34


737 134 34
7 53 9 29 35

73 a 126 37

0 794 177 27
G 767 127 3?
7 79 728 37
7 4 729 37
7 9 179 32
G 76 139 37

73 767 737 3?


7 767 133 ?
7 761 132 3?
S767 13 33 7?




767 135 7


761 135 7?


147 119 32
147 17 36
147 174 33
147 1179 7
147 119 29
146 II9 29


14: II9 3I
146 119 79
14 119 1 9
146 119 D

146 119 39
146 779 37

146 119 S4
146 119 31
149 119 37
156 119 37
117 179 31

16D 119 71
150 119 32
16 II79 3

150 119 73
156 119 43
163 779 33


151 174 33
15 119 343
163 19 43
162 179 34
156 779 34
163 179 36
114 779 36
164 119 39
154 779 32
154 179 35
114 179 36
156 119 39
154 779 36
165 179 36
166 179 37
116 174 37
116 174 37
16B 179 37



197 179 36
167 779 39
16? 179 34

156 179 39

136 179 49
169 119 39
169 779 39
166 179 41
197 179 41
169 779 41
169 179 43
197 119 42
167 179 41

197 179 41

161 179 42
193 191 47









163 121 42

Ei9 11 02


47 113 3
143 113 ,3
147 113 64
147 113 34

140 11 34
148 11 69
148 113 4
148 113 35
146 113 69
149 113
149 113

149 113 37
154 113 63
159 113 36










10 113 3
150 113 37

151 123 37
151 113
751 13 37
1 115 4
12 114 369

152 113 39
763 13 67

713 113 69


763 114 47
167 113 40
194 13 40

14 113 4
1 113 41
154 773 43

155 113 47

115 113 43

115 113 47
16 113 43
16 119 42
156 13 43




156 116 47



15 11 42
168 114 47
158 116 41



15 727 65
159 177 69




9 72J 31 4
710 131 77

161 122 37

76 123 369
762 121 33




762 724 34


S136 1 5

763 1379 4

763 129 34



















Table C-1. Continued

A=Arrivals, D=Departures, Q=Queue: A
Time (sec)

1B 163
811 186
816 164
820 164
621 164

624 155
726 1685
825 165



67B 16
31B 1665
166

831 167
7 16H1
G b7

615 117
831 18 1
B37 167


647 169
641 168





846 115
666 115
646 177
851 175
652 175
651 111
649 171
855 171

856 172

86B 171
561 171
862 172


6 112
685 171



73E 173

688 114

671 174
B72 174
BB 17









171l
615 115
67 173
871 175



666 It



171
667 1 176



B93 G 177
865 1 11
8671 177


666 16 178

6 6 G 116


BOB G 176
694 16 176

g*5 1 16
677 17 1




6s6 16 1


D Q A D Q


130 33
1B
132 34
13 64
136 34
136 33
132 33

132 31
132 32
134 32
134 32
136 31
135 31
135 31
135 32
131 30
132 35
13B 29
139 29

146 27
14 78
140 28
146 16
146 25
146 79
14D 29
146 39
14B 21

146 1"
146 3?
14 3'0
14 301
146 301
146 31
144 31
146 33
14b 31
146 31
14D 32
14G 33
146 32
14 3
14 3
14B 33
14 3
14 3
146 3
14 34
146 15
146 3
146 1

14 34
14 35
14B 36




147 B2





143
144 35
145 35
146 36




14t 3




147 16
147 35
142 3S

143 34

144 34
14& 33







14 31


AD Q AD Q AD Q


I63 132 31
161 132 31
64 16' 1'
664 11 3
64 132 32
664 132 32
W66 132 32
B65 132 33
665 132 33
63 112 33
I6 132 33
M66 132 34
M 123 24
65 132 34
6S6 133 34
66 123 J2
661 134 31
6B6 134 37
67 135 32
61 135 32
667 136 32
IBM 139 92
66 127 21
66 137 31
66 138 31


666 130 35
66 149 30
666 145 27

618 140 27
66 142 26
73 142 26
5t] 142 26
171 142 2

671 142 27
611 142 29
671 142 29
171 142 29
17I 142 350
73 142 30
" 142 315
17 142 20

672 142 30

673 142 31
73 142 31

61 147 31
73 142 31
673 147 91
174 142 32
174 142 32
14 142 32
174 149 3
374 142 32
675 142 33
765 142 33
675 147 34
675 142 33
675 142 33
611 142 35
671 142 34
18 142 34
678 144 24
%78 142 34
677 142 35
679 142 36
66 1462 14

677 142 35
T78 149 3S



178 143 95
179 144 35
I73 144 35
58 14 34
7t 45 34

68E 146 34


S63 130 33

684 130 3
684 1605 4
164 130 34
64 130 34
485 130 24
665 130 35
175 130 31

666 11 12
665 10 35
185 130 32
tS5 130 3B

4 'I I s

66? I ll 16
67 132 35
611 132 31
661 133 35
686 133 35
86 164 34
688 134 34
68 135 34
686 136 33
66 136 33
655 137 32
686 197 62
16 138 32
1 ,13 11

17T 139 3t
176 140 31
676 140 3D
1l1 141 23
611 141 7E
I11 147 76
671 142 29
1 I 143 2
172 143 29

617 143 1
T2 143 97
612 143 26
61 143 2
?13 143 3B
T15 141 31
673 143 36
674 143 37
614 143 32
114 1413 3
114 143 3
1574 143 3t1
174 143 3t
675 143 32
15 143 37
I6S 141 92
616 1416 13
675 143 32
16 143 33
175 143 33
616 14 3 33
61 143 3S
616 143 34
177 143 34

t11 143 34
711 143 4
676 143 37
16& 143 35
61 143 35
15 1413 31
T78 143 35
176 143 32
671 143 3B
t1S 143 36
66 143 3B
615 143 3B
tSO 143 37


AD Q A D Q AD Q A D Q


IB3 135 28
163 11 ?'
164 116 '9

1I6 135 29
14 1 1S 23
165 13 29
175 176 59
155 135 30
165 16 30


I6 12 30
16 135 31
1B7 135 31
151 176 31
156 145 31
1B7 15 321
1B7 16 311
17 145 32

1E7 135 32
IB1 195 34
1a 1S5 34
171 16 33
176 1735 3
176 175 33
IB1 1&3 33





171 1 1
179 135 34
166 176 14
166 165 314
115 163 34
16 113 23
17 176 35
170 135 35
170 135 35
171 163 35
171 14S 3S
11 125 31


SIll 135 36

S 177 132 38
1 72 137 35
1 1I3 13 35
S Il3 136 31
6 173 116 14
1 73 176 34
I 173 140 34

S 174 141 33
5 114 141 133
1 174 147 33
S 174 142 32
1 175 143 32
1 11 143 327
S 115 144 31
G 11 144 111
S1175 145 31
3 16 146 302
: 176 146 30
G iB 148 913
1 116 146 25
1 178 147 39

1 7 148 23
I11f 148 319

I7l 146 26




1I 148 31



179 148 31


















Table C-1. Continued

A=Arrivals, D=Departures, Q=Queue:
Time (sec)




903
904
9054
907


901
911
913
9146

911
9H I


9H
91
92
92
9H7
H2
92
9H
97
9H3
9U
93

9H
9: t


9H
98B


9S
930
4 &


95 !
946
941
96
49
95S
96B
9h
96
97



971

a/
9 34
971
97
9
9g
981
9HU


9H4
H9 S
aHUH
961


AD Q AD Q AD Q AD Q AD Q


C ,, ,
C ... ,,

C ,,, ,


C .

c ,, ,,
c i,,


c *
C. ,


H1 134 H
c Ei l .3



1U 134 3D
13 154 3E
183 154 23
184 14U 3H

184 134 S 3
1U4 14 1E,
184 1 3B


105 154 3t
18 154 31
1H5 1U 31
USE 134 32
la5 I0q 31

E1B 154 -'1
185 154 31
1E5 14 31
EU 154 3U
1 E 154 36

1BE 154 3E




1 B 15 5 3
HBE 154 33

HlE 154 34

1H 151 34
18 156 35
18B 15
S1a 15$ 34
C18 13 34
lEE 154 3I
r 1] 1& 1

C 1E9 134 3I
C Eg 155 34
C 1EH E1 34

C4 19E 1EU UE
C 1E 1EU 32
C D 1ER 3U
C E 1H 3
1C HE 1E 1H
4 1. ED E 31
C 1HE El 31





C 1H E 13
C 1HE 1U 3
C 1H 1UH 3D
I 15H 164 BR


ADQ A DQ ADQ A DQ


S 81









0 81
IDI


591
I81
Egl

C 81
IDI
141


0 82
tS3


0 83
ID?



0 83









6 att
0 83






Q 183
G 184
(5 194










S 81
0 8ta
G 584
S 84

0 8a


S 8
(5 8





9 583
G 194


G Wg5





09 89
0 88


G ig5
G tAS
G 187




6 81

B aa
G 486






G t88
0 88
B 81
G ig?

G IB?






0 48
P 188

0 88M
E US

























( 88
G tus
G 98
I;~















G 198
oE IRE






















G IDa
o9 143























I age


191
191

591
18H
1aE
13 E
o ES
C, I 1
U IH
13 14


















Table C-1. Continued

A=Arrivals, D=Departures, Q=Queue:
Time (sec)
989



99w
995








984
169

998
















ISW
991


990
9914
91018








97
1901



199





ID22
1023
101
107







ID2B








DI0
1013
1m?



101S

117




1043
1024
imB





I D11S
12B
198
I3Q

1i8
1939








104
134



1078
100
194

1MS


1B3
I1M4


119G


1911
193
10855


AD Q AD Q


AD Q AD Q AD Q


AD Q A D Q AD Q A D Q



















Table C-1. Continued

A=Arrivals, D=Departures, Q=Queue: A
Time (sec)

[S IN5204
187 '04
1 0 6 r 2 0 4



1011 0
1 DU G 205

1073 205
1074 I05
1015 05
L 0 2Q0
1 M 5 305

ID9 G 206

108


D84
085




I 1 I
1089


1091 i









i1[ 2 4
9ite3 *






i[34 ,, :






1m0 210
S09
l i l i








:I 11
01 74-1
107 i

I 09 -210

1 20
I 12 411

114 211
I113 12 1


I t5 712





I J26 213
I2G 213
I19B 213


130 "213
1 32 214
IJ3 214
1i34 '14
i 2 G 214
I 3B 214
1 37 214
I38 214



I 43 215

I 114 215
1l4l 218
I 2 7 216


D Q AD Q AD Q AD Q A D Q


204

3204
ll4
204
2*&
285
05


20B8



2DB
iC 2I15
C 201

C 20B
C 2f?0



I. 207

C 217




I.11
C 20B
C 20I



C 2D

C 21W
C 2AB




2 l1
21 B

21
21D
21

213
21
211

211
211


215
211


212
215
211?
212
212
21 ?
21 S
213
312
213

213




213 .

S21
4 14

214$




C 215



4 21B


2ul

2 4
4 34
254
2D5
20S
2D7






2D5








2R1









210
251
211

















211
?BB

2B79







2K7








212
210
2W

















213
i 209








212@
2 H








217
214











2E7
JI$
215
i RT7

















21E
3%
- 2iS















\ 1ia
3 2W9






















2 11
: m
S 215
3 7ti 9
- 210
:> 21 0
: 21 0
3 71 0
:* 11





21
51 ?
21 3
-1 ?

5 1 ?
Gl 3
23
23
?ll
13



21

214

214



215
215
Ti





21 G


AD Q A D Q AD Q A D Q


204
"704
*3Q4
204
205
206












3 21191
205
*305

205

210
206














2111
v 20611


G 207

GL 207
G 711
2107
0 307
G 207
G 21308
S 308

213
0 700

G 201
G 208





214
S 109
















214



214


214

21$
215
215
2151




214

215
Qi 209

G 2QS


















G 2111

210
210


















718
210

21



21




12


21
2
21


911
24
21

24
714






21



216


















Table C-1. Continued

A=Arrivals, D=Departures, Q=Queue: A
Time (sec)
I4B 216
1149 216
I 80 ?18B
1151 16
152 216

I 5S '17
I 56 '





1 50 21

I 21 1 4
I162 1 i
I 63 218
I 64 21B
11BB 11
1 67 ., i
1168 *
I69 219
1 113 219
171 219
I16 2219
I1?3 219
i14 720


It?7 .. i

1 i ll ,I
91 6 .,

I123 2 21
184 221


I 88 222
I 69 222
I 91 2 2

183 2 22
I3J4 1 22

96 5 223
1697 222
I292 223
l 5m 223

1202 2234


126 224



1209 -6
1211
1212 22
1212
1?14
1215 .
121.
?1B ,
1219
12"2 228
1I21 226

224 222)
1 4 227
I 5 227
1226 227
32"


1230 .


D Q A D Q


C 21B
C 216

C 216
C 218
C 211

C I17
C 212
,C 217
,. 211

C 216

C 2IB

218


21B
218











2182
2B8


215
268

218



226

228
22G


22B


227
221
221
212


221


2221



2t


222
'28



2234

C223


S22E






1 226



C 22i
C 226

4 22S
1- 8
C 2 5


AD Q AD Q AD Q


216
18

218
218
216
216
216
i- t
21 f


21
i 7t

217
?l f

21
*i :tt
7219
218
26

'16
7t B

2 1 9

218




'16
321





2219
5 216
2 218


, 220
26 226


3 221
2 226
.8 2216

S 221
3221
S221




$ ??-
223

- 223






22
3 112




S 222
: 222
: 226


3 2N



224
224
'24


224

224

226
i 2263
2 2




















i228,
i 2).
22
232
223

226










226
6214
222
'21









226
226








223
225
325


325
3267

226
23S

^2

227
^221

22


AD Q A D Q AD Q A D Q


6 21
718
218





S216
G 219





G 219
G 218







221g
Ci 219
G 21U


0 912
G 912
S 219
G 219
G 202
G 720
Go 219

G 220
G 220





222
0 220
222





222
221
223
221
QE 73?



G 221
224
224

224
3??

222
22
222

226
722
222
223
223
723
223
226
223



22,
2254
224


224




228
922
2?4
22S
225
225
?35S
*3?5
225
227
226

226
2256
^??7
322
257
227
227


^?ai
226


















Table C-1. Continued

A=Arrivals, D=Departures, Q=Queue: A
Time (sec)
1231 228

12834 I
1235 2
1232 .2

12.80 I-
1239 2
1240 I


1 244 230
1285 2 30
1246 230

1282 ,4 21
1250 231
: 1 2310
152 231

12 3 231


I G0 232
129 23 2
12S0 '
181

1264
12 '
1285
1267 -!

1111 1128
1271 -'
12711
1273

1278

IMIS
1280 .

1383 ,235
1284 2*







124
1287 G

1238 236

1292


1286
1297

1288 I
1300
13228 238

12;1114 2 238
122 730 22
SB G 239
10 288 2 239
129 G 239
1310 G ,,
311 I 24

1313 G 240


D Q A D Q


228
'118
2281
228
2118
9228
228
2118
228
288
2181


228

232
228
230

2?3


218
2211

2381
2311
2381
?St
23

232
211


2322
232
232
2 2811
S 131

C. 282
C 233
C 232
S2338
S- a
C *22


C 228
4. 234
8 2331
C 2334
C 22
C 224
C 2384
C 2281
C 2328
C 235

220

2328
235

2281

2328
2116




2331
2321
2321
2811
237
211
232
2388
228
2388
28181
233



248
2481
233
23B
239


23S
230


AD Q AD Q AD Q


224
I28
229
1228
228
229




228
229
228



,i
219
279
729
231
2231



2301
232
la


.3
28 '110
2 231
2 2311

:2 231
3 221

2 232
8 2311
3 T3
5 232

2 222
28 282
28 222
S 232








222
'- US?

. 2238

3 222
S22333
S 2331
2. 224J
5 733
3 130
28 234

22
23M
234
236
281
2)81
'28
235



235
228
236
2S

"SBB






11818
237
3?SI
237
237

237
238
221
223
2811


228

229
239
23
239
?h0
71180
248


AD Q A D Q AD Q A D Q


24 ,228
226
2 332
*330
228
G 229
CE229
G 229
G 229

G 229
G ??9



24 221
G 230
G 220
2 230

G 31
G 232
12311
S 23
G 231
G 231
2231
*334
6 731
G 231
G 232

G 232

21731










2628
S232










2118
232
232
2331
233
233
I

234
22341
234
834
224
3134
234
335
238

235
235
730
2335
336
236
236

?BB
237
237
237
337
7237
337
237
23U
238
336
238

238

239
239
239
,il

240



















Table C-1. Continued

A=Arrivals, D=Departures, Q=Queue: A
Time (sec)

1314 240
1315 14 0
1318 1 40

1318 G240
319 5 '1
1231 241
1322 1
1S324 *

I B '42


1 329 242
1330 412


1333 .4 !

I35 243
100 243
1367 243
123301

1239 243

1340 244



13541 240
1043 244









354
13 t *, *

349
1350 245
1 ,240





12"4 248


103 24?
1%4 74
3@00 .8


1301 47
1S4 2471

136 5 N24O
1I0B 240

139 0241
130 248
1371 248
13?3 7U
1 33 "4, a
1314 0 40
1375 249

1307 249


130 N"49
13aO N0

1382 I

1084 I
109 .




1001
1392 251



1398@ 252


D Q A D Q


24
CNI
240
4
240
240
41
C 24t




C4 234
L 24t


C. 04
C 24


C 243
C 3



244
C N4?
c 242
C 243
C 24


243
N O
244




244
4 5
245

245


245



NIE
24 8
248



245
NB





248
24 E,
N3





24B
24 0
045



247
3240



24B










C 205

C 000
L 255
4. 050
C. 200
C 5001
C 250
I. 0
C 251
4 251
C 251
C 050

4. 252


AD Q AD Q AD Q


240
'40
040
240
240
241
041
I 41
041
041
241

0140
7/41
242
SS;



242
0 342

S 242
-> 74
-; 343
O 243
St43
: 243
G 43
2 743
S744
5 '43

3 244
3 244
? W4
O 244
$ '44
, 344
0 744
2 45


- 245
3 044
5 044


345
246
346
3 B
246
048
JAS
t48
246
040

748
248

347
247
047
248
740

240
248
348
'48
349

348
245


048
249

049
3249
249
040
205
350
250
7&0
200
080
300

081
051
251
'01
001
051
202


AD Q A D Q AD Q A D Q


















Table C-1. Continued

A=Arrivals, D=Departures, Q=Queue:
A D Q
Time (sec)
1099 252 230 252
1095 253 320 22
1400 752 '30 2?
401 253 230
1402 253 230 23
1403 253 230 23
14004 253 '30 22

1407 ?53 230
140 253 220 23
0141 254 230 4
410 254 230 24

1412 25 230 04
1413 214 230 24
1414 254 230 24
1410 254 20 23

141 ?25 320 20


142G 255 20 31
121 C 2?56 S2 1
1420 255 2332 4


142U $5 331 3
141' C 0 222 25
1424 C250 222 "24

1425 25 2 34 22

179 250 235 25
1420 200 2136 22
142 250 220 25


143 ?7 20307 3
1434 2157 3? $1


1438 8 ?S 39 8 9
1439 250 240 101
147 ?258 240 1a
1433 258 240 0L
1441 250 240 19

1441 200 240 18
1444 250 240 19

110 05 40 41
1441 20 '20 L1

1444 200 240 10
145B 200 240 10

1460 20 240 210
1453 260 24D 20
1464 26U 24 2U0
1450 260 240 20
14ii 70 740 71?
1402 200 240 20


1400 001 245 01
1401 261 240 21
146 2601 240 21
1457 760 240 72
14n7 71 74in ?1






1400 262 240 22
14fin In1 4 n ii1



141 2061 240 221
141 62 461 4D 221
1462 262 24D 22


14A3 002 214 ?2
14f6 7 12 2 10 7

1470U BA 2413 '1
1471 262 240 2
14fG 109 240 73




1466 262 24D 22
I 40A in 241 n 12


A D Q


ADQ A DQ ADQ A DQ


A DQ A D Q


252


252
254

253
20 5










2S 2
0 200

- 20
S 254
S 254





S205


G 25
S256
0 255
Q B
3 250
G 250
0 207
1 5 25



0 20



e E
3 257

0 200



S 259
3 25



G E59
3 209
3 20
3 20
3 25B




3 250




3 200





325
320
3 21
S20




3 260


A D Q





















Table C-1. Continued

A=Arrivals, D=Departures, Q=Queue:

A D Q A D Q A D Q


Time (sec)

1480
481
1482
1483
1484
1485
1486
487
1488S
1489
140
14 1
142
1493
1494
1495
1498S
1497
14B8
1499
150
501
582
153
1584
505
158
157
1503
159
510
1511

512
513
514
515

1517
1518
1519
1520
1521
1522
1523
524
1525
1526
1529
1528
1529

1530
531
532
1533
1534
1535
153B
1537
1538
1539
1540
541
1542
1543
1544
1545
154 B
1547
1548
1549
1550
1551
152
1553
1554
1555
15586
1557
1558
1559
1560
561


S26 24 24
S26 241 23
G 26 241 23
G 264 242 23
G 26 242 22
G 26 243 22
G 265 243 22
G 265 244 21
G 265 244 21
G 265 2456 21
G 265 245 20
G 265 246 20
G 26B 246 20
G 266 247 19
G 266 247 19
G 26B 248 10
G 266 248 19
G 266 249 18
G 26B 249 17
267 258 17
267 258 17
267 25 17
267 258 17
267 25 17
267 258 17
267 260 17
268 250 18
268 250 18
268 260 10
268 250 19
268 258 18
268 260 10
268 258 18
269 250 19
269 250 19
269 26058 19
269 250 19
269 258 19
269 260 19
269 250 19
270 258 20
270 250 20
270 258 20
270 250 20
270 260 20
270 258 20
270 250 20
271 260 21
271 250 21
271 250 21
271 258 21
271 260 21
271 250 21
271 258 21
227 256 22
272 258 22
272 250 22
272 250 22
272 258 22
272 250 22
272 258 22
G 273 251 22
G 273 251 22
G 273 252 21
G 273 262 21
G 273 23 21
G 273 253 20
G 273 254 20
G 274 254 20
G 274 255 19
G 274 255 19
G 274 256 19
G 274 256 19
G 274 257 10
G 274 267 17
G 275 250 17
G 275 258 17
G 275 259 1G
G 275 269 16
G 275 210 '1
G 25 260 15
275 268 15
276 268 16


264
264
264
264
264
264
265
265
257
265
265
265
266
26B
265
266
26B
2H6
O26
267
267
267
G 267
2G7
G 267
G 27
G 267
G 28
G 268
G 26B
G 26B
G 268
G 268
G 269




















2740
G 29
G 29






















27.
G 269
G 279
G 269
G 2.9
G 270
270
270
270
270
270
270
270
271
271
271
271
271
271
272
272
272
272
272
272
272
272
274
274
274
274
274
271
274
274
274
274
274
274
274
2 7 4 5

275
275'




G 275
G 275


264
264
264
264
S264
264
2 66
265
2 65
2 '5
265

G 266
G 2
G 266

G '266
G 266
G266
G 266
G 267
S28
G 27
G 27
G 27
G267
G 267
G 268
7 2
G 268
G 28
S27
268























& 7
2C
26S
26S


267
269
2 9
26S
26S
278

2 '76
2G 8
278
270
278
278
271

271
271
271
272
271

2712

272
271
273



273
273
273
273
274
274
G27
274
274
275
275
G 275
G 274
G 274
G 275
G 275


G 275


G276


AD Q AD Q AD Q AD Q AD Q


2B4 24
264 27
2B4 27
2B4 247
264 27
284 247
265 247
265 247
265 249
265 27
265 247
265 247
288 247
2BB 247
266 247
288 247
2BB 247
266 247
28 24'
257 247
267 247
287 24
267 247
267 27
2B7 247
268 247
2BB 247
2BB 247
260 247
29 247
2B9 247
260 247
268 247
2 47
269 247

2B9 24
269 247
279 247


270 24
270 247
269 2,
270 247
270 247
270 247
27 27
27 24
27 27
270 247
27 24
271 247
27 27
271 247
271 27
27 247
27 27
272 24
272 28

272 2

272 24'
272 249



272 2'6
272 250
273 21





27 21


27 262
273 283
27 25S






273 254
27 2 4




2 5




275 256
274 2556
254 2-6
2764 2E
274 257
2 74 257
27 5 280
275 258
275 259
275, 2H
275 2 '0
275, 260
275 211
S78 28_


264
264
2264

o 264
S264
G 264
G 266
G 265
G 265
G 265
G 265
G 265
G 266
G 257

G 267
G 267
G 267
G 268
G 26
G 268
G 267
G 267
G 267
G 267
G 267
G 268

G 27


S266
G 268
G 298









26S
26S


271




271
269
271
267
278
276



275
278
27
278
27M

273
27
27
27
272
270
270
27N
270
271







27
276
272
272
272
272





273






271
274
278
278
276
278
278
274
275
274
G 274
G 274


A D Q



















Table C-1. Continued

A=Arrivals, D=Departures, Q=Queue:
A D Q A D Q A D Q A D Q A D Q A D Q A D Q A D Q A D Q
Time (sec)

53276 26 G 7 256 2 G 76 25 17 76 26 6 G 7 54 2 G 276 258 8 2 243 33 2 254 2 276 240 7
564 276 260 1 G 27 256 20 G 76 25 17 76 260 6 2G 67 254 2 76 25 1 27 243 3 27 5 22 276 24 2
5 85 2 6 0 6 6 2 776 26 0 7 25 76 7 76 2 16 G 27 1 6 259 7 27 6 243 33 27 254 22 276 24 2
156 27 260 1 G 27 257 9 G 76 2 16 1 76 26 6 G 276 255 1 G 2762 7 26 276 243 33 27 254 22 276 243 2
57 276 260 1 G 7 2 8 9 G 76 21 6 7 2260 76 G '127 256 1 G 276 22 7 27 2 7 2644 1 277 254 23 276 24 2
563 276 260 16 G 276 2 S13 G 276 26 5 76 260 6 G 2 56 2 G 2B6 260 1 G 276 244 32 276 254 23 276 243 20
59 277 260 17 G 277 21 277 262 1 277 26 7 G 7 57 2 G 277 261 277 245 32 277 254 23 277 246 2
570 27 7 260 17 G 277 2' 8 267 15 '77 20 1 G 27 5 2 2 G 27 261 16 277 245 32 277 254 23 277 24 28
1 277 26 17 G 77 2 7 77 26 2 5 77 256 1 G 277 256 1 G 277 282 5 277 246 31 27 254 23 277 24 29
572 2 7 260 1 G 27 26 7 7 1 7 28 G 27 258 G 27 22 5 27 24 312 277 254 23 277 24 29
573 2 7 260 17 G 277 261 7 278 262 15 277 260 G 27 5 19 G 7 263 15 27 247 31 276 254 23 277 24 28
54 277 260 17 G 7 7 2 5 277 2 7 G 277 259 277 2 15 27 247 3 27 254 24 77 24 29
55 277 260 17 G 777 26 15 77 20 7 G 7 6 8 26 5 27 7 24 3 277 254 24 277 248
157 6 27 8 267 22 78 22 7 278 20 8 G 27 2 278 2 5 27 243 3 2 2 54 24 278 24 28
157 27 80 26601 G 278 26 5 2 126 G 27 261 17 278 263 15 27 24 2 2 27 254 24 278 24 29
5 9 28 260 1 G 78 263 5 278 22 18 7 20 28 26 1 278 263 1 5 27 242 9 2 27 25 24 278 24 0
579 278 260 1 G 76 24 5 78 26 78 260 8 G 27 26 28 5 2 250 26 275 254 24 278 24 30
1580 28 60 8 G 78 24 20 260 27 262 27 2 28 62 6 28 263 6 28 250 2 2 2 8 254 24 278 248 30
581 2 8 260 1 278 24 4 278 2 1 27 2 C0 s G 2 266 15 28 8 251 23 27 254 25 278 243 30
2512 260 19 79 264 5 79 26 7 1 79 20 19 G 29 63 1 279 263 6 2 2 25 4 25 279 240 30
53 279 260 9 79 264 15 27 22 7 27 2 60 1 G 9 26 2 286 29 252 27 29 254 25 7 249 0
594 29261 260 I G 272 4 2 295 28 6 279 252 27 279 254 25 278 243 0
585 279 260 9 279 264 5 79 262 7 79 260 19 G 89 5 263 16 279 253 26 279 254 25 279 24 31
520 27 269 19 27 64 2 78 7 7 26 1 G 2 5 9 26 2 2 26 26 2C 27 25 25 78 243 31
157 27 9 260 19 279 264 5 27 262 7 79 260 5 19 26 7 29 254 22 26 279 254 25 27 24 31
58 279 60 19 279 264 5 79 26 7 79 2 9 279 26 1 2 2 7 279 254 253 279 254 2 279 24 1
159 2 26 1 28 26 8 28 2 02 2 2 26 28 266 14 280 263 1 7 2 255 25 2 25 22 238 243 2 1
590 280 260 2 24 2 22 2 20 2 2 14 2 23 17 228 255 25 28 254 2 29 24 31
591 280 260 20 290 264 56 2 20 6 22 39 280 260 20 200 265 4 260 263 7 200 256 24 280 254 21 20 24 31
592 280 0 2 26 20 264 0 26 20 2 60 20 80 26 15 280 26 17 20 5 24 2 0 254 26 2 0 245 2
593 280 260 20 280 28 280 280 260 20 280 26 7 1 280 257 24 280 254 26 280 248 3 2
594 280 26 0 32 264 9 6 30 6 13 2 3 2 6 2 28 2651 280 2 8 2 22 268 2 54 2 G 23 249 31
595 280 260 20 280 264 6 1 28 15 2 45 23 280 254 27 G 0 24 1
596 281 0 2 26 2 2 7 22 2G 2 1 2o 21 28 265 1 281 263 18 21 259 23 281 254 7 G 21 25 31
1597 281 260 2 21 264 7 6 1 26 9 2 260 2 8 5 2 266 2 263 18 21 259 22 21 254 7 G 1 25 36
1 59 21 60 2 1 21 264 7 1 26 9 8 26 1 65 21 266 8 2 281 259 2 228 2 54 27 G 21 256 0
1599 28 20 21 231 264 7 31 2 19 2 21 26 15 28 26 8 21 21 21 254 27 G 231 25 0
21 600 281 260 15 285 26 9 6 '8 6 19 6 20 21 8 266 281 26 2 285 270 21 285 254 27 G 281 256 29
117 281 6 20 21 281 264 7 8 1 26 9 2 2 1 28 266 1 281 26 9 25 201 21 25 254 27 G 281 25 29
62 G 281 21 6 231 264 7 31 22 69 21 26 1 266 19 28 21 2 21 254 2 3 G 231 25 3
03 G 282 62 2 264 2 2 0 2 267 2 8 265 2 28 26 2 28 62 2 28 254 25 G 32 25 1
64 G 282 62 0 232 264 3 832 26 260 15 28 2 9 2 2 20 2 2654 23 G2 282 36
605 G 28 63 19 291 2 27 5 260 27 28 265 1 286 26 9 22 203 1 2 2 54 25 G 29 25 27
166 G 2 28 2 1 28 2 86 0 82 2 2 28 26 286 263 19 28 23 19 28 254 2 G 28 25 7
607 G 28 264 1 32 24 3 320 72 29 2I 20 2 2 2665 1 2 23 9 0 2 04 1 201 254 23 G 232 25 27
10' G 282 264 52 282 264 2 .2 22 20 202 2 22 282 65 282 263 0 2 24 18 22 254 28 G 282 253 20
609 G 22 65 13 202 264 1 2g2 286 20 202 260 22 202 265 16 202 263 20 9 282 265 18 202 254 29 G 22 254 20
610 G 28 265 3 2 179 328 2 26 21 2 267 2 28 2665 17 28 263 0 2 5 18 2 2 6 254 29 G 283 25 26
611 G 28 267 17 23 274 9 23 22 20 28 260 23 28 267 0 28 1 7 28 2 2 2 254 29 G 23 27 5
612 G 202 2B6 17 233 264 9 G 233 262 0 203 260 23 202 265 13 2 3 26 0 203 26 17 203 254 29 G 233 25 25
613 G 28 67 3 24 9 G 3 2 9 8 21 2 8 6 17 282 27 0 79 27 7 16 292 254 29 G 23 251 25
614 G 2 67 16 26 9 3 26 19 8 2 2 2 28 265 18 28 2 1 28 277 16 28 254 29 G 283 259 24
615 G 283 263 1 233 264 9 G 233 26 19 1 3 262 2 23 265 1 283 263 1 23 63 1 23 254 29 G 233 25 24
616 G 28 3 264 17 283 264 13 G 283 26 2 21 282 28 283 265 17 283 263 21 283 263 1 283 254 30 G 283 260 24
1617 G 26 9 269 17 203 244 20 G 214 2,6 29 203 260 23 284 2665 1 263 263 1 22- 269 17 203 254 30 G 24 250 23
61 G 28 269 24 2G4 0 G 24 26 5 263 21 280 2G5 19 2 26 1 G 22 2G9 12 2 254 30 G 2 4 25 23
619 G 283 270 1 284 24 0 G 2834 267 G 284 264 22 283 265 19 283 263 1 G 28 270 1o 283 254 30 G 234 259 22
20 G 28 70 1 234 264 9 G 234 26 7 G 203 264 22 284 265 10 28 263 1 G 23 279 1 23 254 39 G 234 259 22
1621 28 3 70 G1 284 2G5 0 G 2 4 268 G 254 265 21 283 268 263 222 25 270 14 G 203 254 30 G 2 4 260 22
6122 284 70 2 G 284 265 9 G 284 268 18 81 265 2 284 265 19 284 2 0 21 G 28 24 2 29 G 234 26 21
23 26 70 1 G 24 266 29 G 284 269 10 2 4 26 16 284 2665 19 264 263 21 284 2. '1 284 255 29 G 294 23 21
6124 284 270 G 235 26 19 G 235 269 G 214 264 20 24 2695 19 285 263 27 15 G 24 256 29 G 235 261 21
6125 285 2700 14 204 287 20 204 27 17 205 267 4 8 2 8 265 20 285 263 22 285 270 1 G 205 256 25 G 325 2o 20
1626 2885 270 10 284 2 12 G 826 17 285 267 1 26 65 02 G 284 263 22 28 270 1 G 28 217 23 G 25 26 2
627 28 70 14 55 G 284 208 13 G 235 271 15 G 20 265 18 284 265 1 G 285 26 13 235 270 1 G 20 2 7 23 G 235 263 2
621 28 2761 G 2g5 28 17 G 528 271 16 8 265 17 28 255 19 254 2 20 270 14 G 2 218 27 G 25 26 19
629 26 70 G 35 269 7 G o 5 27 G 8 269 7 6 65 0 G 26 265 2 1 285 270 1 G 28 256 27 G 25 26 9
630 28 7 G 35 2 35 272 1 2 26 1 2 2695 6 G 28 263 22 235 270 1 G 23 256 27 G 235 24
631 28 270 G 285 27 18 G 26 27 15 G 26 270 16 285 26G 20 2 5 263 20 205 270 16 G 205 2 5 28 G 286 2G 15
1632 286 270 15 G 238 270 18 326 27 15 G 28 270 16 288 265 21 G 286 263 20 285 270 15 G 285 260 28 G 236 26 8
633 286 270 G 235 271 15 236 271 15 G 2 271 5 8 B5 1 G 28B 267 29 205 270 16 G 205 2BO 20 G 236 260 27
634 28 270 1 G 296 260 15 629 271 14 G 2 271 5 28 25 21 G 28G 267 19 295 270 1 G 295 2G1 25 G 216 2 17
635 285 270 1 G 285 272 5 1 6 28 272 14 G 26 21 7 285 25 20 2 56 265 29 285 270 1 G 265 261 25 G 236 26E 17

1637 286 270 16 G 236 273 14 236 272 14 203 273 64 28 5 2 1 G 28B 269 28 203 270 16 G 205 22 24 G 236 27 6
613 28 270 G 2 7 273 4 27 272 l 4 s 273 4 287 265 1 G 287 269 27 2H 270 1 G 2H 26s 24 G 27 27
639 276 270 1 G 27 274 3 27 27 45 20o 273 14 286 266 1 G 267 270 17 23 270 17 G 203 260 23 G 287 271 15
1640 287 270 16 G 237 274 13 237 2721 287 273 14 G 87 6 1 G 287 270 97 206 270 16 G 206 2B0 23 G 237 272 15
641 286 270 16 7 2 274 3 27 27 45 2 273 4 G 28 7 0 G 287 271 19 25 270 17 G 286 261 23 G 297 27. 5
1642 276 270 17 G 237 274 3 237 272 l 4 G 290 2772 14 206 265 2 287 271 16 289 270 1 7 G 289 265 22 G 237 273 14

643 287 270 7 237 274 3 237 272 15 206 273 4 G 287 268 2 G 287 272 10 206 270 17 G 206 2B6 22 G 237 273 14
644 287 270 17 297 274 13 297 272 15 297 273 14 G 287 28 21 G 287 272 15 625 270 17 G 297 2G 22 G 2o7 271 4
145 288 70 2 8 8 274 4 38 27271 2688 273 15 0 211 2 1 2 287 273 15 288 270 18 G 288 26 21 G 28 272 13
41573 o 72727 5 2723 4 G2726 0 G 2721161 272 2 G 2 7
1715g742
1645757 2724 3 2gI ,2 1273 G2I 6 0 2722 62120 1 0 2126 g 7
1657B73 2
41577 272 4 02728 1 2722 61 87262 8 7
21577G9 1
145 2 70 1 8 74 1 2. 22 1o 2. 23 1 .0 26 9 G 2 7
20170 1.5 0 6o 2 12 7





















Table C-1. Continued

A=Arrivals, D=Departures, Q=Queue:

A DQ A D Q A D Q A D Q


Time (sec)

1649
647
1648
1649
1650
651
1652
653

1654
1655
1656
657



656
16 59
1660
1 661
1662





16372
1664
16 75
1676





16 7
1669




670
671

1672

1675

1677
1678







1680
1681






1682
1683
I684







1685
1686








1687
17683
1689
1699







691
1692

















170.
1673

1695
1696









697
17 69
1699








1710
1701
702

1723
704
1705











1706
1707
1708
1709
1710
171 1
1712
1713
1714
1715
1718
1717
1718
1719
1720

1722
1723
1724
1725
1728
1727


299 270 19
288 270 8
88 270 8
288 270

299 270 9
289 270 9
289 270 9
299 270 9
209 27 9
389 20 19
389 20 19
289 27 0 19

290 271 920
290 270 20
290 271 19
90 273 9
290 272
290 275 7
290 273 1
291 273 1
91 274 7
291 274 17
291 275 7
21 275 1
91 286 3
292 7 5
92 7
92 2706


292 278 4
92 9 4
92 9 3

93 280 3
93 280 3
293 280 3
93 28O 13
293 280 13
293 28O 13
293 2SO 13

94 280 4
294 280 14
294 280 14
94 280 14


295 280 514
294 280 14
295 280 15
95 280 15
295 280 15
295 280O 15
295 280 15
295 280 15
295 280 15
296 280 16
96 280 9
298 280 18
296 280
296 282 6
299 282 7
96 280 6
297 280
297 280 7
297 280 7
297 20O 7
297 20 7
297 280 7
297 20 7
297 280 U17
298 280 1
290 20 0180
2 9 8 O 2 U
298 280 38


298 280
290 20O 18
299 201
299 281 8
99 72 7
99 S2 7
99 83 17
99 2S3
99 S4 6


288
238
288
2H8
2 "
288
288
2.9
289
289
239
29
290


290
290
290
290
290
290
290








293
290
29



293
293
29
29
29
29
295
292




S29
S292
292
2 29
292
296
293
G 293
G 29
G 29
G 293
G 29
G 294
G 294
G 294
G 294
G 294
G 294
G 294
G 295



S299
G 29
S295
G 29
295
0 295
298



298
298
29
298
29H




298
297
297
297
297
297





297
297
299
299
29H
298

2 98
29H
299
299
29
299
299
29
299
300


2 16
27 16
72 16

272 6
272 16
272 18
72 7
2 17
272 7
72 17
2 17
27 17
2 2
2 S
72 1
72 1

72 1
2 S
82 1
2 2 1

22 12
272 1
272 1

2 I
273 1












282 13
78 I9
23 16
74 1S
24 18

7

7

8 1






280 I
278 1
271
2
2 71 14
280 1 4
201 1 3
281 13
282 12
202 12
202 12
282 12
282 13
282
2 2 13
S22

282 1
S22

282 13
2
82

282 1 4
282 14
282 14
282 1


8 27
8 273
88 273
27
288 273


28 273
20 273
286 271
2. 27
28C 273
2I 273
281 273
2I 273
289 27
290 273
290 273
27
2 273




8 273












91 273
260 273
6 277
29 278


















2E 273
29 273
262 271
27 28
292 273
272 286
8.. 278
283 278
28 278





















28 2786
2932 273


282 286
293 273






















23 273
293 273
214 273
. 27 3
293 273
214 273
2 94 273
29 4 274




















28 286
294 273
2954 27 4


296 278
29 276


296, 286


29 6 271
297 282
297 2




297 2.8

2 '7 282
297 2M4
297 284
287 2854
297 28 4
297 2086
298 286
211 286
298 2N6


290 2


299 286
M6

































29 28

29 N6
291


AD Q A D Q ADQ


288 25
288 67
28 276
28 276
2 7E021
289 26
28 276
289 26
288 276
289 276
289 76

29 276
290 276
20 276
290 276
290 276
20 26
291 76
2 9 276
29 6













29 67
2 9 276

















29 6
261 276

291 276
292 276
22 276 '
292 276
22 276 '
2 127
292 276
292 276
293 276
2 3 276
29 276
2 3 27 '
293 276
291 276

2 4 276
294 276

2 14 276
2 4 276
294 276

28 4 276
294 276
294 276

295 276
259 276
295 276
295 276

2 '5 27 '
295 276
298 276

266 276
29 276

288 276
2 276
298 277
298 277
297 27.
297 278
297 2

217 2760
297 260
297 286

287 281
290 202
298 68
29Q 7~


270 20
270 20



'71 2
27 21
270 20
270 20
270 20
270 20
270 2
270 2
270 2
27 21
270 2
270 2
270 22

27 2
270 2
270 2
270 2
270 2
270 23
278 23
270 23
270 21
270 23
270 23
270 21
270 23
270 24
270 24
270 24
270 24
270 24
270 2
270 2
270 25
270 25
270 25
270 25
270 25
270 25
270 25
278 28
270 28

270 28
270 28
270 29
270 29
270 29
270 27

270 2 7
270 21
270 29
27 0 27
270 27'
278 27
270 28'

271 28'
27 0 28


27 0 20
270 29'
27 0 2 9
278 29
270 2 9
27 0 29
270 29


A D Q



288 287 21
288 267 21
288 288 20
288 268 20
288 269 20
288 269 19
289 270 1
289 270 18
289 271 18


2 1
94 I
42 1
292 2 1

292 8
262 291 12

293 281 2
213 2 1 12
293 281 12
253 20 12









2 8

2 8
9 28 2
29 28
24 28 3
29 I8
2 1
2 1
29 2 1
24 281 1
249 4 8 13
94 8
295 281 14
295 201 1 4
295 201 1 4
295 291 1 4
295 281 1 4
295 201 154
295 201 151
... 281 15
29 28
29 281 15
26 6 281 1
298 281 15
298 281 1
287 28 18
29 28 18
297 2 1 18
279 7 8 1
297 281 16
297 28 1
297 2 1 17
29 8






298 281 17
280 2 1 1 7
'g Z







2 '8 281 17
29 89







298 21 1 7
29 8 2 1 1 0

218 281 19
299 281 10
29 28

2 99 281 1
299 281 19
211 281 1
29 281 9
9O 281 9


A DQ


97 15


288 275
238 275
288 276
288 27
28 27-
29 276
289 276
289 27
29 276
29 275
239 275
29 275
289 275
290 275
290 275
290 275
290 275
290 27
29 276
29 27
29 276
291 275
29 275
291 275
29 27
29 27

293 275



293 275
29 275


293 275
29 275
29 275
29 275
294 27
295 275
294 275



294 27
294 27
294 276
296 27
29 275
294 27
294 276
295 27
29 276
295 276
29 27
294 275
295 276
295 276
29 27
298 276
29 275
299 275
299 275
29 275


297 275
297 275
297 275
297 27'
29 275
298 275
297 275
29 275
297 275
29 275
297 275
29 275
29 275
29 275
299 275
299 275
299 275
298 275

299 275
299 27
299 27


300 27


















Table C-1. Continued

A=Arrivals, D=Departures, Q=Queue:
A D Q
Time (sec)

172? an ?fl 15
730 G 300 285 15
1 I 1 S Yuu 1uu 15
1732 G 300 20B6 14
1733 G 300 207 14
1734 G 300 207 13
1735 301 28B 13
1 11 0 3U1 2UH 13
1 1d/ 0 3U1 2U 12
1730 6 301 209 12
17339 301 290 12
174A Gn61 2An 11
1741 301 ?an 11
1 4) Yu1 JuD 14
1 43 30U2 2U 12
1741 302 290 12
1745 302 ?2R 17
1745I 302 2fl 17
W14/ 35U 22U 12
1 14H 35) 25U 17
1l M3 3 2UyU 13
1753 3703 n ? 13
71 3643 ?92fn 13


1757 303 2SD 13

175 33 2Ufl 13

16U2 30U 29U 14
1760 304 290 13
1761 304 2SD 14
1757 3AU4 2fln 15

1728 U4 2Un 1'A
1777 304 290 15



1765 304 290 11
1761 304 29D 1$


1767 305 2a0 1'
17FA 35 29rf 15
176i 300 290 16

1766 305 290 16
1772 305 29D 156
1773 305 29D 16
1774 305 29D 1I
1 790 3UH 2 U 169
1771 3UI 29U 16
1777 306 290 16
1770 3D6 290 16
177" 30A 29D 16

17fH 3U1/ 2.0 17
1777 3D7 299 17

177 307 290 1
177M 3n07 7B 1
1 iu 3Uf 2U4 1/
1/17 3Uf 291 14

17 1 a 3D9 7292 1
1792 a 3N) 723 15
1793 a 3uN 297 12
7U9 G 33A 2971 2
1706 0 3D9 294 12


171 0 3D 298 13
1792 G 3DS 298 13
1793 G 3DS 297 12
1794 G 30D 297 12
1705 G 3D9 2DB 12
1('iS 3U 23J 11
1(1 31U 23u'J 11
1796 G 310 299 11
17A9 G 3111 'nn If
1HU a 31w 36n 1(


A DQ A D Q


A D Q


AD Q


A D Q


s nn
3DD

300
3D0

30D


3D
3D1


nUl


3Ul
30D
30n
30h
30u
306


30u
30D
3UM
3D4
30D
30
3u0


306
3an
L30
304


3D4
3n1
3n6
3U4



3DB
304
3n5
3n5



3D5
3D5
365


3n,
306


3DS
30U










3 310
307
3D?
3D 7
3UH
30 3
G 33l
G RflR
G Rfl




G an
G 305
G 3D0
6 3D9





G 310
3 sin


AD Q A D Q


AD Q


ann
3DO
3DO
300
300
300


3D1
3D]
3.I
3D1
301
G ni
3U1



A30
3U3
3062

S 3D
30n
3u3


3uD3


G 30

d 303
3UA




6 3
S 36M

G 3M4
G 3M9


G 3J4












3D5
G 307
G 305
S 305
305
G 3D5

S 3Uf
G 3f0
G 306
G 306

G 3UH

G 307

G 307
0 33f
o 3Ali

G 3B
S38
3 3B





G 30B
G 3B8

G 30S



G 310

| 13









REFERENCES


1. P. Tarnoff and J. Ordonez, Signal Timing Practices and Procedures, Institute of Transportation
Engineers, March 2004, pp. 1-3,

2. Hurdle, V.F., Signalized Intersection Delay Models A Primer for the Uninitiated,
Transportation Research Record 971, TRB, National Research Council, Washington, D.C.,
1984, pp. 96-105.

3. Dowling, R. G., Definition, Interpretation and Calculation of Traffic Analysis Tools Measures
of Effectiveness, Final Report, Dowling Associates, Inc., November 2006

4. Highway Capacity Manual. TRB, National Research Council, Washington, D.C., 2000

5. M. Saito, J. Walker and A. Zundel, Use of Image Analysis to Estimate Average Stopped
Delays Per Vehicle at Signalized Intersections, Transportation Research Record 1776, National
Research Council, Washington, D.C., 2001, pp. 106-112,

6. Engelbrecht, R.J., Fambro, D. B., Rouphail, N.M. and Barkawi, A.A., Validation of
Generalized Delay Model for Oversaturated Conditions, Transportation Research Record 1572,
TRB, National Research Council, Washington, D.C., 1997, pp. 122-130

7. Potts, I. B., Bauer, K. M., Harwood, D. W., Ringert, J. F., Zeeger, J. D., Gilmore, D. K.,
Relationship of Lane Width to Saturation Flow Rate on Urban and Suburban Signalized
Intersection Approaches, Transportation Research Board 2007 Annual Meeting Compendium of
Papers

8. Lin, F., Chang, C., Tseng, P., Errors in Capacity and Timing Design Analyses of Signalized
Intersections in the Absence of Steady Queue Discharge Rates, Transportation Research Board
2007 Annual Meeting Compendium of Papers

9. Maddula, S. Monitoting the Performance of a Signalized Intersection by Video Image
Detection. Masters Thesis, August 1994, Department of Civil Engineering, University of
Florida.

10. Lall, K.B., Berka, S., Eghtedari, A.G. and Fowler, J. Intersection Delay Measurement Using
Video Detection Systems. In proceedings of the Third International Symposium on Highway
Capacity, Volume 2, Copenhagen, Denmark, June 1998, pp.711-727.

11. Quiroga, C.A. and Bullock, D. Measuring Control Delay at Signalized Intersections.
July/August 1999, Journal of Transportation Engineering, Vol. 125, No. 4, pp. 271-280

12. Zheng, J., Wang, Y., Niahan, N.L., Hallenbeck, M.E., Detecting Cycle Failures at Signalized
Intersections Using Video Image Processing, Transportation Research Board S4 Annual
Meeting, TRB, National Research Council, Washington, D.C., January, 2005









13. Hoeschen, B., Bullock, D., and Schlappi, M., A Systematic Procedure for Estimating
Intersection Control Delay from Large GPS Travel Time Data Sets. Transportation Research
Board S4 Annual Meeting, TRB, National Research Council, Washington, D.C., January, 2005

14. Sun, C., Ritchie, S.G., Tsai, K., and Jayakrishnan, R. Use of Vehicle Signature Analysis and
Lexicographic Optimization for Vehicle Reidentification on Freeways. In Transportation
Research, Part C, 1999, pp.168-185.

15. Palen, J., Coifman, B., Sun, C., Ritchie, S., and Varaiya, P. California Partners for Advanced
Transit and Highways (PATH) Enhanced Loop-Based Traffic Surveillance Program. In ITS
Quarterly, Fall 2000, pp.17-25.

16. Liu, H.X., Oh, J. and Recker, W. Adaptive Signal Control System I i/h On-Line Performance
Measure. December 2001, Institute of Transportation Studies, University of California, Irvine

17. Sun, C., Arr, G., Ramachandran, R.P. and Ritchie, S.G. Vehicle Reidentification Using
Multi-Detector Fusion. July 2002, IEEE Transactions (forthcoming)

18. Oh, C. and Ritchie, S.G. Real-Time Inductive-Signature-Based Level of Service for
Signalized Intersections, December 2001, Institute of Transportation Studies, University of
California, Irvine

19. Coifman, B. and Ergueta, E. Improved Vehicle Reidentification and Travel Time
Measurement on Congested Freeways, September/October 2003, Journal of Transportation
Engineering, Vol. 129, No. 5, pp. 475-483

20. Coifman, B. and Dhoorjaty, S. Event Data-Based Traffic Detector Validation Tests,
May/June 2004, Journal of Transportation Engineering, Vol. 130, No. 3, pp. 313-321

21. Jeng, S., Ritchie, S. G., Tok, Y. C., Freeway Corridor Performance Measurement Based on
Vehicle Reidentification, Transportation Research Board 2007 Annual Meeting Compendium of
Papers, Washington D.C.

22. Washburn, S. and Nihan, N., Estimating Link Travel Time i/ i/h the Mobilizer Video Image
Tracking System, January/February 1999, Journal of Transportation Engineering, Vol. 125, No.
1, pp. 15-20

23. Grenard, J.L., Bullock, D. and Tarko, A.P. Evaluation of Selected Video Detection Systems
at Signalized Intersections, November 2001, School of Civil Engineering, Purdue University

24. Bonneson, J and Abbas, M. Video Detection for Intersection and Interchange Control,
September 2002, Texas Transportation Institute, Texas A&M University

25. Oh, J. and Leonard, J.D. Vehicle Detection Using Video Image Processing System:
Evaluation of PEEK VideoTrak, July/August 2003, Journal of Transportation Engineering, Vol.
129, No. 4, pp. 462-465










26. Riley, W. R. and Gardner, C. Technique for Measuring Delay at Intersections,
Transportation Research Record 644, TRB, National Research Council, Washington, D.C., 1977,
pp. 1-7

27. Bonneson, J, Modeling Queued Driver Behavior at Signalized Junctions, In Transportation
Research Record 1365, TRB, National Research Council, Washington, D.C., 1992, pp.99-107.

28. Fambro, D. and Rouphail, N. Generalized Delay Model for Signalized Intersections and
Arterial Streets. In Transportation Research Record 1572, TRB, National Research Council,
Washington, D.C., 1997, pp. 112-121.

29. Tarko, A.P. and Tracz, M., Uncertainty in saturation Flow Predictions, In Transportation
Research Circular E-C018, TRB, National Research Council, Washington, D.C., June 2000,
pp.310-321.

30. Li, H. and Prevedouros, P.D., Detailed Observations of Saturation Headways and Start-Up
Lost Times. In Transportation Research Record 1802, TRB, National Research Council,
Washington, D.C., 2002, pp.44-53.

31. Cohen, S.L., Application of Car-Following Systems to Queue Discharge Problem at
Signalized Intersections. In Transportation Research Record 1802, TRB, National Research
Council, Washington, D.C., 2002, pp.205-213.

32. Mousa, R.M., Simulation Modeling and Variability Assessment of Delays at Traffic Signals,
March/April 2003, Journal of Transportation Engineering, Vol. 129, No 2, pp. 177-185

33. Rakha, H. and Zhang, W., Consistency of Shock-wave and Queuing Theory Procedures for
Analysis of roadway Bottlenecks, Transportation Research Board S4 Annual Meeting, TRB,
National Research Council, Washington, D.C., January, 2005

34. Perez-Cartagena, R.I. and Tarko, A.P., Calibration of Capacity Parameters for Signalized
Intersections in Indiana, Transportation Research Board S4 Annual Meeting, TRB, National
Research Council, Washington, D.C., January, 2005

35. Kebab, W., Dixon, M., and Abdel-Raheim, A., Field Measurement of Approach Delay at
Signalized Intersections Using Point Data, Transportation Research Board 2007 Annual Meeting
Compendium of Papers, Washington D.C.

36. Brilon, W., Geistefeldt, J., and Zurlinden, H., Implementing the Concept of Reliability for
Highway Capacity Analysis, Transportation Research Board 86th Annual Meeting, TRB,
National Research Council, Washington, D.C., January, 2007

37. Jiang, Y., Li, S., and Zhu, K.Q., "Traffic Delay Studies at Signalized Intersections with
Global Positioning System Devices", ITE Journal, August 2005









38. Ko, J., Hunter, M., and Guensler, R., Measuring Control Delay Using Second-By-Second
GPS Speed Data, Transportation Research Board 861 Annual Meeting, TRB, National Research
Council, Washington, D.C., January, 2007

39. Comert, G. and Certin, M., Queue Length Estimation from Probe Vehicle Location:
Undersaturated Conditions, Transportation Research Board 2007 Annual Meeting Compendium
of Papers, Washington D.C.

40. Wunnava, S., Yen, K., Babji, T., Zavaletta, R. Romero, R. and Archilla, C., Travel Time
Estimation Using Cell Phones (TTECP) for Highways and Roadways, Final Report, Florida
Atlantic University, January 29, 2007

41. Buckholz, J. and Lee, S., TSDViewer Program Documentation, Unpublished Research
Report, University of Florida, October 3, 2007

42. Buckholz, J. and Lee, S., DTDiagram Program Documentation, Unpublished Research
Report, University of Florida, October 3, 2007

43. Buckholz, J, BuckTRAJ Program Documentation, Unpublished Research Report, University
of Florida, October 3, 2007

44. A Policy on Geometric Design of Highways and Streets, Fifth Edition, 2004, American
Association of State Highway and Transportation Officials

45. Courage, K. G., Fambro, D. B., Akcelik, R, Lin, P. S., Anwar, M., & Viloria, F., Capacity
Analysis of Traffic-Actuated Intersections, December 1996, University of Florida, NCHRP
Project 3-48 Final Report

46. Long, G., Startup Delays of Queued Vehicles, Transportation Research Board S4 Annual
Meeting, TRB, National Research Council, Washington, D.C., January, 2005

47. Akcelik, R, Time-Dependent Expressions for Delay, Stop Rate and Queue Length at Traffic
Signals, October 1980, Australian Road Research Board Internal Report AIR 367-1

48. Courage, K. G., Summary of HCM Procedure Inputs, Process and Outputs, University of
Florida Transportation Research Center, Draft Working Paper 385-2, January 2007

49. Tarko, A. P., Perez-Cartagena, R.I., 2005, Variability of Peak Hour Factors at Intersections,
Transportation Research Record: Journal of the Transportation Research Board No.1920, pp
125-130

50. Hellinga, B., and Abdy, Z., Impact of Day-to-Day Variability of Peak Hour Volumes on
Signalized Intersection Performance, Transportation Research Board 2007 Annual Meeting
Compendium of Papers, Washington D.C.









51. Zhang, L. and Prevedouros, P.D., User Perceptions of Signalized Intersection Level of
Service, Transportation Research Board 64- Annual Meeting, TRB, National Research Council,
Washington, D.C., January, 2005









BIOGRAPHICAL SKETCH

Jeffrey W. Buckholz is president and chief traffic engineer for JW Buckholz Traffic

Engineering Inc of Jacksonville Florida as well as an adjunct professor at the University of North

Florida. Mr. Buckholz holds a masters degree in civil engineering with a transportation major

from the University of California at Berkeley and both a masters of business administration

degree and a bachelor of science degree in civil engineering from the University of Toledo. He

is a registered civil engineer in Florida, Georgia, Massachusetts, Ohio, Michigan and California

and is a state-certified unlimited electrical contractor in Florida and Georgia

Mr. Buckholz is a senior level traffic engineer with 26 years of wide-ranging experience in

the transportation profession. This experience includes transportation planning and traffic

impact analysis, highway capacity analysis, advanced signal system design, traffic signal

construction, traffic signal timing, and ITS design. He is a court certified expert witness in the

field of traffic engineering and is also certified by the International Municipal Signal Association

as a Level II Traffic Signal Technician. In addition, he has authored three training manuals on

traffic signal design, construction, and inspection for the International Municipal Signal

Association.





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1 REAL-TIME ESTIMATION OF DELAY AT SIGNALIZED INTERSECTIONS By JEFFREY W. BUCKHOLZ A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007

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2 2007 Jeffrey W. Buckholz

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3 To my dogs Zack and Sweet Pea, who always provided me with free fuzz therapy. I wish that service was still available.

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4 ACKNOWLEDGMENTS Special thanks go to Mr. Seokjoo Lee for hi s programm ing assistance and to Mr. Petra Vintu for checking the mathematical derivations

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........8 LIST OF FIGURES.......................................................................................................................10 ABSTRACT...................................................................................................................................14 CHAP TER 1 INTRODUCTION AND PROBLEM STATEMENT............................................................16 Background Discussion.......................................................................................................... 16 Problem Statement.............................................................................................................. ....19 2 OBJECTIVES AND RESEARCH APPROACH................................................................... 22 3 CURRENT STATE OF THE ART........................................................................................ 30 Real-Time Measurement of Intersection Delay...................................................................... 30 Vehicle Reidentification via Inductance Loops......................................................................36 Performance of Video Detection Systems.............................................................................. 45 Signalized Intersection Queuing and Delay........................................................................... 52 Probe Monitoring............................................................................................................... .....65 Extending the Body of Knowledge......................................................................................... 67 4 ESTIMATING NON-VISIBLE DELAY............................................................................... 68 Data Analysis Programs......................................................................................................... 68 Prediction Algorithm fo r Non-Visible Delay ......................................................................... 80 Non-Visible Queue Estimation Technique...................................................................... 80 Non-Visible Queue Adjustment Technique:................................................................... 82 Non-Visible Queue Re-Adjustment Technique:............................................................. 83 Examples.................................................................................................................................84 Queue Prediction....................................................................................................................87 Stopped Delay Prediction.......................................................................................................88 Control Delay Prediction........................................................................................................90 Variability Considerations..................................................................................................... .91 Limitations to the Dela y Prediction Procedure ....................................................................... 92 5 THEORETICAL BOUNDS FOR DELAY ESTIMATION................................................. 129 Derivation of the Bounds......................................................................................................131 Derivation of the Upper Bound.....................................................................................134 Derivation of the Lower Bound.....................................................................................138

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6 Analysis of Bounds Summary.......................................................................................146 Derivation of Delay for Upper and Lower Bounds..............................................................148 Derivation of the Bounds with Visible Period 1 Queue.......................................................166 Derivation of Upper Bound w ith Visible Period 1 Queue ............................................ 166 Derivation of Lower Bound w ith Visible Period 1 Queue ............................................ 172 Analysis of Bounds Summary with Visible Period 1 Queue........................................173 Derivation of Delay with Visible Period 1 Queue................................................................174 Derivation of the Bounds When Queu e is Visible During Three Periods ............................ 176 Derivation of the Bounds When Analysis Tim e Frame is Greater Than One Hour............. 176 Derivation of the Five Period Upper Bound.................................................................. 177 Derivation of the Five Period Lower Bound................................................................. 183 Five Period Analysis of Bounds Summary................................................................... 197 Generalized Analysis of Bounds Summary.......................................................................... 200 Historical Peak Hour Factors................................................................................................ 203 Limitations to the Theoretic al Bracketing Procedure ...........................................................205 6 COMPARISONS WITH VEHIC LE TRAJECTORY ANALYSIS ..................................... 228 Trajectory Example............................................................................................................. .230 Cumulative Arrival/Depa rture Curve Exam ple....................................................................232 Reconciling the Difference Between Cu m ulative Curves and Trajectories......................... 233 Calculating Trajectory-Based Delay Com ponents for the BuckQ Examples....................... 236 Calculating Cumulative Curve De lay for the BuckQ Exa mples..........................................237 Bracketing the Stopped Delay Prediction Results................................................................ 240 7 PERIOD ISSUES DURING OVER-SATURATED FLOW................................................ 276 Simplified Example of Cycle-Pe riod Issues in Calculating d3.............................................276 Residual Queue Discrepancy................................................................................................281 Detailed Example of Cycle-Pe riod Issues in Calculating d3................................................283 8 CONCLUSIONS, APPLICATIONS, AND RECOMMENDATIONS................................ 299 Research Findings.................................................................................................................299 Application of the Research.................................................................................................. 301 Example 1: Signal System Retiming Evaluation.................................................................. 302 Example 2: Real-Time Traffic Signal Control..................................................................... 303 Example 3: Signalized Inters ection Capacity Analysis ........................................................ 304 Potential Areas of Extended Research..................................................................................304 APPENDIX A DATA SETS FOR BUCKQ TESTING............................................................................... 308 B TYPICAL PEAK HOUR FACTORS................................................................................... 331 C GENERALIZED CYCLE-PE RIOD DELAY EXAMPLE: ................................................. 353

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7 REFERENCES............................................................................................................................376 BIOGRAPHICAL SKETCH.......................................................................................................381

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8 LIST OF TABLES Table page 4-1 Example summary volum e a nd capacity....................................................................... 122 4-2 Example summary queue discha rge, delay check and goodness-of-fit .........................123 4-3 Queue prediction........................................................................................................... ...124 4-4 Stopped delay prediction..................................................................................................125 4-5 Control delay prediction.................................................................................................. 126 4-6 Comparison of variation in act ual and predicted stopped delay ...................................... 127 4-7 P-value determination for difference in median values................................................... 128 6-1 Calculation of cumula tive curve delay conversion factors, volum e pattern 625_700_650_350vph......................................................................................................249 6-2 Calculation of cumula tive curve delay conversion factors, volum e pattern 700_725_625_350vph......................................................................................................251 6-3 Calculation of cumula tive curve delay conversion factors, volum e pattern 700_700_700_350vph......................................................................................................253 6-4 Calculation of cumula tive curve delay conversion factors, volum e pattern 725_700_700_350vph......................................................................................................255 6-5 Cumulative curve delay fo r standard 4-period case ......................................................... 257 6-6 Cumulative curve delay w ith m ultiple visible periods.................................................... 258 6-7 Stopped delay prediction re sults for 700_725_625_350vph volum e pattern..................259 6-8 Average stopped delay predicti on results for 700_725_625_350vph volume pattern .... 262 6-9 Stopped delay prediction re sults for 700_700_700_350vph volum e pattern..................263 6-10 Average stopped delay predicti on results for 700_700_700_350vph volume pattern .... 266 6-11 Stopped delay prediction re sults for 725_700_700_350vph volum e pattern..................267 6-12 Average stopped delay predicti on results for 725_700_700_350vph volume pattern .... 270 6-13 Stopped delay prediction re sults for 625_700_650_350vph volum e pattern..................271 6-14 Average stopped delay predicti on results for 625_700_650_350vph volume pattern .... 274

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9 6-15 Prediction comparison.....................................................................................................275 7-1 Generalized example of cycle-pe riod delay discrepancies data .................................... 292 7-2 Generalized example of cycle-pe riod delay discrepancies summary ...........................293 7-3 Detailed example of cycle-period delay di screpancies, residual queue determ ination... 296 7-4 Detailed example of cycle-period de lay discrepancies, delay com parison...................... 297 7-5 Detailed example of cycle-period delay discrepancies, delay com parison with modified d2 term.............................................................................................................. 297 7-6 Detailed example of cycle-period delay discrepancies, delay com parison with d3 adjustment..................................................................................................................... ...297 B1 US 1 machine counts (Southern St. Johns County)......................................................... 334 B-2 US1 Machine counts (northern St. Johns County)........................................................... 339 B-3 Atlantic Boulevard machine counts................................................................................. 342 B-4 University Boulevard mach ine counts (Jacksonville)...................................................... 345 B-5 SR A1A machine counts (Crescent Beach)..................................................................... 348 B-6 SR A1A machine counts (Ponte Vedra) PDF 17 KB ...................................................... 351 B-7 Appendix B data summary...............................................................................................352 C-1 Generalized example of cycle-pe riod delay discrepancies data. ..................................354

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10 LIST OF FIGURES Figure page 4-1 Queue relationships........................................................................................................ ....94 4-2 Signalized intersection delay components.........................................................................95 4-3 Measured versus estimated delay.......................................................................................96 4-4 Visible and non-vi sible variables ....................................................................................... 97 4-5 Relationship between v/c ratio and rati o of control delay to stopped delay ...................... 98 4-6 Re-queuing that result s in sim ultaneous queues................................................................ 99 4-7 Re-queuing that does not re sult in simultaneous queues ................................................. 100 4-8 Example of a blind period................................................................................................ 101 4-9 Example of adjacent blind periods................................................................................... 102 4-10 Counters and queue status................................................................................................103 4-11 Base case for P, C and X; stopped delay comparison...................................................... 104 4-12 Effect of increasing the power constant on stopped delay com parison........................... 105 4-13 Queue propagation example............................................................................................ 106 4-14 Actual vehicle queues..................................................................................................... .107 4-15 Average queue length comparison................................................................................... 108 4-16 Maximum queue length comparison................................................................................ 109 4-17 98th percentile back of queue comparison....................................................................... 110 4-18 Vehicle re-queuing........................................................................................................ ...111 4-19 Stopped delay comparison............................................................................................... 112 4-20 Stopped delay prediction, 12 FOV...................................................................................113 4-21 Comparison of actual and predicted stopped delay......................................................... 114 4-22 Adjacent blind period counter v. stopped delay............................................................... 115 4-23 Control delay comparison................................................................................................ 116

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11 4-24 Ratio of control delay to stopped delay........................................................................... 117 4-25 Graphical control delay comparison,............................................................................... 118 4-26 Control delay estimates................................................................................................... .119 4-27 Control delay composition............................................................................................... 120 4-28 Ratio of control delay to stopped plus m ove-up delay.................................................... 121 5-1 Cumulative arrival-departur e curves and overflow delay ................................................ 207 5-2 Critical time and volume points for period 4................................................................... 208 5-3 Overflow delay in period 4..............................................................................................209 5-4 Maximum reasonable cumulative arrival curve............................................................... 210 5-5 Minimum reasonable cu mulative arrival curve ............................................................... 211 5-6 Minimum overall reasonable cum ulative arrival curve................................................... 212 5-7 Minimum reasonable cumulativ e arrival cu rve (minimum V4 for minimum V1 and V2)....................................................................................................................................213 5-8 Minimum reasonable cumulativ e arrival cu rve (minimum V4 for minimum V1)...........214 5-9 Period 1 delay for the upper bound.................................................................................. 215 5-10 Period 2 delay for the upper bound.................................................................................. 216 5-11 Period 3 and period 4 delay for the upper bound.............................................................217 5-12 Reasonable overflow delay region for 600 vph capacity and 0.75 m inimum PHF......... 218 5-13 Reasonable overflow delay region for 600 vph capacity and 0.80 m inimum PHF......... 219 5-14 Reasonable overflow delay region for 600 vph capacity and 0.85 m inimum PHF......... 220 5-15 Maximum delay estimation error for 0.75 minimum PHF.............................................. 221 5-16 Maximum delay estimation error for 0.80 minimum PHF.............................................. 222 5-17 Maximum delay estimation error for 0.85 minimum PHF.............................................. 223 5-18 Maximum reasonable cumulative arri val curve with period 1 visible ............................. 224 5-19 Minimum reasonable cumulative arri val curve w ith period 1 visible............................. 225 5-20 Maximum reasonable cumulative arrival curve with 5 periods ....................................... 226

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12 5-21 Minimum reasonable cumulative arrival curve w ith 5 periods....................................... 227 6-1 Trajectory example A) Complete chart B) Detailed view of circled area in upper right corner. .................................................................................................................. ....244 6-2 Cumulative arrival-de parture curve exam ple................................................................... 246 6-3 Trajectory conversion of cumulative curve example....................................................... 247 6-4 Delay and travel time components................................................................................... 248 7-1 Cycle v. period initia l queue delay analysis..................................................................... 294 7-2 Cycle v. period "control delay" analysis.......................................................................... 295 7-3 Upward bias in HCM residual queue calculation ............................................................ 298 A-1 Queue discharge headway histogram............................................................................... 309 A-2 Start-up lost time histogram............................................................................................. 310 A-3 Comparison of control delay and sto pped delay by cycle length (g/C =0.30) ................. 311 A-4 Comparison of control delay and stopped delay (g/C =0.30) .......................................... 312 A-5 Comparison of control delay and stopped plus queue m ove-up delay by cycle length (g/C = 0.30)......................................................................................................................313 A-6 Comparison of control delay and stopped delay plus queue m ove-up delay (g/C =0.30)...............................................................................................................................314 A-7 Relationship between v/c ratio and stopped delay...........................................................315 A-8 Relationship between v/c ratio and stopped delay by cycle length ................................. 316 A-9 Relationship between v/c ratio a nd stopped plus queue m ove-up delay.........................317 A-10 Relationship between v/c ratio and st opped plus queue m ove-up delay by cycle length................................................................................................................................318 A-11 Relationship between v/c ratio and control delay............................................................319 A-12 Relationship between v/c ratio a nd control delay by cycle length................................... 320 A-13 Relationship between vehicle re-queues and control delay ............................................. 321 A-14 Relationship between v/c ra tio and vehicle re-queues ..................................................... 322 A-15 Relationship between v/c ratio a nd vehicle re-queues by cycle length ...........................323

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13 A-16 Relationship between v/c ratio and cycles with phase failure .........................................324 A-17 Relationship between v/c ratio and cycl es with phase failure by cycle length ................ 325 A-18 Percentage of cycles in 1 hour with phase failure by cycle length .................................. 326 A-19 Percentage of cycles in 1 hour with phase failure ............................................................ 327 A-20 Linear relationship betw een ABPC a nd stopped delay....................................................328 A-21 Exponential relationship betw een ABPC and stopped delay ........................................... 329 A-22 Relationship between ABPC and control delay............................................................... 330 B-1 US 1 S. PM peak hour fa ctor, southbound (outbound) flow ...........................................332 B-2 US 1 S. PM peak period f actor, southbound (outbound) flow......................................... 333 B-3 US 1 N. PM peak hour factor, northbound (outbound) flow........................................... 337 B-4 US 1 N. PM peak period factor, northbound (outbound) flow........................................ 338 B-5 Atlantic Boulevard PM peak hour factor, eastbound (outbound) flow ............................ 340 B-6 Atlantic Boulevard PM peak pe riod factor, eastbound (outbound) flow .........................341 B-7 University Blvd. PM peak hour factor, northbound (outbound) flow .............................343 B-8 University Blvd. PM peak peri od factor, northbound (outbound) flow .......................... 344 B-9 SR A1A S. PM peak hour factor, southbound (outbound) flow ...................................... 346 B-10 SR A1A S. PM peak period factor, southbound (outbound) flow ...................................347 B-11 SR A1A N. PM peak hour factor, southbound (outbound) flow ..................................... 349 B-12 SR A1A N. PM peak peri od factor, southbound (outbound) flow ..................................350

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14 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy REAL-TIME ESTIMATION OF DELAY AT SIGNALIZED INTERSECTIONS By Jeffrey W. Buckholz December 2007 Chair: Ken Courage Major: Civil and Coastal Engineering To evaluate improvements at signalized intersec tions it is important to know the resulting change in vehicular delay. Howeve r, it is difficult to collect delay data during over-saturated conditions even though this is when knowledge of delay levels is critical. Extensive peak hour queuing thwarts our ability to collect key data, su ch as arrivals at the back of queue. This incomplete information makes it impossi ble to calculate the resulting delay. The research presents a real-time procedur e for estimating delay during over-saturated conditions with limited information. The proce dure utilizes a series of adjustments to the measured arrival rate entering the field of view to estimate the true arrival rate at the back of the queue. An advantage of the pr ocedure is that estimated queues and associated delay are calculated on a second-by-second basis in real tim e. A disadvantage is that no theoretical relationship exists between the measured arrival rate and the real arrival rate. Fortunately, it is possible to calculate a se t of theoretical upper and lower bounds on the solution space by using historical minimum peak hour factors. The theoretical bounds take the form of cumulative arrival curves. Delay is ob tained through considerat ion of the area between these arrival curves and the associated depart ure curve. Trajectory analysis during over-

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15 saturated conditions is used to reconcile the difference between stopped delay and the area between the curves. This research also demonstrates that the Highway Capacity Manua l (HCM) definition of an initial (residual) queue is incorrect. To id entify the true residual queue, the situation must be evaluated at the end of the red in terval and thruput during the subs equent green interval must be deducted. Failure to do so leads to overestimation of both the initial queue and the corresponding delay. Another finding is that the random component of the HCMs incremental delay term incorrectly contributes to delay during over-satur ated periods preceded by an initial queue. A remedial modification to the d2 term is proposed. Finally, it is demonstrated that the HCM s period-based queue accumulation procedure has drawbacks that can produce su bstantial errors in delay duri ng over-saturated conditions. A remedial cycle-based counti ng technique is proposed.

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16 CHAPTER 1 INTRODUCTION AND PROBLEM STATEMENT Since the efficient operation of signalized intersections is a pertinent topi c throughout the world, providing a real-tim e evaluation system that allows such intersections to be operated at maximum efficiency has the potential for tremendous benefit. Reductions in travel time would be the primary benefit, along with associated reductions in fuel usage and vehicle emissions. The benefits would accrue "24/7" in that signa lized intersections function around the clock. In the United States alone there are approximately 265,000 signalized intersections and the delays at these signalized intersections contribute an es timated 25% to total highway system delay [1]. Background Discussion To properly evaluate improvem ents made at a signalized intersection it is important to know the resulting change in various Measures of Effectiveness (MOEs), including what may be the most important MOE, vehicular delay. De lay is a particularly attractive measure of effectiveness because, as discussed by Hurdle [2], it can: be measured; it has obvious economic worth; and it is easily unders tood by both technical and non-technica l people. As recognized by Dowling [3], many MOEs (such as queue lengt h, speed, stops, and density) are relatively invariant during highly over-saturat ed conditions where little vehi cle movement occurs. Delay, on the other hand, continues to in crease under such conditions, whic h is a highly desirable trait. The benefit of corridor re-timing programs, signal phasing changes, and intersection geometric improvements can be properly evaluated only if a realistic assessm ent of the change in overall vehicular delay is determined. Collectin g delay data by hand, as described in Chapter 16, Appendix A of the 2000 Highway Capacity Manual [4 ] is a labor-intensive task that must, by practical necessity, be limited to brief data collection periods. As Saito, et al. [5] put it: Manual field observations require large number of personnel and large amounts of other resources if delay estimates must be done freque ntly, such is the case if delay estimates are

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17 needed for Advanced Traffic Management Systems (ATMSs). The method is meant for occasional checks of delays at signalized in tersections; it is not meant for continuous monitoring of the LOS (level of service) of signalized inters ections. A more advantageous method would be to create automated methods of estimating delay fr om direct observation of queued vehicles. This significantly redu ces the amount of data that needs to be collected and (eliminates) unnece ssary assumptions. When such methods work, they allow traffic engineers to continuously monitor the LO S at intersections and estimate the arterial LOS In addition, it is particularly difficult to co llect delay data during over-saturated conditions even though this is exactly when knowledge of de lay levels is most critical. Consequently, under congested conditions, delay calculations that are based on manual information can be considered both piecemeal and of dubious accuracy. As Engelbrecht, et al. [6] explain From a practical point of view it is very difficult to accurately measure over-saturation delay in the field. Long queues and restricted sight distance may make the actual counting of queued vehicles impossible. Also, counting a large numbe r of vehicles in a short 10second interval may be very di fficult. Furthermore, not all vehicles in the queue may be stationary at a single point in time, as intern al shock waves due to the stopping and starting of traffic at the stop line may travel through the queue continuously. Because of the presence of non-stationary vehicles in the queue, transformation of the measured stopped delay into the overall delay predicted by most of the delay equations may be the most difficult task of all. A properly automated method for collecting delay data, either on a cycl e-by-cycle basis or on a periodic basis, could provide the needed evaluation data for all pertinent periods. Such a system would also provide reasonable estim ations of delay, even during over-saturated conditions. Resulting delay data could then be used for proj ect evaluation or for real-time modification of controller settings. Using real-time delay obtained from intersec tion-based field measurements for project evaluation purposes (such as signal retiming ev aluation) provides an im portant supplement to traditional before and after travel time runs, which completely ignore the delay experienced by side street motorists or main street left turn mo torists. A rather large leap forward in project

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18 evaluation could be taken if we are able to de velop a widely applicable, robust procedure for calculating vehicular delay on the fly. Video detection systems, vehi cle re-identification systems using inductance loops, and probe monitoring all offer the potential of bei ng able to calculate (or reasonably estimate) vehicular delay in real time. Unfortunately, direct measurement of stoppe d delay via video detection or inductance loops falls prey to a number of practical limitations, ranging from detection inaccur acies to field of view limitations. The accuracy of any in tersection-based delay measurement system is essentially limited by the detection technology available at the approaches under study. For example, if an intersection approach has video de tection oriented to see from the stop bar to a point far upstream (the best case scenario) then th e resulting estimation of delay can be expected to be relatively good whereas if the approach only has a stop bar loop (other than no detection, the worst case scenario), then the delay estimation will be relatively poor. In addition, the accurate estimation of approa ch delay is of most interest during peak periods when traffic demand is at its greatest. It is during these critical periods that extensive queues typically form; queues that can extend we ll beyond the field of view of any intersectionbased detection system. Consequently, when we most need an accurate estimation of approach delay is exactly when we are least likely to obtain it from conventional detection systems. Theoretical delay models for signalized intersection approaches, such as those described in the Highway Capacity Manual (HCM), offer anot her means of determining delay. One would expect that these models could be used in a r eal-time manner to obtain r eal-time delay results. However, to produce reasonable re sults the models must be base d on reasonably ac curate input data. If this needed data cannot be accurately ob tained, then the models are of little value. This

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19 brings us right back to the problems associ ated with obtaining accurate data under peak hour conditions. Extensive peak hour queuing essentiall y thwarts our ability to collect key approach data, such as the rate of vehicle arrivals at the back of the queue. The use of probe vehicles provides a fresh alte rnative for collecting de lay data. However, a host of challenging technical and privacy issues still need to be worked-out before probe vehicles can provide the needed detail to accurately estimate approach delay. On the technical side, a team of researchers in Florida recently discovered that cell phone technology, a promising probe alternative, is not accurate in congested traffic conditions and that the level of accuracy decreases rapidly as congestion increases. Problem Statement The latest e dition of the Highway Capacity Manual provides a well-recognized analytical procedure for calculating contro l delay at signalized intersections, with control delay being defined as the sum of deceler ation delay, stopped delay, queue move-up delay, and acceleration delay. This procedure has been automated in the form of the signalized intersection module of the HCS+ software suite. The HCS+ software offers a direct, user -friendly procedure for calculating lane group, approach, a nd intersection control delay and their associated levels of service. However, the HCM methodology assumes that, on a given approach, certain average conditions apply over the entire analysis period (sat uration flow rate, startup lost time, g/C ratio, arrival type) and that the vehicle arrival rate on the approach remains constant within each of the four 15-minute periods. In reality, conditions change on a cycle-by-cy cle basis depending on random fluctuations in approach volumes and driver composition. For example, the considerable variation in cycle-by-cycle saturation flow rates at signalized intersections was documented in two recent papers, one citing data from the Unite d States [7] and one ci ting data from Taiwan [8].

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20 In addition to this cycle-by-cycle variation in conditions on a given approach, variations also occur between different approaches due to unique characteristics of the approach. For this reason, the HCM recommends collecting field data to establish such items as ideal saturation flow rate. The HCM recognizes th at true site-specific delay ca n only be evaluated accurately by field measurement. Unfortunate ly, the field measurement of de lay requires knowledge of the entire extent of the queue, and survey techniques required to capture the entire extent of the queue must utilize costly resources such as aerial surveillance or multiple coordinated ground observers. Less expensive observation techniques such as a video camer a located at a single point, can estimate delay only if the back of the que ue is always in sight, which is typically not the case when peak hour congestion occurs. Recognizing these limitations, a new procedure is needed that can reasonably estimate delay over a wide variety of conditions, including grossly over-saturated conditions. In order to properly measure delay during over-saturated conditions, multi-per iod analysis becomes a must in order to ensure that that no initial queues exist e ither at the start or at the end of the analysis. Keeping track of the various components of control delay (stopped delay, move-up delay, acceleration delay prior to the stop line, accelerat ion delay beyond the stop line, and deceleration delay) becomes more difficult as volume exceeds capacity for any significant length of time. Predicting control delay in real-time with limited information, and being able to do so even with over-saturated conditions, is the challe nge addressed in th e research at hand. Key to this problem statement is the idea of limited information. Obviously, if we have perfect knowledge of each and every vehicle traj ectory then we can rather easily compute a complete set of arrival rates, departure rates, queue lengths, and the resulting control delay. However, detailed vehicle trajectory information can be very difficult to obtain and trying to

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21 secure it for more than a few locations quickly becomes cost-prohibitive given current technology. The crux of the problem is to find a me thod that uses more easily obtainable data to approximate the same delay information that a complete set of accurate vehicle trajectories would produce. The most easily obtainable data are usually data that occurs in proximity to the stop line. Current vehicle detection systems, in cluding most video and inductance loop systems, are best suited to obtaining data at this location. The quest is to develop a practical, real-time delay estimation system that is supported by theo retical considerations and which also makes use of readily obta inable data.

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22 CHAPTER 2 OBJECTIVES AND RESEARCH APPROACH The following objectives were es tablished for the research. OBJECTIVE 1: Develop a m ethodology and associated re al-time procedure th at can reasonably estimate delay associated with vehicles that ar e beyond the reach of the detection system. The procedure should function during both under-saturated and over-sat urated, obtaining reasonable estimates of vehicular delay even when queues are long and multiple phase failures occur. OBJECTIVE 2: Identify variables to be used in the procedure that are important in the prediction of delay beyond the det ection area (non-visible delay). OBJECTIVE 3: Establish and clearly define any ne w terminology needed to document the methodology. OBJECTIVE 4: If the proposed procedure is empirical in nature, deve lop theoretical limits on the solution space that can be establishe d using readily available information. OBJECTIVE 5: Ensure that all delay estimates are c onsistent with trajectory analysis and reflect the true nature of control delay. OBJECTIVE 6: Ensure that all delay estim ates are reconciled to the procedures contained in the 2000 Highway Capacity Manual and the current version of the HC S+ software. Document any needed modifications to the manual or the software based on the research. OBJECTIVE 7: Provide examples of how the procedure could be used to address real-world traffic analysis or traffic control issues. OBJECTIVE 8: Indicate areas of future research. Objectives of the research would best be achieved using actual field data. However, detailed field data are not only expensive and time consuming to collect; one cannot safely or expeditiously manipulate field data in order to experiment at co ntrolled volume levels or cycle lengths. Analyzing substantially over-saturated systems is also ve ry difficult using actual field data as queue lengths can become quite exte nsive; spilling over into adjacent signalized intersections Therefore, theoretical research work was conducted in the laboratory using the CORSIM micro-simulation model. CORSIM allows us to quickly simulate a va riety of real-world conditions in a relatively r ealistic manner and to accumulate important measures of

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23 effectiveness, including delay. CORSIM was used because it is a we ll-accepted and wellunderstood model that has the cap ability to accommodate a wide range of input variables, including variable combinations that produce grossly over-saturated conditions with multiple phase failures. CORSIM also allows the user to vary the set of random number seeds to order to investigate changes in the results that occur due to random fluctuati ons. This ability is important since the stochastic nature of micro-simulation m odels can result in a le vel of variation that masks cause-and-effect relationships. CORSIM was specifically used to examine how measured dela y differs from actual delay when queues exceed the limits of the detection syst em. In order to investigate such differences, it was necessary to assume a certain field of view for the simulation runs. The field of view is defined as the number of vehicl es on an intersection approach lane that can be accurately measured by the detection system when the vehicl es are queued at the stop bar. A field of view of 12 vehicles was used in most of the examples associated with this theoretical work. This would be a reasonable field of view for a modern video detection system. Using various fields of view and cycle lengths, a reasonably accurate method for estimating actual stopped delay was developed. Fo r example, the back-of-queue on a single lane approach might extend to 20 vehicles whereas a video detection system may only be able to accurately see a queue extent of 12 vehicles. If this happens, the delay associated with the remaining 8 vehicles (the vehicles queued in the blind area) cannot be measured and must instead be estimated in some reasonably accu rate manner. Knowing the time during which a queue existed in the blind area, which may extend over multiple cycles, and knowing the number of vehicles that come into sight after such a period of blind queuing, the procedures developed in this endeavor allow us to obtain a wo rkable estimate of the non-visible delay that

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24 occurred. The procedure deve loped is capable of handling both under-saturated conditions (having little or no blin dness) and over-saturated conditions (with blind periods occurring over multiple cycles; referred to in this document as adjacent blind periods). The development of this procedure is one of the primary contributions to the literature dealing with signalized intersection delay. A limited field of view produces a situation wher e arrivals at the back of the queue cannot be observed. This incomplete information make s it impossible to calculate the resulting delay. However, using the methodology contained in th is dissertation, the delay can be reasonably estimated under a rather wide variety of cond itions. The procedure that was developed in response to the challenge of estimating non-visi ble delay begins by calculating an "estimated arrival rate" (which is actually the departure rate). If th e back end of the queue is not visible, the procedure modifies the estimated arrival rate upward using a power function in an attempt to predict the real arrival rate. This power function adjusts the rate in a manne r that varies with the amount of time during which the back end of the queue is not visible. A major advantage of this approach is that the resulting estimated queues a nd associated delay are immediately calculated on a second-by-second basis, in re al time. A major disadvantage of the approach is that there is no theoretical relationship between the departure rate and the re al arrival rate. Hence, two different arrival patterns that result in the same number of vehicles cros sing the stop line during the analysis period can produce si milar delay results. This problem is most evident when the length of time that the end of the queue is no t visible covers most of the analysis period. Fortunately, it is possible to calculate a se t of theoretical upper and lower bounds on the solution space by using information obtained at the end of the analysis period, when all queues are visible and the arrival rate equals the departur e rate. In order to make any type of reasonable

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25 delay estimation, all queues must dissipate prior to the end of the analysis period. Once queues become fully visible, an accurate calculation of the arrival rate can be made. Knowing this arrival/departure rate and knowing the total number of vehicles that have crossed the stop line during the entire hour we can, by assuming a reasonable minimum peak hour factor, work backwards through the period to identify minimum and maximum cumulative arrival curves. From these curves we can then calculate both lower and upper bounds on the overflow delay. These theoretical bounds can be us ed, in an ex post facto manner, to bracket the previously discussed real-time delay estimat ion procedure. They can also be used to identify an independent most probable arrival pattern by selecting an intermediate curve between the upper and lower bounds that minimizes the maximum percent error between the estimate and the actual delay The development of these theoretical bounds is another importa nt contribution to the literature dealing with signalized intersection delay. The theoretical upper and lower bounds on th e delay solution are calculated using cumulative arrival and departure curves. Vehicu lar delay is obtained th rough consideration of the area between these curves. W ithin this document it is demons trated that, contrary to popular belief, the area between the arrival and departure curves is not the delay incurred by approaching vehicles. An evaluation of trajectory analys is during over-saturated conditions is used to reconcile the difference between the true delay and the area between the cumulative arrival and cumulative departure curves so that a consiste nt set of upper and lower bounds are provided. This reconciliation is another contribution to the literature dealing with signalized intersection delay. The multi-period signalized intersection analysis procedure that is currently contained in the 2000 Highway Capacity Manual is codified as pa rt of the HCS+ version 5.21 software suite.

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26 The period-based procedure for queue accumulation that is described in this manual has certain drawbacks that can produce substantial erro rs when calculating control delay during oversaturated conditions. A descrip tion of these errors and the presentation of a cycle-based technique for eliminating them is yet another contribution to the literature dealing with signalized intersection delay. The following detailed work tasks were develo ped in order to carry out this research approach: TASK 1: Select a micro-simulation model for conduc ting the research a nd develop tools to extract needed information from the model. TASK 2: Develop a comprehensive softwa re tool that will facility the evaluation of real-time second-by-second delay estimation procedures for a one-hour analysis timeframe. TASK 3: Develop data test sets for use in identify ing the preferred delay estimation procedure. Various v/c ratios, cycle lengths, and fields of view should be reflected in this test set. TASK 4: Using the test sets, identify the preferred delay estimation procedure. TASK 5: Use the delay estimation procedure to analyze multiple replicates of four examples and document the results TASK 6: Examine statistical variabili ty issues by using a large num ber of replicates of a single example. The first 6 tasks are documented in Chapter 4. TASK 7: If the delay estimation procedure is em pirical in nature, de velop a theoretical technique for constraining the solution space. Task 7 is documented in Chapter 5. TASK 8: Develop a software tool for extracting trajectory information from the selected microsimulation model. TASK 9: Develop a software tool that will analyze all components of control delay associated with vehicle trajectories. The tool should summarize the resulting delay by 15-minute period for a one-hour analysis timeframe. TASK 10: If necessary, modify the delay estimation pr ocedure or the theoretical constraints to reflect true control delay concepts.

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27 Tasks 8 through 10 are documented in Chapter 6. TASK 11: Compare the results obtained with resu lts produced by the 2000 Highway Capacity Manual and reconcile all differences. Task 11 is documented in Chapter 7 TASK 12: Summarize the results and identify potential areas for further research. Task 12 is documented in Chapter 8. The end result of this research is the deve lopment of a theoretically constrained delay estimation procedure that is based on limited in formation. The delay estimation procedure makes use of available data to predict arrivals at the back of the non-vi sible queue as well as departures from the front of the non-visible queu e at each point in time, information that would otherwise be unknown. Knowing the arrivals and departures we can pred ict the length of the non-visible queue at each point in time. This pr edicted non-visible queue le ngth is then added to the measured visible queue length to obtain the total queue length with stopped delay being obtained directly from the queue length. Theoretical bounds based on historical minimum peak hour factors are then imposed on the delay estimate to ensure a reasonable result. Use of the procedure to estimate control delay on an over-saturated intersection approach for a one-hour analysis time frame would proceed as follows: 1. Using the vehicle detection equipment for th e approach of interest, real-time secondby-second data are collected on the number of vehicles crossing the stop bar, the number of vehicles entering the field of vi ew, the length of the visible queue, and the presence or absence of a stationary vehicle in the last queue position of the field of view. 2. This data set is entered into the delay estimation software, which measures the length of the visible queue and estimates the lengt h of the non-visible que ue at every second of the one-hour analysis time frame. S econd-by-second cumulative stopped delay is then calculated using this queue information. 3. The stopped delay prediction is converted to control delay using a series of conversion ratios that vary by cycle length a nd v/c ratio. The conversion ratio varies

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28 between 1.2 and 1.4 with 1.3 being a typical value. The predicted control delay is considered the final control delay fo r use in real-time traffic control. 4. The time during the last 15-minute period at which the end of the queue becomes visible is recorded, as is the cumulative num ber of vehicles that have crossed the stop bar at that time. At the end of the one -hour analysis time frame, the cumulative number of vehicles that have crossed the stop bar is also recorded. This information is used to calculate the arrival rate during the last 15-minute period. 5. The minimum reasonable Peak Hour Factor (PHF) for the approach and time period in question is obtained from historical traffic counts. The analysis software constructs a theoretical set of minimu m and maximum cumulative arrival curves using this minimum PHF and the calculated arrival rate during the last 15-minute period. 6. The analysis software then calculates the cumulative curve delay (overflow delay) associated with the minimum and maximum cumulative arrival curves. 7. The cumulative curve delay is then converted to stopped delay by th e application of a correction factor (approximately 0.77) de rived from trajectory analysis. 8. The corrected maximum theoretical stopped delay is used as an upper bound for the predicted stopped delay and the corrected mi nimum theoretical stopped delay is used as a lower bound. If the pred icted stopped delay falls outside of the theoretical bounds during any of the four 15-minute pe riods, then the predicted delay is appropriately adjusted to remain within the bounds. The resulti ng hybrid stopped delay is considered the final stopped dela y prediction. Note that the theoretical bracketing of the predicted st opped delay is carried-out in an ex post facto manner, after the analysis time frame has expired. 9. The hybrid stopped delay result s are converted to control delay using a series of conversion ratios that vary by cycle length a nd v/c ratio. The conversion ratio varies between 1.2 and 1.4 with 1.3 being a typical value. Th e hybrid control delay is considered the final control delay pred iction for project evaluation purposes. By using the maximum amount of informa tion available and by recognizing the true characteristics of overflow delay, this procedur e produces, for over-saturated conditions, a delay estimate that is generally superior to that f ound in the Highway Capacity Manual and does so in real time.. The proposed delay estimation t echnique should prove useful for both real-time traffic control and project evaluation. It is e nvisioned that the eventual end product of this theoretical research will be a se lf-contained delay estimation module that could be attached to

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29 either a closed-loop or centralized signal control system, or could be inserted within the software of a local traffic signal controller.

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30 CHAPTER 3 CURRENT STATE OF THE ART A literatu re review was conducted to identify both past and ongoing research efforts affecting the area of interest. The studies obtaine d from this search can be segregated into the following general areas: Real Time Measurement of Intersection Delay, Vehicle Reidentification via Inductance Loops, Performance of Video Detection Systems, Signalized Intersection Queuing and Delay, and Probe Vehi cle Monitoring. Quite a bit is known about intersection control delay, especia lly for under-saturated conditions and for situations where all of the information needed to calculate delay is known. The current state of knowledge with respect to over-saturated conditions is more primitive and the results less tested. Real-Time Measurement of Intersection Delay In 1994, Maddula [9] studied signalized inte rsection delay using an AUTOSCOPE 2003 video detection system This system is based on a tripwire approach and has count, presence and speed detectors. The system can provide inte rval data (from 10 seconds to 1 hour) and event data. The computational model developed makes us e of a mandatory detec tion pattern that has 4 detectors in each lane. The first upstream detect or (position 1) is located as far upstream as possible such that section length includes all dela y associated with the signal and identifies the beginning of the Approach Delay Section (defined as the section wh ere most, or all, of the approach delay is incurred) and reports arrival events. Position 2 is an additional upstream detector located between position 1 and the stop line. This detector accounts for vehicles changing lanes. It is used to estimate any missi ng data at other positions. Position 3 is at the stop line and defines the end of the approach delay section and reports departure events. Position 4 is beyond the stop bar and is used to determ ine the signal indication. Position 4 houses a directional detector.

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31 The first step is the identification of each ev ent in their chronological order. This step includes the removal of all ev ents that lead to unrealistic headways (FILTER I). The second step in the process is the use of the data from detector positions 3 and 4 to determine the signal status associated with ever y recorded event. The following user input is required to conduct the search: 1) beginning of red i ndication for first cycle, 2) limits of travel time between positions 3 a nd 4, and 3) limits of red indication for the phase. Each event is associated with a signal indicati on (red or green) and a cycle number. This step includes the removal of all events that lead to departur es when there is no right-of-way (FILTER II). The third and final step is the computation of the MOEs (throughput, stops, saturation headways, and saturation flow rate). Volume is computed from thr oughput and the estimated green time is treated as effective green time. Delay is then calculated using the 1985 HCM formula and LOS is identified via the HCM signaliz ed intersection LOS table. The calculations are done using a computer prog ram written in C called ADELAY. The inputs to ADELAY are an ASCII detection file from the video system w ith extension TXT (the events) and a text file with extension VXT (other required informa tion) from the VIADET user interface program. The report defines the Approach Free Flow Time as the time used by an unimpeded vehicle to traverse the approach delay section and defines the Approach Time as the time used by an impeded vehicle to traverse the approach delay section. Approach Delay (defined as the Approach Time minus the Approach Free Flow Time) is converted to Stopped Delay (defined as the time that the vehicle is stopped with stationary wheels) for comparison to field observations by dividing by a factor of 1.3 The raw data are converted to usab le data using three filters: FILTER I. False detections (glare, reflections, turn signals) resulting in unrealistic headways (1 second is used as a minimum realistic headway)

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32 FILTER II Detections at position 3 that lead to departures when there is no right of way (detections during red produced by pedest rians, crossing vehicles, etc.) FILTER III. Unrealistically high throughput (continuous detection due to shadows, turn signals) Maximum Throughput = Green Time / Minimum Headway All vehicles that arrive on the approach dela y section and depart be fore the end of the green of the current cycle are re ported as throughput for the cycle. If a vehicle could not clear the intersection before the end of the green, it is reported as thr oughput for the next cycle. When the throughput reported at various positions in th e lane is different (due to lane changing or detection errors), the maximum number of vehi cles reported at any position is taken as the throughput for the cycle. Every vehicle that arrives befo re the beginning of the green indication, minus the free flow travel time within a current cycle, is automatically treated as a stop. The fr ee flow travel time for the vehicles that arrive after the stated time is calculated at 5 miles per hour (mph). If the travel time of the vehicle is more than th is time, it is treated as a stop for that vehicle. (i .e. a vehicle is defined to have stopped if the actual travel time is more then the free flow travel time calculated at a speed of 5 mph.) Reported departure times are used for determ ining saturation headwa ys and calculating the saturation flow rate. Headways associated with the first 3 vehi cles in the queue, and headways of more than 3 seconds, are not used. If the nu mber of vehicles in the queue never exceeds three throughout the study, then default saturation flow ra tes are used that vary by lane type (1756 for a thru lane, 1946 for a single left turn la ne, and 1651 for a dual left turn lane). A preliminary study for a limited number of ob servations indicated that, for queues of passenger cars, average distance headway (front bumper to front bumper) is 25.1 feet and average spacing between cars is 9.0 feet. This yields an average car length of 16.1 feet.

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33 The report defines the Time-in-Queue Delay (a.k.a. Time-in-Queue) as the time from the vehicles first stop to the vehicles exit across the stop lin e. The report also defines Percentage of Vehicles Stopping as the number of vehicles incurring Stopped Delay divided by the number of vehicles crossing the stop line. Of the traffic parameters investigated, vehi cle count, delay and level of service were obtained accurately from the data reported by VI DS (Video Image Detection System). However, throughput and stops were not. Minor changes in detector size, placement and orientation caused noticeable variation in the results. Data missi ng at a particular dete ctor location was often available at another detector location, which argu es for the use of multiple detection systems for evaluation. The basic limitation of this work with respect to the research at hand is that it relied on a relatively optimum detection configuration and was not used for estimating delay during oversaturated conditions (a time when de lay estimation is most critical). In 1998, Lall, et al. [10], developed a speed-based procedure for calculating delay on a signalized intersection approac h. For a 15 minute study period, traffic volumes and average speeds were recorded every 10 seconds using AUTOSCOPE at 5 distances from the stop bar (20 ft, 65 ft, 88 ft, 267 ft & 500 ft). Free-flow speeds (for vehicl es not stopping) and prevailing speeds (for vehicles stopping) we re calculated and associated trav el times compared to estimate delay. The comparison checked well with control delay calculated for the approach using the HCM. If posted speed is used instead of prev ailing speed the delay calculated is substantially higher and probably corresponds to t otal delay, wherein total dela y is defined as the difference

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34 between the travel time actually experienced and the reference travel time that would result during ideal conditions. 1 The authors noted that the longer the lens focal length (view more zoomed in), the easier and more robust is vehicle tracking and detection. The shorter the focal length of the lens, the smaller the objects are on the image, but the larger the field of view If the vehicle image is smaller than 5 pixels of the image that is analyz ed by their video system, the tracking of vehicles becomes rather unreliable. Two types of shadow problems were revealed. The first problem occurs when a tree, tall building or some other tall object is close to th e section of roadway bein g monitored. On sunny days, the objects shadow will cover the monitored roadway at certain times of the day. If a vehicle enters the shadow, it may become barely visi ble, especially if the vehicle is dark. If a detection zone is located in th e area covered by the shadow, the de tection performance from this zone may be seriously impaired. A second type of shadow problem occurs due to vehicle shadows. A shadow of a moving vehicle in one lane may sweep over the detection zone in another lane. This sweeping shadow may be taken for a vehicle. The authors solved the problem of thru lane vehicles activating left turn lane detection thr ough the use of a 1.2 second detector delay setting (for a 6 foot detector length). However, experience with this site indicate s that the accuracy of video detection is adequate (the average maximum error is only about 5%). It is better than the accuracy of loop detectors at this location, which gave a maximum error rate of 10%. 1 The important delay calculations contained in Tables 2 and 3 of this report cannot be followed given the information contained in the report and I contact ed the primary author for clarification. Unfortunately, the author did not provide a response.

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35 In 1999, Quiroga, et al. [11], developed a proce dure based on linearly referenced GPS data that can be used to accurately measure both control delay and stopped delay. Algorithms were developed which accurately detect when a GPS-e quipped probe vehicle eith er begins or ends acceleration or deceleration. More than 100 floati ng car travel time runs were made along two coordinated corridors having a background cycle length of 150 seconds. In addition to establishing the viability of this procedure for accurately determining stopped delay and control delay, the following was discovered: 1. A linear relationship exists be tween stopped delay and control delay. However, the line does not pass through the origin. It was found that control delay = (stopped delay + 19.3 seconds) x 1.04, which is quite different than the control delay = 1.3 x stopped delay formulation provided in the Highway Capacity Manual. The authors caution that other independent variables, such as length of th e red interval, may be needed to properly generalize this equation. 2. An average end-acceleration distance of 427 feet downstream of the stop bar was established. An average begin-deceleration distance of 951 feet ups tream of the stop bar was also established, but this distance obviously depends on the extent of queuing at the intersections. 3. Approximately 5% of the inters ection control delay occurred after the vehicle crossed the stop bar. In 2001, Saito, et al. [5], estimated stopped delay using simulated vehicle images generated by CORSIM and two image analysis methods: the gap method and the motion method. A simulation duration of 15 minutes was used. Th e simple algorithms that were developed produced promising results. The authors defined Percent Deviation using the following formula: Percent Deviation = [Delay Estimated by Model Delay Estimated by CORSIM]/(Delay Estimated by CORSIM) x 100 In 2004, Zheng, et al. [12], developed a met hodology for using video image processing to accurately detect queue lengths and phase fail ures on a signalized intersection approach. A Trafcon video system was used to test the pro cedure on an actual inters ection approach with a

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36 field of view of about 18 vehi cles. The camera was mounted 26 feet above the ground and was oriented at a 30-degree downward angle. Th e video algorithm extracts stopped vehicle information from the traffic stream, tracks the e nd of the queue, and identifies phase failures. Zheng concludes that: The program based on this algorithm may pr ovide reliable and accu rate [phase] failure detections in real time for many traffic mana gement and operation purposes if the camera that provides the video stream is correctly positioned to see the stop bar and a sufficient number of queued vehicles. We can safely assume that, if the camera cannot see a suffici ent number of queued vehicles (with a sufficient number obviously being to the end of the queue ) then Zhengs technique will provide erroneous results; hen ce, the need for the extension provided in this research. In 2004, Hoeschen, et al. [13], developed a procedure for using travel time between intersections (expressed as segment delay) to approximate control delay. The approximation was found to be much better than using stopped delay to estimate control delay, especially for higher delay values. Control delay was appr oximated by subtracting mid-block delay from segment delay. The authors cautioned that queue spillback from a downstream intersection or non-recurring delay could negativel y affect the results. The segment lengths for the research varied between mile and 1 mile in length. 300 feet was selected as the distance from the upstream intersection at which most vehicles ha d accelerated to running speed. 300 feet was also selected as the distance from the downstream in tersection at which vehicles began decelerating. Vehicle Reidentification via Inductance Loops In 1999, Sun, et al. [14], exa mined the vehicl e re-identification pr oblem on freeways. A vehicle waveform pair can be formed by us ing one downstream waveform and one upstream waveform. The vehicle re-identif ication problem is to find the ma tching upstream vehicle from a set of upstream vehicle candidates given a downstream vehicle.

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37 Inductive loop detector manuf acturers are incorporating the ability to monitor and output vehicle inductance values (or waveforms). Dete ctors that output vehicle waveforms include detectors manufactured by: Peek/Sarasota, In tersection Development Corporation (IDC), and 3M. The authors concluded that solution of the vehicle re-identifica tion problem has the potential to yield reliable secti on measures such as travel times and densities. Implementation of their approach used conventional surveillance infrastr ucture; 6 by 6 freew ay inductive loops spaced 1.2 miles apart on a 4 lane westbound stretch of freeway with no intervening ramps. Typical 6 x 6 loops produce a le ss distinctive waveform that is more difficult to re-identify compared with shorter (3.3) European loops. Th e 13 to 14 ms detector sampling period of most detectors is also problematic in that it misses sharp corners of the waveform. Previous approaches that uti lized sequences (Bohnke and Pf annersstill, 1986) are suitable for the case when sequences of vehicles are preserved from upstream to downstream. The preservation of sequences occurs when there is very little lane changing and the speeds across all traffic lanes are similar. The approach used in this study is suitable for cases where there is significant difference in lane speeds. This appr oach also has the potential to yield partial origin/destination demands and indi vidual lane changing information. This paper formulates and solves the vehicle re-identification problem as a lexicographic optimization problem using goal programming. Goal Programming is an optimization method wherein target values are set for each of th e multiple objectives and then a single global objective, which is the sum of th e deviations from the target values over all objectives, is optimized. Lexicographical Goal Programming is a goal programming pr ocedure wherein the multiple objectives are introduced in a specified hi erarchical order. The lexicographic method is

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38 a sequential approach to solving the multi-obj ective optimization problem where each objective is ordered according to its importance. Multi-Objective Optimization is defined as the discovery of optimum points x* within a feasible set x th at are as good as can be obtained when judged according to multiple criteria. A Pareto Set (a.k.a. an Efficient Frontier) is the optimum solution for multi-objective problems in that it contains all points (efficient points) for which there does not exist any other point that would be uniformly better on all objectives. The results of the prior level of optimization constrain the feasible set for the current level of optimization. A lexicographic method has adva ntages over the traditional weighted average method in that the problem of specifying relevant weights when the multiple objectives are measured in different units is avoided and, by introducing the multiple objectives sequentially, the individual effect of each objective can be identified. Five levels of optimization (multiple objectives) are used. The first three are implemented as goal programs. They are used to reduce th e feasible set by eliminating unlikely waveform pairs. Level 1: travel time Level 2: vehicle inductance magnitude (the induct ance magnitude is inversely proportional to the height of the vehicle) Level 3: vehicle electronic length (derived from occupancy time) Maximum tolerances must be set for each leve l and a minimum toleranc e must also be set for travel time. Level 4 uses a traditional weighted average utility function of the change in inductance magnitude, lane changes, and change in vehicle speed between the upstream and downstream detection points. Level 5 has a stocha stic objective that is solved using Bayesian analysis. Calibration of the algorithms was performed with training data.

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39 This research shows that the direct measurem ent of section measures of traffic system performance such as travel times and densities av oids the inaccuracies as sociated with estimating such values from point speeds and occupancies. This research also shows that values of point and section measures derived fr om freeway data differ significantly. The authors also concluded that congestion causes more variab ility in the traffic stream which translates into more mismatches. Th e authors also cautioned that, when a higher percentage of trucks are matc hed (which often happens since they are longer and have more distinguishable features), sp eed results could be biased. In a 2000 paper, Palen, et al. [15], discusse d three phases of Caltran s detector research dealing with vehicle re-identific ation. Phase I initially used ex isting detectors with bivalent output only. Bivalent Output is defined as a detector output wherein just the presence or absence of a vehicle is reported. Vehicle lengths (calculated fr om loop-based time and distance data) and headway sequences were used to match platoons of vehicl es. Vehicle lengths can only be calculated plus or minus 10% using conventional loop detection so additional sequence information based on headway distributions was n eeded to obtain useful results. Since model 170 traffic signal controllers lack the computa tional power needed to carry out the matching calculations for the sequence information, bivalent loop data was brought back to a web server via a wireless Internet Protocol (IP) modem. A stretch of I-80 ne ar San Francisco currently uses this technique to obtain performance measures. Phase II used commercially available scanning detector cards to obtain loop signatures. These signatures were used to match vehicles. This technique was applied to an intersection approach in Irvine, California having a 2070 controll er. This process is mo re accurate than the Phase I process and loops can be spaced further apart.

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40 Phase III examined new loop geometries. In a 2001 study, Liu, et al. [16] used a vehicle re-identification algorithm developed at UCIrvine to estimate the average and total delay by movement during each cycle at a signalized intersection, and these estimates were then fed to an on-line signal control algorithm to find the optimal green splits. Vehicle re-identificatio n based on inductive loop signatures was used to estimate the delay. Knowing the prevailing free flow speed for the approaches, and the distance between detector stations, the minimum travel time for each movement can be derived. The delay of each vehicle was calculated by deducting this minimum travel time from the vehicles actual travel time. The analysis was conducted at the Alton/Irv ine Center Drive intersection in Irvine, California with the microscopic simulation program Paramics used for online signal optimization as a complementary module to the existing signal controller. Paramics provides a framework that allows the user to customize many featur es of the underlying simulation model with access provided through an Application Pr ogramming Interface (API). I nductance loops were used for both vehicle detection and delay estimation in Paramics. Thirty simulation runs were made for each scenario with each run comprising a 2-hour period. The use of multiple simulation runs perm its statistical evaluation. Three measures of effectiveness were evaluated: total intersection dela y, total throughput and average delay. The average delay-based on-line control algorithms pe rformed better than the off-line case for both pre-timed and actuated signal control (as evidenced by a 10% reduction in delay). In 2002 Sun, et al. [17], invest igated the use of video cameras to improve the accuracy of vehicle re-identification using i nductance loops. In this researc h, color information from video cameras was used to augment the inductive signat ure obtained from inductive loop detectors to

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41 track individual vehicles. When inductive loop signatures alone ar e used, vehicles of the same model or even different models on the same body frame can be mismatched. On the other hand, the use of video alone can be sensitive to change s in illumination levels (night, dusk, dawn, rain, glare, etc.) The test section was located in one direction of a 4-lane arteri al. The two lanes of arterial traffic for the test section were treated separate ly; lane changing was ignored. Detector stations, each of which consisted of a speed trap (double inductance loops), were located 425 feet apart. A traditional method of vehicle re-identification is license pl ate matching. Other potential methods of vehicle re-identificati on involve GPS, cellular, toll tags or tracking beacons. Section measures can also be obtained via video using tripwire systems or through vehicle tracking. The advantages of using vehicle colo r are that it is not correlated with vehicle signatures (i.e. represents an independent iden tification measure), it can be extracted from imperfect video images, and it can be verified visually. Linear feature fusion with six features was used in this study. The features used were: 1) vehicle signature, 2) vehicle velocity (distance between loops divided by turn-on time), 3) platoon traversal time (time between first and last vehicle in platoon cro ssing loop), 4) maximum inductive amplitude (inversely proportional to the cube of the distance from the ground to the vehicle undercarriage), 5) electronic length (length of metallic components only but includes the length of the magnetic field gene rated by the loop), 6) RGB triplet (color). The combined classifier score due to li near fusion is calculated us ing the following formula: Dlinear = i=1,n wi di Where i is an index from 1 to 6 for the six feat ures and di are the feature values. The fusion weights (wi) are determined using an exhaustive s earch such that the re-i dentification accuracy is

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42 maximized. The candidate upstream platoon that achieves the smallest D is matched to the downstream platoon. A time window constraint with upper and lower bounds is applied to identify candidate platoons. The research concluded that the use of de tector fusion provides system redundancy and yields better results than the use of either inductive signature information or vehicle color information alone. A re-identification rate of over 90% was obtained using multi-detector fusion whereas the rate was 87% for inductive signature information alone and only 75% for color alone. The authors postulated that the re sults would be even better if the vehicle re-identification system could be tied into the arterials signal co ntrol system since this would allow the direct estimation of lost time associated with star ting and stopping. The tie-in would improve the accuracy and possibly yield real-tim e estimates of startup delays an d saturation flow rates. The authors added that it is difficult to compute arterial travel times accurately using point measures (speed, occupancy, counts) since lost times a ssociated with starti ng and stopping are not measured directly. The authors provided the following definitions in the report: Point Traffic Parameters traffic parameters that pert ain to a particular point on the roadway (volume or flow, point speed, presence, occupancy) Section Traffic Parameters traffic parameters that pertai n to a section of roadway (link speed, travel time, origin/destination information) Platoon Matching a method of vehicle re -identification that matc hes groups of vehicles rather than individual vehicles. In 2002, Oh and Ritchie [18] used inductance l oop signature data to track vehicles form upstream approach loops to receiving lane loops at a signalized inte rsection. Features used in the lexicographic optimization were maximum magnit ude difference between front and back loops

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43 (relates to vertical clearance), vehicle speed, an d lane information. The matching rate was 32.5% for vehicles turning right, 51.7% for thru vehicles, and 62.5% for vehicles turning left, for an overall match rate of 46.7%. Left turns were elim inated from the analysis due to low absolute volume. Cluster analysis was used to determine LOS categories based on reidentification delay (RD). Reidentification Delay is defined as th e difference between the actual time required to traverse vehicle reidentification st ations at a signalized intersect ion and a base travel time (such as that calculated from the speed limit). Two di fferent aggregation methods were investigated, cycle-length based average (CBA) and fixed time av erage (FTA). A fixed interval of 60 seconds was used for FTA. K-means clustering, fuzzy clustering, and Self Or ganizing Map (2 layer neural network) methods were us ed in the clustering analysis. Wilks lambda was used to compare the results: Wilks lambda = |W|/|B+W| W = pooled within-group variance B = between group variance A lower Wilks lambda value indicates better clustering. K-means clustering produced the best results, with the most appropriate number of clusters being 5. When compared to ground truth, reidentification delay errors were on the order of 26% A rolling average RD based on 3 signal cycles was recommended to avoid signal control related stability problems associated with single cycle delay reporting. A recommended RD LOS classification system is presented with LOS I (excellent) through V ( poor). The LOS table stratification values are similar to those containe d in the HCM if LOS F is eliminated. Slightly different LOS stratification values are provi ded for right turn and thru movements. Mean Absolute Percent Errors were calculated using the following formula:

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44 MAPE = [ i=1,N(ARDi-AADi/AADi) x 100 ]/ N MAPE = Mean Absolute Percent Error ARDi =Average Reidentification Delay at time step i AADi =Average Actual Delay at time step i N = total number of time steps In a 2003 paper, Coifman and Ergueta [19] presented an improved algorithm for vehicle matching at a freeway inductive loop detector st ation having dual loops. This new algorithm, which includes four separate tests, performe d significantly better than older algorithms developed in previous work by the authors. Th e algorithm should be applicable to any detector technology capable of extracting a reproducible vehicle si gnature. In this study, vehicles were matched based on length and lane changing was accounted for. The algorithm matched between 35% and 65% of the vehicles, depending on lane. The authors noted that other researchers have es timated that matching 20% of the population is sufficient for travel time measurements. Ma tching percentage is improved as the speed decreases. The report defined a False Positive as a collection of incorrect matches and Effective Vehicle Length as Physical Vehi cle Length plus Length of the Dete ction Zone. The algorithm is attractive in that it utilizes existing surveill ance equipment and performs well under congested conditions. In 2004, Coifman and Dhoorjaty [20] presented eight detector validation tests for freeway surveillance. Five of these tests can be applied to single-loop detectors while all of the tests can be applied to dual-loop detectors. The tests are used to compar e the performance of different detector models and to identify permanent or transient hardware problems such as crosstalk between loops and shorts in the loop wire. Thr ee of the tests could be applied to arterial loop detectors and these tests could be incorporated into the cont roller software for continuous

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45 monitoring. The authors discovered that some dete ctor units stay on a fraction of a second after the vehicle passes and some are prone to flicker (turning on and off multiple times as a vehicle passes). A large variability in detector operation was noticed from one mode l to the next and, in the case of one of the detectors, from one software revision to the next within the same model. In a 2007 paper, Jeng, et al. [21] described an inductance loop based vehicle reidentification algorithm (RTREID2) that produced excellent results when compared to GPS information from control vehicles. Performance of Video Detection Systems In 1999, Washburn and Nihan [22] evaluated the Mobilizer, a video im age detection system based on vehicle tracking developed by Condition Monitoring Systems. Preliminary results indicated that the Mobilizer is capable of matching vehicles in su ccessive fields-of-view with a reasonable degree of accuracy and that the travel time estimates provided by the system are statistically valid. Two sites were evaluate d, one on an arterial a nd one on a freeway. For both of these sites, a departing FOV (Field of View) was used. The ar terial had 76% correct matches while 78% of the freeway matches were co rrect. The system can be instructed to not consider matches that fall outside of dynamic travel time ranges, ranges that are adjusted in realtime by the system, however, the system does not currently utilize color information and the system does not consider matches of vehicles that change lanes. The system was only evaluated under free flow conditions. In 2001, Grenard, et al. [23], evaluated va rious video detection systems (Autoscope, VideoTrak and Odetics) for signalized intersections. They discovered that: The effective length of the detection zone incr eased from an average of 23.7 feet during the day to an average of 67.7 feet at night, wh ich could cause the signal to operate less efficiently. The percentage increase in effective detection length at night due to headlight glare ranged between 50% and 500%; this a dds 2 seconds of detection time.

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46 False video detections became slightly larger at night with rain due to headlight glare. Video detection frequently only de tects the headlights at night so the call is lost if the video detection zone ends just a few feet in front of the stop bar. Extending the video detection zone somewhat past the stop bar would help to remedy this situation, but at the expense of detecting additional pedestrians or crossing/lef t turning traffic. This produces both safety (due to missed calls) and efficiency problems. Illuminating the intersection eliminates this problem. The video detection systems tested sometimes stuck on for substantial periods of time. During dawn and dusk, sunlight causes so much glare that the camera is often unable to distinguish between the absen ce and presence of vehicles. Wet pavement does not significantly impact th e likelihood of a T0L1 error (loop on when no vehicle is present) but traffic volume does (probably due to spillover). Neither wet pavement nor traffic volume significantly impact the likelihood of a T1L0 error (loop off when vehicle is present). Under base (optimal) conditions, the video detec tion system has a false detection rate of 2% to 6% and a missed vehicle presence of between 7% and 8% The authors distinguished between Error, defined as video results compared to actual or ground truth and Discrepancy defined as video results compared to another type of detection system (such as loops). Discrepant calls include false calls and missed calls (discrepancies of less than 3/10 of a second were not recorded). Discrepant Call Frequency is defined as the number of discrepant calls per cycle. Error Rate is defined as the ratio of discrepant calls to true calls and Relative Error Rate is defined as the ratio of the error rate to the average error rate. Under worst-case conditions (rain, night, wet pavement, average count, heavy camera motion) video detection misses between 16% a nd 20% of vehicle presence time and indicates false detection during about 40% of the vehicle absence time. The authors defined Activation Distance as the distance a vehicle is from the stop bar when it is detected by the video detection system, and Blanking Band as a process used to remove all discrepancies smaller than a user-defined value. Due to the imprecision of night detection, the authors recomme nded that video detection not be used to provide dilemma zone protection. The authors cited past work in this area: MacCarleys 1992 evaluatio n of video detection found that several conditions caused significant de gradation in video detection performance: non-optimum camera placement, day-to-night transition, headlight reflections on wet pavement, shadows, fog, heavy rain with error rates of 20% to 40% for most tests performed. MacCarleys 1998 evaluation of video detectio n found that several additional conditions caused significant degradation in video detection performance: transverse lighting, low

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47 lighting and vehicles that have a low contrast to the pavement. 65% of all vehicles were detected correctly with an 8.3% false detection rate. 64.9% of all red-green transitions would have been actuated corre ctly if video were used instead of properly functioning loops. Middletons 1999 evaluation of vide o detection found that video de tection: 1.) consistently over-counted by as much as 40% to 50% at night, 2.) at dawn and dusk sun angles produced glare that caused undercount rates of 10% to 40%, 3.) undercounted by 6% to 8% during heavy rain. The most consistent period of error was between midnight and 5:00 am. Middleton and Parkers 2000 evalua tion of video detection found that video detection: 1.) over-counted both day and night during wet pavement conditions because of headlight reflections, 2.) had reduced accuracy at night and when long shadows occurred. The authors provided the following formulas for calculating detection errors: Missed Detection Rate (MDR) = Number of Ac tual Detection Events Missed By Loop/Total Number of Actual Vehicle Arrivals (discrete definition) P(L=0|T=1) = D(L=0 & T=1)/D(T=1) where D=Duration, T=Ground Truth, L=Loop, 1=On, 0=Off (continuous definition) False Detection Rate (FDR) Number of Fa lse Detection Events Reported By Loop/Total Number of Inductive Loop Events (discrete definition) P(L=1|T=0) = D(L=1 & T=0)/D(L=1) where D=Duration, T=Ground Truth, L=Loop, 1=On, 0=Off (continuous definition) P(L=1|T=0) = D(L=1 & T=0)/D(T=0) where D=Duration, T=Ground Truth, L=Loop, 1=On, 0=Off (revised co ntinuous definition) For the likelihood (probability) of a detection discrepancy the following formulas apply: The probability of video detection being o ff when loop detection is on = P(V=0|L=1) = D(V=0 & L=1)/D(L=1) where D=Du ration, V=Video, L=Loop, 1=On, 0=Off The probability of video detection being on when loop detection is off = P(V=1|L=0) = D(V=1 & L=0)/D(L=0) where D=Du ration, V=Video, L=Loop, 1=On, 0=Off For the likelihood (probability) of a detec tion error the following formulas apply: The probability of video detection being off when a vehicle is present = P(V=0|T=1) = P(L=1|T=1) x P(V=0|L=1) + P(L=0|T=1) x [1-P(V=1|L=0)] The probability of video detection being on when a vehicle is not present = P(V=1|T=0) = P(L=1|T=0) x [1-P(V=0|L=1)] + P(L=0|T=0) x P(V=1|L=0) In 2002, Bonneson and Abbas [24] investigated th e operation of Video Imaging Vehicle Detection Systems (VIVDS) in Texas. It was es timated that about 10% of the intersections in

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48 Texas were using VIVDS and th at Texas DOT was installing VI VDS at about of all newly constructed intersections. They identified the following VIVDS manufacturers: Image Sensing Systems (Autoscope system used by Econolite), Iteris (Vantage system used by Naztec and Eagle), Peek Traffic Systems (VideoTrak sy stem), Traficon, Nestor Traffic Systems and Transformation Systems. A review of VIVDS pr oduct manuals revealed that these manuals do not describe techniques for the effective use of delay, extend, or passage time settings in conjunction with a VIVDS installation. Their report made the following points: Detection zones can be linke d via Boolean logic functions (AND, OR, NOT, etc.) VIVDS can provide reliable presence detecti on when the detection zone is relatively long (say, 40 ft or more). However, its limite d ability to measure gaps between vehicles compromises the usefulness of several controller features that rely on such information (such as volume-density control). A VIVDS system is sometimes used to provide advance detection on high-speed intersection approaches. However, some engi neers are cautious about this use because of difficulties associated with the accurate detec tion of vehicles that are distant from the camera. Of those agencies that use a VIVDS for advance detection, the most conservative position is that it should not be used to monitor vehicl e presence at distances more than 300 feet from the stop line. The minimum camera height (in feet) for advanced detection is calculated using the formula: Ha = (xl + xc)/R Where x1 is the distance in feet between the stop line and the upstream edge of the detection, calculated as: xl = 1.47tbzV95, and: xc = distance in feet be tween camera and stop line R = distance-to-height ratio (17 in Texas) Tbz = travel time from the start of the d ilemma zone to the stop line (5 seconds) V95 = 95th percentile speed in mph (= 1.07 x V85) Table 4-2 in the report provides the re sulting minimum required camera heights for advanced detection. The required height varies between 24 feet and 36 feet.

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49 A cameras field of view is impacted by th e following factors: camera height (distance from ground to camera), camera offset (lateral distance from camera to the lane or lanes being monitored), distance (longitudinal distance from the detection zone to the camera), pitch angle (angle of downward tilt of th e camera relative to the ground), and focal length (which determines the relative size of objects in the cameras field of view). Detection Design is defined as the selection of came ra location and the calibration of its field of view whereas Detection Layout involves locating detection zones, determining the number of detection zones, and identifying th e settings or detecti on features used with each zone. The ft to 1 ft rule states that, if ca mera set up is optimal, one should be able to extend out 10 feet for every 1 feet of cam era elevation to a maximum distance of around 300 feet. However, Texas DOT staff indi cated acceptable operations using 17 feet instead of 10 feet. Detection accuracy will improve as camera he ight increases within the range of 20 to 40 feet. Increased height improves the cameras fi eld of view of each approach traffic lane by minimizing the adverse effects of occlusion. Three types of occlus ion are present with most camera locations: adjacent-lane, same-lane and cross-lane. Increasing camera height tends to decrease call error, provided there is no increase in camera motion. Cameras mounted above 34 feet may expe rience unacceptable camera motion unless located on a stable pole. Adjacent-Lane Occlusion (Horizontal Occlusion) occurs when the blocked and blocking vehicles are in adjacent lanes, which can result in false detections in adjacent lanes. Table 4-1 of the paper provides minimum required camera heights to reduce adjacent-lan e occlusion. The required height depends on the lateral offset, whether the offset is to the left or to the right, and the la ne configuration, and varies between 20 feet and 63 f eet. The minimum required hei ght is lowest for a camera mounted in the center of the approach, 20 feet. Same-Lane Occlusion (Vertical Occlusion) occurs when the blocked and bloc king vehicles are in the same lane, which can result in a low vehicle count. The extent of this problem increases as the distance from the stop line increases. Same lane occlusion is associated with an increase in the effective length of a vehicle. Consequently, passage settin gs must be reduced to yield operation equivalent to that obtai ned with an inductance loop. Cross-Lane Occlusion occurs when a vehicle crosses between the camera and the intersection approach being monitored, which can result in false detections. The optimal field of view for a camera is one that has the stop line parallel to the bottom edge of the view and in the bot tom one-half of this view. The optimal field of view also includes all approach traffic la nes. The focal length should be adjusted such that the approach width, measured at the stop line, represents 90% to 100% of the horizontal width of the view. The view must exclude the horizon. Detection accuracy is significantly degraded by glare from the sun and, sometimes, from strong reflections from smooth surfaces. Sun glare typically caus es problems for the eastbound and westbound approaches. A larger pitch angle can reduce the im pact of sun glare and a camera equipped with an automatic iris (or electronic shutter) will minimize

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50 the adverse effects of reflection. An infrared filter can also reduce the adverse effects of glare. VIVDS processors have the ability to detect excessive glare or reflection and automatically invoke maximum recall for the troubled approach. Detection Accuracy is defined as the number of times that VIVDS reports detection when a vehicle is in the detection zone, or reports no detection when a vehicle is not in the detection zone. Most VIVDS have separate image-proces sing algorithms for daytime and nighttime conditions. The daytime algorithm searches for vehicle edges and shadows. During nighttime hours, the VIVDS searches for the vehicle headlights and the associated light reflected from the pavement. Research ha s found that the nighttime algorithm is less accurate than the daytime algorithm and also has a tendency to place calls before the vehicle actually reaches the detection zone. Intersection lighting can minimize the extent of this problem. The detection design should a void having pavement markings cross the boundaries of a detection zone since camera movement combined with high-contrast images may confuse the image processor and trigger false calls. The following equations are prov ided for determining the required length of a stop line detection zone: lsl = vq (MAH-PT) lv lv* = (lv-lro) + xc(hv/hc) lsl = length of stop line detection zone in feet vq = maximum queue discharge speed at the stop line (use 40 ft/sec) MAH = Maximum Allowable Headway (use 3 seconds) PT = controller Passage Time in seconds lv = effective length of vehicle in feet lv = length of design vehicle (use 16.7 feet) lro = distance from back axle to back bumper of design vehicle (use 4.3 feet) xc = distance between the camera and the stop line in feet hv = height of design ve hicle (use 4.5 feet) hc = height of camera in feet The detection zone length should be approximately equal to th e length of a passenger car in order to maximize sensitivity. Stop line detection typically consists of multiple detection zones. For reliable queue service, detection zones should ex tend at least 40 feet from the stop line Zone Location is defined as the distan ce between the upstream edge of the detection zone and the stop line. The camera field of view should be established to avoid inclusion of objects that are brightly lit in the evening hours, especially those that flash or vary in intensity. If these sources are located near a detection zone, th ey can trigger false calls. The light from

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51 these sources can also cause the cameras to reduce its sensitivity by closing its iris, which results in reduced detection accuracy. Each VIVDS detection zone has a directional mode that allows it to recognize calls only for traffic moving in a specified direction. However, this mode appears to reduce the sensitivity of the detection zone. During daytime hours, swaying power lines, su pport cables or signa l heads can trigger false calls as they move into and out of the detection zone. The performance of VIVDS is adversely affected by environmental conditions such as fog, precipitation, and wind. Condensation and dirt buildup on the camera lens can further degrade VIVDS operation. Shadows can extend into a de tection zone and trigger fals e calls or compromise the VIVDS ability to detect vehicles. Delay settings are sometimes used to reduce the frequency of false calls. For example, a few seconds of delay is often set for st op line detection zones on the minor street approach. The delay eliminates false calls at night caused by right-t urning vehicles from the major road whose headlights sweep across the detection zone. It also eliminates false calls due to cross-lane occlusion caused by tall vehi cles on the major road. A lens adjustment module is an essential VIVDS-related installation device. It connects to the back of the camera and is used during camera installation to adjust the cameras zoom and focus settings. Having this device facilitates camera replacements or adjustments. Enough room is needed in th e controller cabinet to house the needed VIVDS equipment. Standard RG-59 coaxial cable is good for up to a distance of about 500 feet for connecting the camera to the hardware in the controller cabinet. Satisfactory operation of a VIVDS requires verifi cation of the initial layout and periodic on-site performance checks (at least every 6 months is recommended). A review of some existing VIVDS installations in Texas indicated that there was more than one discrepant call each cycle with about 1.8 discrepant calls per true call. About 80% of the discrepant calls averaged less than 2 seconds per call and were typically associated with the VIVDS registering a call s lightly before or after its true arrival or departure time. Wholly missed or false calls were less frequent and often had a duration in excess of 2 seconds. During approxima tely 20% of the signal cycles, a phase experienced about 4 missed calls with the tota l duration of these missed calls being about 25 seconds per cycle. In 2003, Oh and Leonard [25] obtained validat ion results for the PEEK VideoTrak 900 image processing system. The test site was on I-75 in Atlanta. The te st results showed huge

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52 volume errors in some case, especially at night. The system also provided lower speeds than true speeds at night. The farther the lane was from the camera, the more inaccurate was the count. Signalized Intersection Queuing and Delay In 1977, Riley and Gardner [26] investigated various techniques for measuring delay at signalized intersections. Four possible techniques were listed: Point Sample 1st Advantage: self-correcti ng, each sample is independe nt of the previous one 2nd Advantage: not depende nt upon signal indications Disadvantage: accuracy redu ced when counts become high (an upward bias exists such that an adjustment fact or of 0.92 is recommended) Input-Output (a.k.a. Interval Sample) Disadvantage: field data must be corrected for vehicles that ente r or leave the study area between the input and output points (at driveways or cross streets) Path Trace Disadvantage: a very large samp le of vehicles is needed to provide an estimate of delay having reasonable confidence Modeling As part of their work, the authors concl uded that; Once the recommended field data corrections have been made, stopped delay per vehicle multiplied by 1.3 will yield a good estimate of approach delay per vehicle. In 1984, Hurdle [2] proposed the use of delay m odels that take more account of variations in travel demand over time. Hurdle noted that: any steady-state model that does not assume completely uniform arrivals will predict that th e queue length, and therefore the delay, approach infinity as the v/c ratio approaches unity. This is, of course, the reason that systems with a high v/c ratio take a long time to settle into a steady state; it simply takes a long time for such long queues to form, particularly sinc e vehicles keep leaking through the signal. As a result, one seldom sees real delays as large as those predic ted for high v/c ratios. Th is discrepancy is not a

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53 result of faulty mathematics but of the unrealistic assumption that the system is in a steady state. If vehicles continued to arrive at a rate v nearly equal to the capacity c, the giant queues really would form, but in reality the peak period ends and v decreases long before a steady state is reached. As a result, steady-state models are usef ul for predicting delays only at lightly loaded intersections. Hurdle added: there is one group of models the steady-state queuing models, that work well when v/c is considerably less th an one and another type, the deterministic queuing models, that work well when v/c is consider ably more than one. In between, there are problems. He also stated: W hat modeling approaches make very clear is that the development of the queue is very dependent on the details of the arrival pattern more information about arrival patterns must be provided than is now customary. In 1992, Bonneson [27] developed a discharge headway model for signalized intersections that was based on non-constant acceleration beha vior. Bonneson mentions that, in 1977, Messer & Fambro found that, except for the first position, driver response by queue position was fairly constant at 1.0 second. The firs t driver experienced an additiona l delay of 2 seconds. Messer & Fambro also found that the average length of ro adway occupied by each queue position is about 25 feet. Bonneson found this distance to be 25.9 feet. Bonneson used regression analysis to obtain an approximate equation for the Standard Deviation (SD) of delay: SD = 0.42 x (mean delay)0.7. The Maximum Error (ME) in the calculated delay at the 95% c onfidence interval is then: ME = 1.96 x SD = 0.82 x (mean delay)0.7. Bonneson concluded that the minimum discharge headway of a traffic movement is a complex process that is dependent on driver res ponse time, desired speed, and traffic pressure. The discharge headway model developed in his research indicates that the minimum discharge

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54 headway of a traffic movement is not reach ed until the eighth or higher queue position. Bonneson also concluded that: A rather strong inverse linear relationship exists between vehicle acceleration and stop line speed. For the driver acceleration model develope d, the maximum acceleration ranges between 6 and 8 ft/sec/sec with an averag e of 6.63 (this is similar to a value of 6.0 found by Evans and Rothery). For the stop line speed model developed, stop lin e speed increases with queue position in an exponential manner to a maximum value between 46.7 and 51.0 ft/sec with a median value of about 49 ft/sec (33 mph). Traffic pressure (vehicles per lane per cycle) is a significant factor (p=0.001) in reducing discharge headways. Based on the calibrated model, the start-up lost time for a typical through movement with a common desired speed of 49 fps and a maximum acceleration of 6.63 ft/sec/sec is 3.67 seconds Based on the calibrated model, the minimum discharge headway fo r a typical through movement of an at-grade intersection w ith a common desired speed of 49 fps and a nominal traffic pressure of 5 veh/ln/cycle is 1.81 seconds The following formulas are provided in the report: Briggs Models Based on Constant Acceleration Calibrated Discharge Headway Model: Headway of nth vehicle = hn = T + [2dn/A]1/2 [2d(n-1)/A]1/2 (if nd < dmax = Vq 2/ 2A) Headway of nth vehicle = hn = T + d/Vq (if nd >= dmax) Vq = desired speed of queued traffic (29.4 ft/sec) d = distance between vehicles in a stopped queue (19.65 feet) T = driver starting response time (1.22 seconds) A = constant acceleration of queued vehicles (3.67 ft/sec/sec) dmax = distance traveled to reach speed Vq n = queue position

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55 Bonneson Models Based on Non-Constant Acceleration Calibrated Stop Line Speed Model: Stop Line Speed for vehicle n = Vsl(n) = Vmax (1 e-nk) k = -0.290 + 24.0/Vmax Calibrated Discharge Headway Model: Headway of nth vehicle = hn = (tau)N1 + T(d/Vmax) + 0.357[(Vsl(n) Vsl(n-1)/Amax] 0.0086v 0.23AGI Calibrated Minimum Discharge Headway Model: Minimum Headway = H = T + d/Vmax 0.0086v 0.23AGI Calibrated Start-Up Lost Time Model: Start-Up Lost Time = Ks = 1.03 + 0.357Vmax/Amax n = queue position tau = additional response time for first queued driver (1.03 sec) d = distance between vehicles in a stopped queue (25.25 feet) T = driver starting response time (1.57 sec) v = traffic pressure in vehicles per cycle per lane Vmax = common desired speed of queued traffic in feet per second Amax = maximum acceleration in feet per second per second N1 = 1 for first queued vehicle, 0 otherwise AGI = 1 for at-grade intersection, 0 for single point urban interchange In 1997, Fambro & Rouphail [28] proposed a new se t of delay equations that were, for the most part, incorporated into the 2000 Highway Cap acity Manual. The only difference is that the formulas recommended for the d3 term were replaced by different formulas included in Appendix F of Chapter 16 of the 2000 HCM. Simulation (TRAF-SIM) data were used to validate the over-saturation and variable demand component of the generalized delay mode l because of the difficulty in measuring oversaturation delay in the field The following parameters are defined in this study:

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56 I = parameter for variance-to-mean ratio of arri vals from upstream signal. Isolated signals have the highest I value (I=1.0 Variance=Mean Poisson Distribution). The I value varies between 0.09 and 1.0 at coordinated intersections. The k value produces less delay for actuated sign als with snappy extens ion intervals (down to 2 seconds). The amount of the delay decrease depends on the degree of saturation, with greater decreases experienced when the degr ee of saturation is low (toward 0.5) and no decreases experienced when the degr ee of saturation is high (at 1.0) Including a T parameter in the generalized delay m odel to account for the duration of the analysis period improves delay estimates under oversaturated conditions. Longer periods of oversaturation and higher degrees of oversaturation result in longer delays. It is important to note that part of the estimated delay during oversatura ted conditions occurs after the analysis period. The following definitions are given in the report: Stopped Delay = the time an individual vehicle spends stopped in a queue while waiting to enter an intersection. Average Stopped Delay = the total Stopped Delay experi enced by all vehicles arriving during a designated period divide d by the total volume of all vehicles arri ving during the same period (used to determine LOS in 1985 and 1994 HCM). Signal Delay (a.k.a. Control Delay) = deceleration delay + que ue move-up delay + Stopped Delay + acceleration delay The following formulas are provided in the report: Control Delay (delay per vehicle for each lane group) = d1 (Uniform Delay) + d2 (Incremental Delay due to Random and Overflow Queues) + d3 (Incremental Delay due to Oversaturation Queues at the start of the analysis period) d1 = PF[0.5C{1-(g/C)}2]/[1-(g/C)min(X,1.0)] PF = (1-P)fPA/[1-g/C] (from 2000 HCM) X = v/c for lane group (aka degree of saturation) C = average cycle length (seconds) G = average effective green time (seconds) d2 = 900T[(X-1) + {(X-1)2+8kIX/Tc}1/2] I = upstream filtering/metering factor obtained from Exhibit 15-7 of 2000 HCM k = incremental delay factor obta ined from Exhibit 16-13 of 2000 HCM c = capacity of lane group (vph) T = duration of analysis period (hours)

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57 d3 = (See Appendix F of 2000 HCM) In 1997, Engelbrecht, Fambro, et al. [6] propos ed a generalized dela y model that handles over-saturated conditions at signalized intersections. The delay equations calculate delays consistent with the more accurate path-trace met hod of delay measurement rather than the less accurate (but easier to carry-out) queue-samp ling method. Delays estimated by the proposed generalized model were in close agreem ent with those simulated by TRAF-NETSIM. The path-trace method measures individual vehicl e delays from arrival to departure, even if the departure occurs after the end of the an alysis period. Delay measurement using this technique is typically complicated. However, advances in intelligent transportation system technology may reduce the difficulty a ssociated with this technique. The queue-sampling method records the number of stopped vehicles at periodic intervals (such as every 10 seconds), multiplies this by the length of the sampling period, and then divides by the number of vehicles arriving during the analysis period. For the path-trace method and queue count methods to be compatible, two conditions must hold: 1.) There must not be a residual queue at the start of the analysis period, and 2.) Queue counts must continue until all vehicles that a rrived during the analysis period have cleared the intersection. All vehicles joining the back of the queue after the end of the analysis period should be excluded from this count. TRAF-NETSIM calculates delay by subtracting the free-flow travel time from the actual travel time to yield overall delay. However, th e actual travel time includ es not only intersection, or control delay, but also some delay as a result of interactions betwee n vehicles on the link itself, or traffic delay. In the analysis, the author s decided to ignore this discrepancy, as it is very

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58 difficult to separate control and traffic delay, and the error is assumed to be small, especially under over-saturated conditions. The following TRAF-NETSIM input values (rep resentative of over-saturated conditions) were analyzed: Analysis Period (T) = 15 & 30 minutes Cycle Length (C) = 60, 90, 120 seconds Saturation flow (s) = 1800 & 3600 vphg G/C ratio = 0.3, 0.5 & 0.7 Degree of Saturation (X) = 1.0, 1.1, 1.2, 1.3 & 1.4 (0.9 was also included) The authors point out that equilibrium (in TRAF-NETSIM) can never be reached for oversaturated conditions, as capacity is less than demand and outflow will always be less than inflow. The initialization will terminate before equilibrium can be reached, leaving an initial queue of unknown size. This queue will delay vehicles wh en it clears, increasing the delay experienced by vehicles that arrive during the analysis period. Therefore, the authors decided to use 3 periods in the analysis: an initial 60-second period with very low flow; the actual analysis period of duration T; and a final period of duration T, ag ain with very low flow (TRAF-NETSIM can not handle zero flow). The first peri od is needed to initialize the network without transferring a queue to the second period, the second period is th e actual analysis period, and the third period dissipates the over-saturation queue that built up over the second period. Not all of the input scenarios yielded usable results. In some scenarios, the simulated delays were incorrect be cause of queue spillback In 2000, Tarko and Tracz [29] investigated uncer tainty in saturation flow predictions and concluded that standard errors reached 8 to 10%. They identified three prim ary sources of error: temporal variance, omission of one or more capacity factors in the predictive model, and inadequate functional relationships between model variables and saturation flow rates. The data

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59 were collected on Polish highways but the aut hors conclude that th e results should be transferable to other countries. Using data from over 1100 signal cycles, Tark o and Tracz discovered that the saturation flow rate increases rapidly durin g the first 6 seconds of the green indication to a value of about 1400 pcphg (headway of 2.6 sec/veh), then slowly increases to a value of about 1600 pcphg (headway of 2.2 sec/veh) after another 20 seconds. Past this 25 second mark the rate stabilizes. This type of behavior occurred in all of the la nes investigated although the length of the periods varied somewhat. Consequently, the length of th e counting period has an e ffect on the saturation flow rate that is obtained. Tarko and Tracz also found that the percent of heavy vehicles in the traffic stream has an effect on the headway of passenger cars, with the headway varying between 2.2 sec/veh when no heavy vehicles are present to 2.6 sec/veh when the traffic stream is composed of 30% heavy vehicles. Heavy vehicles also have longer headwa ys than passenger cars, which is another factor that reduces the saturation flow rate. Tarko and Tracz recommend the use of a Passenger Car Equivalence (PCE) factor of 2.4, which is substan tially higher than the va lue of 2.0 used in the 2000 Highway Capacity Manual or the 1.2 default factor used by CORSIM. Tarko and Tracz proposed various predictive models for saturation flow that included the following statistically significant independent vari ables: ratio of heavy vehicles, lane width, turning radius (infinite for straight lanes), and lane location (near curb or middle). The authors conclude by stating that: Where possible, th e saturation flow rates should be determined through direct field measurement. This provides more support for the research at hand. In 2002, Li and Prevedouros [30] studied th ree methods for describing the discharge process of a standing queue at an approach of a signalized inte rsection. Method 1 (M1) entails

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60 measurements of headways based on the first 12 ve hicles in a standing queue. Method 2 (M2 or HCM Method) entails measurements of headways based on all vehicles in a standing queue. Method 3 (M3) is the same as M2 except that arrivals which join th e standing queue are included. According to the HCM, the saturation headway is estimated by averaging the headways from the 5th vehicle to the la st vehicle in a standing queue The 2000 HCM suggests a base saturation flow rate of 1900 pc/h/ln for thru lanes, which corresponds to a saturation headway of 1.895 seconds (3600/1900) and 1800 pc/h/ln (a 2 sec ond saturation headway) for protected left turn lanes. StartUp Lost Time (SULT) is de rived from the first four vehicles in a standing queue. The 2000 HCM mentions typical observed va lues of between 1 and 2 seconds for thru lanes. Li and Prevedouros collected data on two lanes of a five-lane approach (3 thru lanes and a dual left turn lane) of a signalized intersection in Honolulu, Hawaii. The outside thru lane and the inside left turn lane were measured. These lanes were considered to be of ideal design and no queues with heavy vehicles were used in the analysis. A vehicle was considered to be discharged when its rear axle passed the stop line. Observations containing fewer than four vehicles at the end of a que ue were not included. Start-Up Response Time (SRT) was defined by the authors as the time from the beginning of green to when the first vehicles rear axle passes the stop line. The following relationship between SRT and SULT was provided: Start-Up Lost Time = SULT = SRT + 4*(H4-h) Saturation Headway = h = (TN-T4)/(N-4) Average Headway = Hi = (Ti Ti-4)/4 Where: Ti = time when rear axle of vehicle i passes the stop line (T0 = SRT)

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61 N = last vehicle in the queue The saturation headways (h ) derived by the three methods (M1, M2 and M3) are statistically different. For thru movements: h = 1.90 sec (s = 1895 pc/h/ln ) for M1, std dev = 0.21 h = 1.92 sec (s = 1875 pc/h/ln ) for M2, std dev = 0.20 h = 1.98 sec (s = 1818 pc/h/ln ) for M2, std dev = 0.22 The minimum headway was not reach until th e 9th to 12th vehicle instead of the 5th vehicle as implied by the HCM. If queue a rrivals are included (M3), both the mean and standard deviations of the headways increase after the 12th vehicle. For protected left turn movements: h = 2.04 sec (s = 1765 pc/h/ln) for M1 ( 1765/1895 = 0.931 LT factor), std dev = 0.23 h = 2.01 sec (s = 1791 pc/h/ln) for M2 ( 1791/1875 = 0.955 LT factor), std dev = 0.23 Headways decrease as queue position increases (motorists may be aware of the limited green time and tailgate so as to not experience a phase failure). After the first 12 vehicles the saturation flow rate remained well above 1800 pc/h/ln. Queues of medium length discharge more efficiently than do short queues. After the 16th vehicle in the queue the saturation flow rates of the left turn movement were larger th an for the thru movement. The Start-Up Response Time (SRT) for left turn movements (1.42 seconds) is less than for thru movements (1.76 seconds), indicating a heightened awareness of left turning driver s to the display of the green. There was a high standard deviation of SRT fo r both movement types (0.61 for thrus and 0.74 for LTs), indicating a big variation amongst drivers. Howeve r, SRT was not sensitive to queue length. The calculated SULT was well above the 1 to 2 seconds of the HCM (2.89 for thrus and 2.38 for LTs under peak period conditions and 3.03 for thrus and 2.53 for LTs

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62 under off-peak conditions.) As with the SRTs, the SULTs also have hi gh standard deviations (1.36 for peak thrus and 1.32 for peak LTs; 1.5 fo r off-peak thrus and 1.3 for off-peak LTs). Linear regression models (one for thru m ovements and one for LT movements) were developed that indicate a negative correlation between SULT and queue length (i.e. long queues produced shorter start-up loss times). Distribution tests showed that thru moveme nt headways were lognormally distributed without a shift and that LT headwa ys were lognormally distributed with a shift of 1 second. SRT was normally distributed for both movements. In 2002, Cohen [31] used the Pitt car-following system to examine the effects of lane changing and a heterogeneous vehicl e mix on queue discharge headways. In the Pitt car-following model, the first ve hicle in the queue begi ns to move across the stop line after the lost time (start-up delay) has expired. The second vehi cle in the queue then responds to the motion of the leader through the car-following system with no additional explicit lost time added. The effect of lost time on subsequent vehicles is modeled through the sluggishness of the car-following system. Based on the results of the study, it can be c oncluded that trucks not only have longer headways than cars, but they also increase the he adways of the vehicles behind them. The closer to the front of the queue that the truck is locat ed, the greater the overall negative effect on queue discharge. In addition, for trucks further back in the queue the major item affecting its equivalency factor is it s greater length whereas, for trucks n ear the head of the queue, the major item is vehicle performance limitations. Queue Di scharge Headway is defined as the difference in stop line crossing times between each vehicle pair.

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63 Lane changing also has a substantial effect on discharge headways, particularly if the lane change takes place close to the stop line. For thru lanes with short ad jacent turn lanes (where lane changing is apt to take place) the saturatio n flow rate will be lowered on the basis of the percentage of turns. The results of the study also suggest that th e start-up wave in a discharging queue will slow down as it progresses upstrea m. Acceleration rates decrease as one progresses upstream in the queue (each vehicle accelerates more slowly th an its leader). Consequently, it takes longer for gaps to open between pairs of vehicles in the queue and the presence of these gaps is the necessary requirement for the fo llower to begin to move. Star t-Up Wave (a.k.a. Green Wave, Expansion Wave) is defined as the rate at whic h vehicles in the queue begin to move. (With movement defined as the time at which a speed of 1 ft/sec is achieved.) In addition, the study re sults indicate that the discharge he adway distribution is almost flat beyond the fifth vehicle in the queue, wh ich is consistent with the HCM. The author notes that the best approach for calibration of the Pitt car-following model is to measure in the field the crossing times of both the front and rear of each vehicle in the queue as it discharges across the stop line. These measurem ents allow the plotting of two curves, the frontto-front time headway curve and th e rear-to-front time spacing curve. Unfortunately, this type of detailed data set is usually not collect in queue discharge studies. The author explains that the NETSIM queue discharge mechanism is limited in that it is based on the assumption that vehicl es in a queue discharge from the intersection at equal time headways (other than stochastic variations) subject to start-up delays applied to the first 3 vehicles in the queue. The effect of lane changing is ignored completely and the effect of commercial vehicles is treated heuristically using vehicle equivalency factors.

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64 In 2003, Mousa [32] presented a microscopic stochastic simulation model developed to emulate the traffic movement at signalized intersections and estimate vehicular delays, including acceleration and deceleration delay. By analyzin g 48 cases with a fixed g/C ratio of 0.475, it was found that the ratio of total delay to stopped delay is directly proportional to both the degree of saturation and the approach speed, and inversely proportional to the cy cle length. The effect is greatest for degree of saturation and cycle length and least for approach speed. For the 48 simulated cases, the saturation flow obtai ned from simulation ranged from 1692 vph to 1807 vph, with an average value of 1770 vph and a standard deviation of 28 vph. Approach speeds ranging from 30 to 50 mph a nd cycle lengths varying between 60 and 150 seconds were considered and tested in this study. Different levels of degree of saturation, ranging between 0.5 and 0.9, are also considered. The ratio of to tal delay to stopped delay was found to be between 1.5 and 3.0 with the minimu m ratio resulting from the longest cycle length (150 seconds) and the lowest degr ee of saturation (0.5) and the maximum rati o resulting from the shortest cycle length (60 seconds) and th e highest degree of saturation (0.9). A sufficient length of appro ach was considered in the an alysis to ensure that all acceleration/deceleration delays incurred by indi vidual vehicles were executed within the simulated length. In 2004, Rakha and Zhang [33] authored a pape r that demonstrated the consistency that exists between queuing theory and shock-wave an alysis and that highlighted the common errors that are made with regard to delay estimation us ing shock-wave analysis. The authors point out that the main difference between shock-wave an alysis and queuing models is the way vehicles are assumed to queue upstream of the bottleneck. Queuing analysis assumes vertical stacking of the queue whereas shock-wave analysis c onsiders the horizontal extent of the queue.

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65 Maximum queue reach (a.k.a. back of queue) can only be identified using shock-wave analysis. The authors show that the size of the queue obtained from shock-wave analysis is the same as the size of the queue obtained from deterministic queuing theory if the queuing theory value is adjusted by a factored equal to total travel time divided by total delay. In 2004, Perez-Cartagena and Tar ko [34] demonstrated that, ba sed on studie s conducted in Indiana, town size and lateral la ne location (right-most lane or not) are important variables in identifying the base saturation flow rate for a si gnalized intersection. Saturation flow rates were estimated using the Headway Method and weight ed regression analysis The authors also discovered that small communities tend to have c onsiderably lower values of saturation flow than large communities, indicating that drivers in large communities are more aggressive than drivers in small communities. The reduction in saturation flow rate was about 8% for medium size towns and 21% for small towns (as compared to large towns). Kebab, et al. [35] developed an efficient field procedure for measuri ng approach delay at a signalized intersection that segregated the de lay by movement. The procedure produced good results in comparison to gr ound truth obtained from video. One section of a 2006 paper by Brilon, et al. [ 36] discussed variation in capacity that occurs at signalized intersections due to the randomness of driver be havior and interaction between vehicles. The authors concluded that their stochastic concep t of capacity provides better plausibility than the assump tion of constant-value capacities and that the implications of random capacities on delay distributions shoul d be investigated by further research. Probe Monitoring The m ost promising alternative method for obtai ning the type of globally applicable delay estimates (estimates applicable to over-saturat ed as well as under-saturated conditions) addressed in this paper is the use of probe vehicles. A considerable body of work is being c onducted in this

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66 area, including the potentia l use of cell phone data to track indi vidual vehicles and the results of the work are starting to show up in the literature. A 2005 article by Jiang, et al. [37] examined the collecti on of signalized intersection delay data using vehicles outfitted with global positioning system (GPS) technology. It was determined that, compared to manually measured delays, the GPS appro ach provided the same accuracy with considerably lower labor requirements. A 2007 paper by Ko, et al. [38] also examined the collection of si gnalized intersection delay data using vehicles outfitted with globa l positioning system (GPS) technology. Their technique included algorithms for analyzing spee d profiles and acceleration profiles in order to automatically identify critical control delay points, such as deceleration onset points and accelerating ending points. This automated process permits the analysis of large data sets and provides consistent results. However, the appr oach experienced some difficulty in handling over-capacity conditions and cl osely spaced intersections. A 2007 paper by Comert and Certin [39] used pr obe vehicles to estimated queue lengths on a signalized intersection approach The best estimate of queue length was provided for high volume, but under-saturated, conditions. The resu lts are subject to sampling errors (a common characteristic of probe use) and the proce dure was not tested under congested conditions. A 2007 Florida Department of Tr ansportation report authored by Wunnava, et al. [40] of Florida Atlantic University investigated cell pho ne tracking. The authors concluded that a host of both technical and privacy issues need to be worked-out before probe vehicles can provide the needed detail to accurately estimate approach delay: the team also found that th e cell phone technology is not accurate in congested traffic conditions, where the data is more important than in the free-flow traffic conditions, and the accuracy decreases rapidly as the congesti on increases Additional issues remain such as: (1) privacy of the cell phone users whos e phone transmissions are being probed by the

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67 cell companies for location data, (2) irregular and transient cell data for travel time and speed computations, especially during congest ed traffic and severe weather conditions, (3) limited capabilities of the travel time providers to follow changes by the cell companies in their data formats and structures, and (4) inco mpatibility of data wh en switching from one travel time provider to another. If these issues, some of which are political in nature, cannot be addressed satisfactorily then obtaining widespread delay information from probes may never occur. Extending the Body of Knowledge Although a num ber of research ers have investigated sampling techniques designed to improve the estimation of travel time and delay along the through lanes of an arterial corridor (such as through vehicle re-identif ication or the use of instrument ed probes), the research effort described herein is unique in that it attempts to estimate delay in a manner that is directly applicable to the minor movements of the inters ection as well as the major thru movements, and it utilizes information from all approaching vehicles not a restricted sample. In addition, none of the previous research has dealt with the real-world probl em of queues that extend beyond the detection system for some period of time; ei ther short-lived queues that occur during undersaturated conditions because of spurts in activity or longer-lived, recurring queues that occur during over-saturated conditions. This appears to be the only research that is attempting to intelligently estimate that which cannot be easily measured with respect to intersection delay. The basic problem that is being addressed is the need to establish a methodology that can intelligently estimate delay associated with vehi cles that are beyond the reach of the detection system. This means obtaining reasonable estimates of vehicular delay even when queues are long and multiple phase failures occur. The use of incomplete information, combined with a concentration on over-saturated c onditions, represent a deviation from the research conducted to date.

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68 CHAPTER 4 ESTIMATING NON-VISIBLE DELAY This chapter describes the m ethodology that wa s established to pred ict non-visible delay under conditions of limited information and th e associated analysis procedure that was developed. Variables important to the procedure are discussed and a series of new technical terms relevant to the procedure are introduced (Objectives 1, 2 and 3). Research activities were conducted using CO RSIM (CORridor SIMu lation) microscopic traffic simulation software and TRAFVU (TRAFf ic Visualization Utility) software that are contained within the TSIS (Traffic Software In tegrated Systems) software package. The CORSIM software, which was developed by th e Federal Highway Ad ministration (FHWA), consists of the FRESIM (FREeway SIMula tion) component and the NETSIM (NETwork SIMulation) component. TRAFVU is an objectoriented, graphics postprocessor for CORSIM that displays traffic networks, animates simu lated traffic and traffic controls, and reports measures of effectivenes s for the network under study. The CORSIM runs made use of a very simple case, the intersection of 2 one-way streets, each having a single approach lane. No trucks were placed into the traffic stream and no turns were allowed. A random (Poisson) arrival pattern was set with arrival rates varying each 15minutes during a one-hour analysis time frame. The intersection was controlled by a 2-phase semi-actuated traffic signal and delay data were co llected and analyzed only for the actuated side street approach. Goodne ss-of-fit testing usi ng the chi-square technique wa s used to ensure that a random (Poisson) arrival distributi on was actually produced by CORSIM. Data Analysis Programs In order to obtain the data ne eded f or analysis, a visual basic program called TSDViewer [41] was developed which reads the output file of CORSIM and produces, on a second-by-

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69 second basis, a variety of information pertaini ng to the number of vehicles crossing various checkpoints and arriving and departing queues TSDViewer automates the data collection process from the CORSIM runs by reading CORSIM s output file (the .tsd file for CORSIM 5.1 and the .ts0 for CORSIM 6.0) and producing an Excel worksheet containing the following information: The time at which each vehicle enters the approach link, The time at which each vehicl e enters the delay zone, The speed of each vehicle when it enters the delay zone, The time at which each vehicle ente rs the Field of View (FOV), The time at which each vehicle arri ves at the Back of Queue (BOQ), The time at which each vehicle departs the queue, The time at which each vehicle cros ses the stop bar (leaves the link), The time at which each vehicle leaves the delay zone, The signal indication (red, yellow, or green) at each time point, If two queues exist simultaneousl y, the time at which vehicles a rrive at the back of queue 2, If two queues exist simulta neously, the time at which vehicles depart queue 2, The number of vehicles e xperiencing 1 phase failure, The number of vehicles expe riencing 2 phase failures, The number of vehicles experiencing 3 phase fa ilures, and so on up to a maximum of 15 This information can be used to calculate on a second-by-second ba sis, both queue length and back of queue position. Stopped delay is then calculated using the queue length. An example that shows the relationship between que ue length and back of queue position is provided in Figure 4-1. It is important to recognize that the back of queue is not itself a length, but rather a position. As is shown in Figure 4-1, the value for the back of queue position can be quite large even if the correspond ing queue length is small. A visual basic program named DTDiagram [42] was also de veloped as part of this research. This program reads the CORSIM output file and produces trajectory information (a series of time-distance points) for each vehicle. The data produced by DTDiagram is read by BuckTRAJ [43], another visual ba sic program that was developed as part of this research to calculate, for each vehicle, all of th e components of control delay.

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70 The programs developed allow the researcher to quickly simulate a variety of real-world conditions in a relatively realistic manner and to accumulate the associated MOEs, such as delay. The researcher can then compare actual delay obtained from CORSIM against the predicted delay obtained from the techniques developed in this research The use of simulation allowed many different scenarios to be run in order to compare actual versus predicted delay, allowing us to see how well our proposed delay estimatio n methodology performed. Essentially, microsimulation provided a source of verification against which our delay prediction methodology could be developed and refined. The pivotal task of the research was the creat ion of an automated analysis procedure that can use the outputs of TSDViewer to produce queue and delay info rmation that is required for proper evaluation of candidate delay estimation pro cedures. The analysis procedure must be able to, on a second-by-second basis, es timate the non-visible queue, a dd this queue to the visible queue, calculate the associated stopped delay, and then compare the result to the true control delay as calculated by CORSIM. For the purposes of this study, stopped delay is defined as the delay experienced by vehicles when they are at a complete stop (z ero acceleration and zero velo city). Also for the purposes of this study, a vehicle is considered queued when it comes to a complete stop (zero acceleration and zero velocity). These are slightly more conservative definitions than those used by CORSIM. CORSIM considers a vehicle to be stopped when its sp eed is less than 3 feet/second and considers a vehicle to be queued when its speed is less than 9 feet/second and its acceleration is less than 2 feet/second/second. The zero-velocity-zero-acceleration complete stop definition was chosen since it is easier to correl ate with both video images of vehicle queues and

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71 queues observed in the field. Much less discretion is needed to determine when a car stops than when its acceleration and speed simultaneou sly fall below a certain set of values. Control delay (DC) is defined, both by CORSIM and in general, as the sum of initial deceleration delay (DD), stopped delay (DS), queue move-up delay (DMU), and final acceleration delay (DA). Acceleration delay can be further subdi vided into acceleration delay that occurs prior to the stop bar (DA1) and acceleration delay that occurs after the stop bar (DA2). Figure 4-2 depicts the delay elements. Total delay is defined as the sum of control delay, which is caused by the presence of the traffic signal, and the delay associ ated with vehicular interactions that occur on the link (called interaction delay in this study) Others have called this crui se delay or traffic delay instead of interaction delay since it is the delay resulting from cruise speeds that are lower than the free flow speed due to the presence of other traffic. Ideally, we would like to have a tool that provides accurate real -time measurement of control delay. However, given the limitations of almost all detection systems, the best we can hope for, and what has been developed in th is research, is a proc edure that provides a reasonably accurate real-time es timate of stopped delay. By applying an appropriate factor (such as the commonly-used 1.3 value) or range of factors, we then scale-up the stopped delay estimate to obtain a reasonably a ccurate real-time estimate of control delay. Absent the instrumentation of every vehicle, control dela y cannot be accurately measured using current vehicle detection systems for the following reasons: 1. Since vehicle detection systems are primarily used to allocate green time at a signal, there is usually no detection in the depart ure lanes. Consequently, final acceleration delay cannot be measured. 2. Queue lengths often extend beyond the limits of the detection system, especially during peak hours. When this happens, we can only measure the stopped delay and

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72 queue move-up time that occurs within the limits (or field of view ) of the detection system. Any stopped delay or queue move-u p time that occurs outside the field of view cannot be measured. 3. Motorists usually begin their initial deceleration far in advance of any signalized intersection queue, often well be yond the field of view of the detection system. So, most of the time, initial deceleration delay cannot be measured either. In order to make use of existing detection systems it becomes necessary to measure that portion of the delay that can be observed and then intelligently es timate what cannot be observed (see Figure 4-3). The result is the methodol ogy produced by this research, a methodology that measures visible stopped delay; stopped delay that occurs within the Field of View (FOV) of the detection system and then uses various analytical techniques to pr edict non-visible stopped delay; stopped delay that occurs outside the FOV. The portion of the queue that is outside the FOV is referred to in this res earch as the non-visible queue (see Figure 4-4) and the period of time during which non-visible queues are presen t is referred to as the blind period. During this research, a set of f actors were identified that can be used to c onvert predicted stopped delay to predicted control delay. Previous research by Mousa [32] suggests that the use of a single 1.3 value is too simp listic. His simulation work suggest s that the ratio of total delay to stopped delay varies from a value of 1.5 to 3 depending mainly on cycle length and degree of saturation. Figure 4-5 summarizes the relationship between this ratio and both the v/c ratio and cycle length for over-saturated conditions. For each CORSIM run, a certain Field of View (FOV) was assumed. Measured visible locked-wheel stopped delay (delay occurring with in this FOV) was added to the predicted nonvisible stopped delay to produce a total value fo r predicted stopped delay. This predicted value was then compared to the actual value of locked-w heel stopped delay assuming an infinite FOV. Finally, the predicted stopped delay was factored-up to obtain a pred icted value for control delay.

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73 This predicted control delay was then compared to the actual value of control delay, again assuming an infinite FOV. As might be expect ed, these comparisons are more favorable when traffic volumes are lower, or when the FOV is larger. In this case, queue lengths seldom go beyond the FOV and most of the de lay can be directly measured Conversely, when traffic volumes are higher, or the FOV is relatively s hort, the delay comparisons are less favorable since, under these conditions, the queue freque ntly extends beyond the FOV requiring most of the delay to be estimated. CORSIM accumulates control delay on a link basis and, by necessity, the link numbering changes at signalized intersections. The unfortunate result is that CORSIMs estimate of control delay does not include any final acceleration delay th at occurs past the stop bar. This forces the development of an alternate measure of control de lay to use as the CORSIM reference value. This was accomplished by setting up a delay zone that begins well in advance of the intersection and ends a few hundred feet downstream of the intersection. The location of the start and end points for this delay zone were chosen carefully The start point was set far enough in advance of the intersection (upstream) so that all initial deceleration delay is accounted for, but not so far in advance that a significant amount of pre-signa l interaction delay occurs. Likewise, the end point was set far enough past the in tersection (downstream) so that all final acceleration delay is accounted for but not so far past that a significant amount of post-signal inte raction delay occurs. The best location for the start point depends on the physical extent of the queuing that is expected and was set in an iterative fashion. Gi ven a fixed g/C ratio, the physical extent of the queuing depends on both arrival volume and cycle leng th. For the range of variables considered in this study, the loca tion of the delay zone start point was located either 1600, 2600 or 3600 feet

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74 in advance of the stop bar w ith corresponding CORSIM upstream link lengths of 2000, 3000, or 4000 feet used. The best location for the end point was determ ined using the acceleration charts contained in AASHTOs Geometric Design of Highways and St reets [44]. For example, using Exhibit 224 in this AASHTO manual we see that, on level terrain, approximately 300 feet is required for passenger cars to accelerate from a stop to 34 m ph. Consequently, a delay zone that ends 300 feet past the stop bar is a reas onable configuration for a road with a posted speed limit of 35 mph. Since the link speeds used in our study were kept constant at 30 mph, 300 feet was chosen as a reasonable downstream distance from the stop bar with a corresponding CORSIM downstream link length of 1000 feet. The resulting delay zone length was either 1900 feet, 2900 feet, or 3900 feet. If the start of the delay z one is positioned far enough upstream then all vehicles should enter the delay zone at their free-flow speed (wit h free-flow speed being defined as the speed at which the vehicle would travel ha d the signal not existed). The time it takes for a vehicle to cover the length of the delay zone at its free-flow speed is defined as its free travel time. With the delay zone boundaries properly established, the control delay is simply the difference between the actual time it takes a given vehicle to traverse the delay zone and the vehicles free travel time. Although some interaction delay may occur near the start point and the end point of the delay zone, it should be relatively minor in nature and should not significantly affect the results. For all CORSIM runs over-capacity conditions existed for at least a portion of the onehour analysis time frame, resulting in substantia l levels of queuing. Such queues behave in a manner consistent with shock-wave theory an d when traffic volumes become very high in

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75 relation to the capacity of the approach in que stion, vehicle re-queuing causes the formation of one queue at the stop bar and another queue further upstream. The resulting simultaneous queues are separated by vehicles moving between th em, as is demonstrated in Figure 4-6. When this occurs, it is often the case th at vehicles arrive and depart both queues at the same time. The analysis programs track both queues in order to provide accurate queuing information. In this research, whenever there are two simultaneous queues, the queue closest to the stop bar is referred to as queue 2 and the one furthest from the stop bar as queue 1. When either of the two queues dissipates, the remaining queue is referred to as queue 1. The analysis programs were designed to handle a maximum of two simultane ous queues since three simultaneous queues are only present under extremely congested conditions, conditions for which almost any prediction methodology would be grossly inaccurate. Re-queuing events are associated with phase failures, which o ccur when a vehicle joins the back of a queue and the next green interval is of insufficient duration to allow the vehicle to pass through the intersection. Phase failures tend to occur under congested conditions, but can also occur during uncongested conditions because of poor signal timings. Poor signal timings might include insufficient maximum intervals, extension intervals that are too short for the detection system, or even insufficient minimum intervals if the approach utilizes an upstream detection system. Re-queuing is a necessary condition for the formation of simultaneous queues; however, it is not a sufficient one. As show n in Figure 4-7, re-queuing may not result in the formation of simultaneous queues. Unusual or atypical events can also result in phase failures and associated re-queuing. For example, a vehicle that does not respond in a reasonabl e time to the green indication (because it is temporarily stalled, the driver is not paying a ttention, etc.) may cause an actuated approach to

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76 gap-out prematurely, forcing this vehicle and all vehicl es behind it to requeue. CORSIM does not model such atypical events, but they do occur periodically in the real world. As Courage & Fambro [45] put it; Simulation models introdu ce a stochastic element into the departure headways based on a theoretical distribution. They are therefore able to invoke premature phase terminations to some extent, but they do not deal with anomalous driver behavior. A phase failure may be either liberal or str ict. A strict phase failure occurs when a vehicle that was queued when the signal turned green is forced to re-queue when the signal turns yellow then red. A liberal phase failure occurs when a vehicle jo ins the back of the queue during the green indication but is forced to re-queue when the signal turns yellow then red. It should be noted that the analysis process developed for this research recognizes both types of phase failures, whereas CORSIM only repo rts strict phase failures. It is worth noting that the number of vehicle re-queues is equal to the number of vehicle stops if the first stop is ignored. When the side street approach under inves tigation receives the re d indication, vehicles begin to queue at the stop bar. The time duri ng which the entire queue is within the FOV and can be seen by the detection system is referred to as the visible period. Eventually, the queue fills-up the FOV and th e detection system can no longer measure the exact queue length. When this occurs, the system transitions from a visible period into a blind period and the prediction proces s must begin for the non-visible queue. Figure 4-8 provides an example of a blind period. During this blind peri od, vehicles attach themse lves to the end of the non-visible queue at some unknown ra te, referred to as the actual ar rival rate. Th e portion of the blind period during which vehicles can attach themselves to th e back of the non-visible queue, but cannot leave the front of the non-visible queu e since the signal has not yet turned green and

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77 there are vehicles queued ahead of them, is refe rred to as the rising que ue blind period (which occurs from time T-7 to time T-34 in Figure 4-8). Eventually the side street approach receives the green indication and vehicles on that approach begin to cross the stop bar. The visi ble queue shrinks from the front until the last vehicle in the FOV begins to move and the visi ble queue becomes zero. At this point, vehicles can begin to depart the non-visible queue from the front while they continue to attach to the back of the non-visible queue at the unknown rate. We refer to this portion of the blind period where vehicles can both attach themselv es to the back of the non-visibl e queue and leave the front of the non-visible queue, as the falling queue blind pe riod (which occurs from time T-34 to time T72 in Figure 4-8). The length of the non-visible queue is typically falling during this period since vehicles almost always depart the front of the queu e at a much faster rate than they arrive at the back of the queue. For example, assume a field of view (FOV) of 12 vehicles. When the visible queue extends to a point where the 12th position is fi lled by a queued vehicle, the rising queue portion of the blind period begins. After some period of time the signal turns gr een and, eventually, the vehicle in position 12 moves forward. When this vehicle moves forward the rising queue portion of the blind period ends and the falling queue po rtion of the blind peri od begins. After some additional period of time, a gap of sufficient duration (such as 5 seconds) is enc ountered between successive vehicles entering the FOV, signaling th at the end of the queue has come into view. When this happens, the blind period has ended (which occurs at time T-72 in Figure 4-8). A review of the Figure 4-8 example reveals that the non-visible queue actually shrinks to zero well before the end of the falling queue portion of the bl ind period (somew here abound time

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78 T-50). However, because of th e limited FOV, we cannot be cer tain that the non-visible queue has dissipated until time T-72. Many blind periods may exist ove r a given analysis time frame, with the number of blind periods depending on the number of times that th e end of the actual queue goes out of, and then comes back into, the field of view. If a vehicle does not enter the queue FOV for so me sufficiently long period of time (for our Figure 4-8 example, 5 seconds), and if another queue does not fill the FOV prior to this 5-second period, then the blind period is cons idered to have ended and the system returns to a visible state where the actual queue length is known. When this occurs it is assumed that there no longer exists a non-visible queue (i.e., the non-visible queue has been flu shed out). However, if this 5-second headway does not occur before the FOV is once again filled with queued vehicles, then the system transitions from one b lind period into anothe r with no intervening pe riod of visibility. When this happens, adjacent blind periods occur (see Figure 4-9). As one might expect, the problem of estimating the length of non-visible queues and their associated delay becomes more difficult (and, hence, more approximate) as the frequency of adjacent blind periods increases. As we shall soon discover, the number of adj acent blind periods is an important variable when attempting to predict the le ngth of the non-visible queue a nd its associated stopped delay. The non-visible delay estimation algorithm containe d within our analysis software makes use of two counters (labeled A and D for Ascending and Descending) that are tied to the rising queue/falling queue status as shown in Figure 4-10. One important variable in the queue formati on/dissipation process is the average time it takes a vehicle to depart the queue once the vehi cle ahead of it has begun to move. This time, referred to by Long [46] as the que ue startup lag time (or by others and in this research, as the

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79 queue departure time), is 1 second in CORSIM. However, field studies by Long at 4 sites in Florida involving 140 queues of at least 16 vehicles in length (for a total sample of 1893) resulted in a slightly longer average startup lag time of 1.15 seconds with a standard deviation of 0.52 seconds. Long also references work by Herm an, et al., in 1971 that indicated an average startup lag time of 1.0 sec and work by Messer and Fambro in 1977 that produced an average startup lag time of 1.1 sec. One must use the 1 second startup lag time when trying to replicate CORSIM behavior, however, th e 1.15 second value measured by Long would be applicable when analyzing actual field data. The necessary computations for carrying-out the delay estimation procedure were incorporated into a software tool called BuckQ. BuckQ is a visual basic application program for Excel which reads the data provided by TSDViewer and pr oduces a variety of useful information based on this data. BuckQ provides, for a one-hour analysis time frame having four 15-minute periods, a second-by-seco nd tabulation of items such as queue length, back of queue position, stopped delay, move-up delay and control delay. It also provides a host of ancillary capabilities, including automated calculation of: start-up-lost-time, saturation flow, and capacity by cycle; HCM queuing and delay information by 15-minute period; and arrival type by 15minute period. In addition, BuckQ allows evalua tion of arrival patterns using a chi-squared goodness-of-fit test and provides ex tensive graphing capabilities. However, the most important feature of BuckQ is its abil ity to accommodate second-by-s econd queue and delay prediction procedures and its ability to compare the results of these procedures to CORSIM results. Using BuckQ, delay prediction algorithms can be tested to see how well they perform and the results presented in a graphical format.

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80 The following information is compiled by BuckQ on a second-by-second basis for the entire 3600-second (60-minute) analysis period: Length of queue 1 Length of queue 2 Actual stopped delay Back of Queue position for queue 1 Back of Queue position for queue 2 Length of visible queue 1 (constrained by FOV) Length of visible queue 2 (constrained by FOV) Visible stopped delay Visibility status = 1 when there is a rising queue blind period = -1 when there is a f alling queue blind period = 0 when there is no blind period Development, testing and refinement of th e various software programs was carried out using a large number of data sets covering a wide ra nge of near-saturated and over-saturated arrival patterns and three cycl e lengths (80, 120 and 160 seconds ). The extensive testing was necessary to ensure that both programs functioned properly over a wide variety of conditions, including grossly over-saturated conditions. Prediction Algorithm for Non-Visible Delay One of the central elements of this resea rch is the development of a predictive algorithm that determines a reasonable value for the delay associated with the non-visible portion of the queue. The first component of the algorithm is an estimation technique that uses the rate of arrivals into the FOV to estimate the arriva l rate at the back of the non-visible queue. Non-Visible Queue Estimation Technique Estim ated NVQ Length = f(vehicles entering FO V during blind period, length of the blind period, departure rate) This technique assumes that vehicles arrive at the back of the queue at a uniform rate during the full extent of the bli nd period. The arriva l rate is calculated using the number of

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81 vehicles that enter the FOV during the blind period. For example, if the blind period last for 32 seconds and 8 vehicles enter the FOV, then the es timated arrival rate is 8 vehicles/32 seconds or 0.25 vehicles/second. All of these vehicles enter the FOV during the falli ng queue portion of the blind period, a time when traffic is freely flowing thru the FOV. Vehicles are also assumed to depart the non-vis ible queue at a consta nt rate of 1 vehicle per second during the Falling Queue Blind Period. Since the departure rate is almost always greater than the arrival rate, the non-visible queue shrinks in size and, if sufficient green time is provided, eventually disappears during this period. As discussed previously, the blind period ends when a 5 s econd (or greater) gap occurs between vehicles entering the FOV since a gap of this size suggests that we have come to the end of the non-visible queue of vehicles. The blind period thus gives way to a period of visibility during which we know for sure what the true queue length is because we can observe it. In reality, it may or may not be true that a 5 second headway signals the end of the blind period. It may be that the la st vehicle in the non-visible que ue passed some time ago or, conversely, it may be that there are more vehi cles in the non-visible queue but that some sleeper (a slow truck, someone fiddling with their radio, etc.) has allowed a large gap to form in front of him or her. The use of a five-second headway is a reasonable compromise between these two situations, at least when we are deali ng with a stream of traffic composed solely of passenger cars. In any event, given a limited fiel d of view, selection of some reasonably prudent headway value that is neither too long nor too sh ort under most circumstances is the best that can be done. Initial experiments have verified that th is particular technique does a good job of estimating non-visible queues and delays when a period of visibility follows the blind period.

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82 However, when traffic volumes intensify, it is often the case that the FOV fills with queued vehicles without a 5-second headway being obse rved. In this case, adjacent blind periods occur. The problem with adjacent blind periods is twofold: 1) The true nu mber of vehicles that arrived during the blind period is unknown because the FOV fills-up and all of the arrivals do not come into the FOV, and 2) One never really k nows where the true end of the queue is, forcing non-visible queue length estimations to be ma de that depend on previous non-visible queue length estimations. Additional adjustments are needed to handle adjacent blind periods. When adjacent blind periods occur, the number of vehicles entering the FOV during the blind period may substantially unde restimate the number of vehicles that arrived at the back of the non-visible queue during the blind period. A second adjustment technique is needed to augment the initial estimation technique when this occurs. Non-Visible Queue Adjustment Technique: Adjusted NVQ Length = f(vehicles entering FOV during blind peri od, length of the blind period, departure rate, adjacent blind pe riod counter) = f(estim ated arrival rate, departure rate, adjacent blind period counter) The adjacent blind period counter increments by a value of 1 whenever a blind period is followed by another blind period, and resets to zer o when a period of visibility occurs. The estimated arrival rate is incr eased using an additive power function of the following form: ARadj = ARest + [(ABPC + C)P]/X Where: ARadj = Adjusted Arrival Rate ARest = Estimated Arrival Rate ABPC = Adjacent Blind Period Counter C, P, X = Constants The longer the end of the queue stays out of view, the higher the ABPC becomes and the more the adjusted arrival rate is increased in co mparison to the estimated arrival rate. Extensive

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83 testing suggests that the following constants prov ide good predictive abilities, even during highly over-saturated conditions where some vehicles experience as many as six phase failures: P = 0.4 C = 66 X = 30 These constants can be varied to change the sh ape of the predicted cumulative delay curve. Figure 4-11 is the base condition where P, C and X equal the values just listed. If P, the power constant, is increased from 0.40 to 0.41 while holding C and X consta nt, the entire curve shifts upward as shown in Figure 4-12. If C, the addi tive constant, is increa sed while keeping P and X at their original values, then the curve both increa ses and flattens out. If X, the division constant, is decreased while keeping P and C at their origin al values, then the tail end of the curve shifts upward. The optimum combination of P, C and X that results in a predicted cumulative stopped delay curve that most closely follows the act ual cumulative stopped delay curve is obtained through trial and error. Non-Visible Queue Re-Adjustment Technique: Re-Adjusted NVQ Length = f(vehicles entering FOV during b lind period, length of the blind period, departure rate, adjacen t blind period counte r, average headway, average free flow speed, average vehicle length) = f(adjusted arrival ra te, average headway, average free flow speed, average vehicle length) As a queue becomes longer the back of the queue propagates closer to its source of arrivals. This tends to increase the effective a rrival rate of vehicles at the end of the queue. Hurdle [2] recognized this fact in his investigation of in tersection delay: Another way of thinking about the model is to say that, in the model, vehicles do not line up along the street but form a ve rtical stack at the stop line. The real queue is always somewhat longer than the model predicts because the queue engulfs some vehicles that the model assumes are still driving to th e vertical stack at the stop line.

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84 Figure 4-13 provides an example. In this exampl e, an additional arrival effectively occurs once every 60 seconds due to queue propagation. This adjustment becomes significant as volume exceeds capacity and queues become extensive. Examples To de monstrate the analysis procedure, four examples based on a 120 second cycle length were developed. Each example us es a different set of arrival ra tes that result in over-capacity conditions at some point during the one-hour analysis time frame. Three runs (replications) were made for each example with a different rand om number set used for each of the three replications: See Table B-29. Tables 4-1 and 4-2 summarize the characteris tics of these examples while Tables 4-3 through 4-5 summarize the predictive results. The first column of each table lists the Random Number (RN) set that was used and the second column provides an abbreviation of the file name that includes the 15-minute volumes that were input into CORSIM. Considering the first row, random number set 1 was used and the 15-minute input volumes were 625 vph, 700 vph, 650 vph and 350 vph. The input volume for the last 15-minute period was always set at a re latively low value so that all residual queues would clear by the end of the one-hour analysis time frame. This ensured that all delay was accounted for. Because of the random fluctuation in arrivals the arrival flow rates input into CORSIM are, in almost every case, not th e same as the arrival flow rates that actually ente r the link. For example, the 625, 700, 650, 350 vph input flow rates a ssociated with random number set 1 (the row 1 values) produce link entry flow rates of 640, 692, 628 and 364 vph. By the time these entering vehicle reach the back of the queue, the arrival flow ra tes have changed once again to the 676, 688, 652, 360 vph values shown in Table 4-1. It is these arrival at BOQ (Back of

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85 Queue) volumes that are of interest because it is these volumes that contribute directly to the formation of queues and the associated stoppe d delay. Arrival at BOQ volumes are also provided for the hour as a whole and for the firs t 45 minutes of the hour (the portion of the hour during which near or over capacity conditions exist). Also provided in Table 4-1 are the approach capacity values for each 15-minute period; along with the capaci ty value for the first 45 minutes of the hour. BuckQ automatically calculates the capacity values by applying the methodology described in Chapter 16, Appendix H of the Highway Capacity Manual [4] to traffic stream information obtained from CORSIM. In order to calculate the capacity our analysis procedure determ ines, for each 15-minute period, the needed intermediate variables such as queue discharge Headway (H), Start-Up Lost Time (SULT), and effective green time (g). The Extens ion of Effective Green (EEG) is determined for the first 45-minutes of the hour by minimizing the sum of the squared deviations between the cycle-by-cycle capacity values calculated using the Highway Capacity Manual procedure and actual cycle-by-cycle thruput. A review of Tabl e 4-1 indicates that th e calculated capacity values show considerable variation. This is not surprising when one c onsiders the substantial degree of variation in driver behavior that has been incorporated into CORSIM, including variations in driver aggressiveness associated with departi ng the queue (which affects both SULT and H) and in making use of the yellow and a ll red change interval time (which affects the EEG). All drivers do not behave the same and CORSIM correctly recognizes this. Volume-to-capacity ratios are calculated for each 15-minute period and for the first 45 minutes of the hour. These values are also provided in Table 4-1. For individual 15-minute periods, the v/c ratio varies from a low of 0.92 (RN set 2 for file 625_700_650_350) to a high of 1.24 (RN set 1 for file 725_700_700_350). For the firs t 45 minutes of the hour, the v/c ratio

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86 varies from a low of 1.02 (RN set 2 for f ile 625_700_650_350) to a high of 1.12 (RN set 1, 2 or 3 for file 725_700_700_350). A review of the average values shows that, for the first 45-minutes, both volume and v/c ratio steadily increase as one moves down the tabl e, while capacity remains constant at 644 vph. The average volume increases from a low of 664 vph to a high of 722 vph while the average v/c ratio increases from 1.03 to 1.12 The first section of Table 4-2 summarizes va rious values used for capacity analysis, including cycle length, green time queue discharge headway, satu ration flow rate, and start-up lost time. Our analysis procedure calculates th ese values on both a cycle-by-cycle basis and a 15-minute period basis as well as for the entire hour, but only the hourly values are presented here. As the v/c ratio increases, the amount of green time (G) increases to its maximum setting of 38 seconds, and the cycle le ngth (C) increases to its maximum value of 120 seconds. This makes sense for an actuated approach. The extens ion of effective green, st art-up lost time, queue discharge headway, and saturation fl ow rate all remain about the same as the v/c ratio increases, which also seems reasonable. The overall aver age queue discharge headway of 1.81 seconds is very close to the 1.80 CORSIM input value. However, the overall average start-up lost time value of 2.7 seconds is significan tly greater than the 2.0 second m ean start-up dela y input into CORSIM. The difference is due to a definition inconsistency. CORSIM only applies the mean start-up delay to the first vehicle, adding additi onal delay (of about 0.7 seconds) to subsequent vehicles. In other words, CORSIMs mean startup delay is not the same as start-up lost time. The next section of Table 4-2 provides a qua lity control check on the results for actual stopped delay and control delay durin g the one hour analysis time frame. This check is made by comparing the values obtained from our analys is procedure to similar values found in the

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87 CORSIM output report. Considering the delay definitions used in CORSIM, we would expect CORSIM Stop Time to approximately equal the actual stopped delay obtained from our procedure, and we would expect CORSIM Queue Delay to be slightly greater than the actual stopped delay. This is true in every case. We would also expect CORSIM Delay Time to approximately equal the actual control delay obtai ned from our procedure, and we would expect CORSIM control delay to be slightly less than th e actual control delay. Once again, this is true in every case. As we might expect, the amount of both stopped delay and control delay increases as the v/c ratio increases. The final section of Table 4-2 summarizes, fo r the Poisson distribution, the chi-square goodness-of-fit test results based on 20-second arrival intervals. During only one of the fortyeight 15-minute periods examined (2% of the time) did the test statistic exceed the 95th percentile reference statistic. CO RSIM 6.0 appears to be generating truly random arrivals. It is important to use 20-second arriva l intervals when conducting this test since the use of longer intervals reduces the number of available data poi nts while the use of shor ter intervals can give rise to truncation effects that distort the resu lts. The truncation effects arise because unsafe headways of less than 1.5 seconds are rarely encountered within the CORSIM traffic stream. Queue Prediction Table 4-3 summarizes the queue prediction result s for our analysis pro cedure as compared to actual queues. Comparisons are made of average queue length, maximum queue length, maximum back of queue position, and 98th percentile back of queue position. Figure 4-14 depicts actual queue length as a function of v/ c ratio while Figures 4-15 through 4-17 compare actual and predicted queue results for the averag e queue length, the maximum queue length, and the 98th percentile back of que ue, respectively. Figures 4-14 through 4-17 all demonstrate that, as might be expected, queue length tends to in crease linearly as a function of the v/c ratio. A

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88 review of Figures 4-15 through 4-17 also indicates that our procedure is fairly good at predicting all of these queues, with the amount of error increasing somewh at as the v/c ratio increases. The procedures contained in the Highway Capacity Manual, provide information on the 98th percentile back of queue. A review of Figure 4-17 indicates that the HCM procedures grossly overestimate the 98th per centile back of queue. Also provided in Table 4-3 is information on the number of (liberal) phase failures, the percentage of cycles experiencing a phase failure and the number of vehicle re-queues. Phase failures are defined in relation to the cycle and, as such, are insens itive to the numbe r of vehicles involved. For example, a phase failure occurs for a given cycle if only one vehicle is forced to re-queue, or if 100 vehicles are forced to re-queue. For this reason, the number of vehicle requeues is a much better indicator of the extent of congestion than the number of phase failures. Figure 4-18 demonstrates that th e number of vehicle re-queues te nds to increase linearly as a function of v/c ratio. Stopped Delay Prediction Table 4-4 summ arizes the stoppe d delay prediction results for our analysis procedure as compared to actual stopped delay. Figure 4-19 i ndicates that the proced ure does a pretty good job of predicting stopped de lay over all v/c ratios. Figure 4-20 shows the relative contribution of each segment of the prediction methodology. For the examples under considerati on, visible delay makes-up about 60% of total stopped delay when the v/c ratio is near 1.02 but only 20% of tota l stopped delay when the v/c ratio climbs to 1.12 This clearly demonstrates the need for this predictive procedure, at least for the rather typical case wh ere the cycle length is 120 seconds and the field of view is limited to 12 vehicles. The first step in the predictive process uses an estimated arriva l rate based on vehicles entering the field of view to predict the non-visible queue. This alteration increases the

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89 percentage of captured stopped de lay to about 80% when the v/c is near 1.02 and to about 30% when the v/c is near 1.12. The results become reasonable for relativel y low over-saturated v/c ratios but not for the higher ratios. The second step in the predictive pr ocess uses an adjusted arrival rate obtained from a power function adjustment that increases the estimated arrival rate based on the number of adjacent blind periods. This alteration increases the percentage of captured stopped delay to about 115% when the v/ c is near 1.02 and to about 65% when the v/c is near 1.12. The results are st ill reasonable for relatively low ove r-saturated v/c ratios, and are greatly improved for the higher ratio s, but the error for the higher ratios is still quite significant. The third step in the predictive process adjusts the non-visible queue length and associated delay due to queue propagation. This alteration has li ttle or no affect on the percentage of captured stopped delay when the v/c is close to one but in creases the percentage of captured stop delay to about 90% when the v/c is high. The results are now reasonable over all v/c ratios although a slight upward bias of about 15% exists near th e lower oversaturated v/c ratios and a slight downward bias of about 10% exis ts near the higher v/c ratios. A tremendous improvement in stopped delay estimation is clearly provided by our procedure. Figure 4-21 provides another way of visualizing the final predictive results. The maximum individual over-estimation of delay is 27% and the maximum individual under-estimation is 17.5%. If the results are aver aged over the three random number replicates, as is documented at the bottom of 4-4, the ma ximum over-estimation is 13% and the maximum under-estimation is 11%. If we graph the sum of the Adjacent Blind Period Counter (ABPC) against stopped delay (either actual or predicted) as shown in Figur e 4-22, a strong linear relationship exists. This

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90 provides rather strong support for our use of the AB PC as the explanatory variable in our arrival rate adjustment process. Control Delay Prediction Table 4-5 s ummarizes the control delay prediction results as compared to actual control delay. Figure 4-23 indicates that the analysis procedure also does a reasonably good job of predicting control delay over all v/c ratios, even if we use a cons tant ratio of 1.3 to convert our predicted stopped delay into predicted control delay. This conversion factor actually varies somewhat by v/c ratio as shown in Figure 4-24. (Pre vious work has demonstrated that this factor also varies by cycle length; but that is not of con cern here since we have restricted our analysis to a single cycle length.) Also included in Figure 4-23 is control delay as predicted by HCM procedures. The HCM procedures tend to over-pred ict control delay for th e lower over-saturated v/c ratios. Figures 4-25 and 4-26 provide two other ways of visualizing these comparisons between actual control delay, predicted control de lay, and HCM calculated control delay. Control delay is composed of stopped dela y, acceleration/decelera tion delay, and queue move-up delay. As shown in Figure 4-27, the percentage of stopped delay for our example remains relatively constant at about 80% of the c ontrol delay. This is consistent with the fact that the control delay/stopped de lay ratio does not change much as the v/c ratio increases. However, the percentage of queue move-up dela y increases dramatically (more than doubles) as the v/c ratio increases and the percentage of acceleration/decel eration delay falls correspondingly. Recurrent cycle failures and extensive re-queuing associated with high v/c ratios produces this steady and dramatic increase in queue move -up delay. Figure 4-28 provides factors that convert stopped delay plus queue move-up delay to c ontrol delay. A review of this

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91 figure reveals that there is much more variation in this new ratio than with a ratio based only on stopped delay. Variability Considerations To investigate the degree of variability associated with the actual cum ulative stopped delay, and with the predicted stopped delay, ten replicate runs were made for the 700_725_625_350 volume pattern using the sets of ra ndom number seeds found in Table B-30. The last number in the set produces vehicle behavior variation associated with various driver aggressiveness characteristics, including driver response to the ambe r interval, the amount of start-up lost time experienced by the first vehi cle in the queue, the discharge headway of the vehicle, and the free flow speed of the vehicle. Table 4-6 provides a comparison between the actual 1-hour cumulative stopped delay and the predicted stopped delay. A review of the embedded graph in this table shows that the variation in the predicted stopped delay is very similar to the variation in the actual stopped delay, with only of the 10 data points (the one as sociated with random numb er set 8) exhibiting a somewhat unfavorable comparison. This similarity in variation provides some reassurance that the prediction procedur e is behaving appropriately. It is also encouraging to discover that, as is shown in Table 4-6, the 95% conf idence interval for the mean actual stopped delay includes the mean predicted stopped delay. Formal statistical testing was conducted to determine whether a significant difference exists between the actual and predicted median stopped delay. The non-pa rametric Fisher Sign Test, which does not require a symme trical distribution, was used to test the null hypothesis that the mean of the differences betw een the actual and predicted medi an delay is zero. Table 4-7 contains the test, which produces a p-value of about 0.11 The p-value is not significant so we cannot reject the null hypothesis th at the mean of the differences is indeed zero, which reinforces

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92 the idea that the prediction procedure does a relatively good job of estimating the total cumulative stopped delay. Limitations to the Delay Prediction Procedure Our analysis procedure incl udes a new technique for pred icting delay on a signalized intersection approach under condi tions of lim ited information. Although the usefulness of the technique is evident, limitations on the use of the t echnique should be understood. These limitations include the following: 1. As the size of the field of view decreases, the accuracy of the technique also decreases. Testing to date has concentrat ed on a field of view of 12 vehicles with additional runs made at a field of view of 8 vehicles. Reasonable results are obtained with these fields of view up to a v/c ratio of about 1.12 for the ove r-saturated periods. More testing is needed to determine the maximum v/c ratio that can be accommodated with smaller fields of view. 2. The current delay prediction technique can pr oduce rather inaccurate delay forecasts if sleepers are present at critical points in the non-visible queue. A sleeper is defined as a motorist that does not exhibit normal car-f ollowing behavior within the queue; leaving a large gap between his or her vehicle and the preceding vehicle in the queue. This type of lethargic driver behavior can be cause d by in-vehicle distractions or by simple daydreaming. Under the current analysis methodology, the abnormally large gap between vehicles caused by sleepers can result in a false conclusion that the end of the queue has been reached. This causes the adjacent blind period counter to be lower than desired which results in a corre spondingly low adjusted arrival rate. The end result is an underestimation of delay. 3. Our analysis procedure is essentially a queue prediction technique that uses predicted queue length to calculate expe cted stopped delay. Consequently, by its very nature, the procedure is relegated to directly pred icting stopped delay, not control delay. The emphasis on stopped delay makes sense when one considers the limited information made available to the program. The progr am assumes no knowledge of various items important in the direct calculation of cont rol delay; including vehicle free flow speeds and delay associated with both decelerati on and acceleration most of which occurs outside the field of view. Changing stopp ed delay to control delay requires the application of a delay ratio. Typical delay ratios (such as the commonly used 1.30 value) will need to be applied and there will be some inherent error in this factoring process. 4. If a motorist joins the queue and experiences delay but then, prior to entering the field of view, becomes impatient and leaves the queue (known in the queuing literature as reneging), the delay experienced by this mo torist will not be accounted for. Any

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93 delay associated with motorists that decide d not to join the queue due to its excessive length (known in the litera ture as balking ) would also not be accounted for. 5. The research to date has concentrated on random arrivals at an isolated intersection. Some initial experimentation was conducted wi th platooned arrivals and, based on that work, it is clear that the delay situation can change quite a bit depending on the relative offsets of the upstream intersection and the intersection under study. This platoon progression effect is well documented in th e literature. Conse quently, the analysis procedure is less suitable for use on coor dinated approaches, es pecially during undersaturated or near-saturated conditions. For over-saturated conditions, platoon progression effects on coordinated approaches tend to be minimized since all approaching vehicle are forced to join the queue. The analysis procedure should perform well under these conditions. 6. Work completed to date is based on a single micro-simulation tool a nd is subject to all limitations and characteristics of the CORSIM software. A final drawback is that the analysis procedure is still in the form of a research tool that is oriented towards evaluating simula tion runs. Converting the proce dure to a practi cal engineering tool that can be field implemented at a real inte rsection is an important extension that will require additional effort.

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94 Figure 4-1. Queue relationships

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95 Figure 4-2. Signalized inte rsection delay components

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96 Figure 4-3. Measured versus estimated delay

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97 Figure 4-4. Visible a nd non-visible variables

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98 y = -0.9609x + 2.3199 R2 = 0.6393 y = -0.5071x + 1.7992 R2 = 0.502 y = -0.3342x + 1.5652 R2 = 0.5042 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 0.960.981.001.021.041.061.081.101.121.141.16 v/c Ratio During 1st 45 MinutesRatio of Control Delay to Stopped Delay 80 second cycle 120 second cycle 160 second cycle The ratio of control dela y to stopped dela y decreases as the v/c ratio increases; the amount of the decrease varies with cycle length with shorter cycles experiencing a more dramatic drcrease. Figure 4-5. Relationship between v/c ratio and ratio of cont rol delay to stopped delay

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99 Figure 4-6. Re-queuing that re sults in simultaneous queues

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100 Figure 4-7. Re-queuing that does not result in simultaneous queues

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101 Figure 4-8. Example of a blind period

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102 Figure 4-9. Example of adjacent blind periods

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103 Figure 4-10. Counters and queue status

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104 Figure 4-11. Base case for P, C and X; stopped delay comparison

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105 Figure 4-12. Effect of increasing the power constant on stopped delay comparison

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106 Figure 4-13. Queue propagation example

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107 Figure 4-14. Actual vehicle queues

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108 Figure 4-15. Average queue length comparison

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109 Figure 4-16. Maximum queue length comparison

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110 Figure 4-17. 98th percentile back of queue comparison

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111 Figure 4-18. Vehicle re-queuing

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112 Figure 4-19. Stopped delay comparison

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113 Figure 4-20. Stopped delay prediction, 12 FOV

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114 Figure 4-21. Comparison of act ual and predicted stopped delay

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115 Figure 4-22. Adjacent blind period counter v. stopped delay

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116 Figure 4-23. Control delay comparison

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117 Figure 4-24. Ratio of control delay to stopped delay

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118 Figure 4-25. Graphical c ontrol delay comparison,

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119 Figure 4-26. Control delay estimates

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120 Figure 4-27. Control delay composition

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121 Figure 4-28. Ratio of control de lay to stopped plus move-up delay

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122Table 4-1. Example summary volume and capacity File Name Random 15 min input volumesNumber Set 1st 45 minutes1 hour v cv/c 625_700_650_350vph 1676688652360 672594 616 1.10 668 1.03 656 0.99 511 0.70 646 1.04 625_700_650_350vph 2608676672364 652580 663 0.92 642 1.05 623 1.08 492 0.74 641 1.02 625_700_650_350vph 3648668688344 668587 643 1.01 649 1.03 642 1.07 499 0.69 645 1.04 700_725_625_350vph 1708740628348 692606 631 1.12 659 1.12 656 0.96 547 0.64 649 1.07 700_725_625_350vph 2720728624368 691610 648 1.11 661 1.10 610 1.02 604 0.61 641 1.08 700_725_625_350vph 3692732632364 685605 648 1.07 635 1.15 646 0.98 556 0.65 643 1.07 700_700_700_350vph 1708704680380 697618 631 1.12 659 1.07 656 1.04 602 0.63 649 1.07 700_700_700_350vph 2720688712344 707616 648 1.11 661 1.04 610 1.1 7 625 0.55 641 1.10 700_700_700_350vph 3692712704360 703617 648 1.07 635 1.12 646 1.09 614 0.59 643 1.09 725_700_700_350vph 1788692708340 729632 637 1.24 648 1.07 661 1.07 618 0.55 649 1.12 725_700_700_350vph 2728724700356 717627 663 1.10 639 1.13 623 1.12 645 0.55 641 1.12 725_700_700_350vph 3752704700364 719630 648 1.16 635 1.11 646 1.08 610 0.60 643 1.12 vph vph vph vph vph Averages 625_700_650_350vph 644677671356 664587 641 1.01 653 1.04 640 1.05 501 0.71 644 1.03 700_725_625_350vph 707733628360 689607 642 1.10 652 1.13 637 0.99 569 0.63 644 1.07 700_700_700_350vph 707701699361 702617 642 1.10 652 1.08 637 1.10 614 0.59 644 1.09 725_700_700_350vph 756707703353 722 630 649 1.16 641 1.10 643 1.09 624 0.57 644 1.12 Calculated Capacity and Volume-to-Capacity Ratio Arrival at Back of Queue Volumes 15 min volumes vph 4th 15 minutes1st 45 minutes 3rd 15 minutes 1st 15 minutes2nd 15 minutes

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123Table 4-2. Example summary queue di scharge, delay check and goodness-of-fit ActualCORSIMCORSIMCORSIMActualCORSIM File NameRandomCycleGreen DischargeSat FlowLo stExt. ofStoppedStopQueueControlControlDelay 15 min input volumesNumber SetLengthTime g/C ratio HeadwayRateTimeGreenDelayTimeDelayDelayDelayTime CGg=G+EEG-SULTHSSULTEEG dS=<<= 625_700_650_350vph 1118.535.7 0.31 1.8020042.743.464.765.066.776.682.681.93.4 16.9 6.42.2 625_700_650_350vph 2116.734.7 0.30 1.7920092.733.465.966.868.778.984.583.56.92.65.62.2 625_700_650_350vph 3117.635.6 0.31 1.8119942.953.377.377.879.991.998.197.12.57.54.32.5 700_725_625_350vph 1118.836.6 0.32 1.8019992.563.4113.9113.4116.7133.4143.9141.95.24.45.40.9 700_725_625_350vph 2119.137.1 0.31 1.7920072.923.3137.2136.9141.4163.6174.4172.43.74.01.48.3 700_725_625_350vph 3118.936.9 0.32 1.8219802.603.3141.3139.8143.8165.4176.1175.45.45.42.13.9 700_700_700_350vph 1119.837.6 0.32 1.8119912.493.4117.0117.4120.9138.6148.1147.35.27.94.63.8 700_700_700_350vph 2119.537.5 0.32 1.8019982.713.3144.3142.4147.1169.8181.3178.83.72.93.12.6 700_700_700_350vph 3120.038.0 0.32 1.8219742.533.3145.0143.9148.2171.1182.1181.35.42.43.71.1 725_700_700_350vph 1120.038.0 0.32 1.8119852.653.4191.8190.8197.0225.8237.9236.53.32.67.80.7 725_700_700_350vph 2120.038.0 0.32 1.7920072.753.3183.8183.4189.5218.6229.5229.16.93.11.04.6 725_700_700_350vph 3120.038.0 0.32 1.8319712.523.3193.3192.3198.5229.4241.1240.61.03.06.56.6 secsec sec/vehvphgsecsec Averages 625_700_650_350vph 117.635.3 0.30 1.8020022.83.469.369.971.882.588.487.5 700_725_625_350vph 118.936.9 0.31 1.8019952.73.3130.8130.0134.0154.1164.8163.2 700_700_700_350vph 119.837.7 0.31 1.8119882.63.3135.4134.6138.7159.8170.5169.1 725_700_700_350vph 120.038.0 0.32 1.8119882.63.3189.6188.8195.0224.6236.2235.4 ALL 119.137.00.321.8119932.73.3 Chi-Square Test Statistic Queue Discharge Data, Average Over All Cycles 95% Ref. Statistic = 9.49 BuckQ Delay Check Goodness-of-Fit Test (20-sec arrival intervals) sec/veh

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124Table 4-3. Queue prediction Actual% ofActual File NameRNVol PhaseCyclesVehicle 15 min input volSet45minv/cAP%ErrAP%ErrAP%ErrAP%ErrHCM%ErrFailw/ PFRe-Q's 12FOVvv/c 625_700_650_350 1 6721.04 1113 18% 3238 19% 4558 29% 4158 41% 99 141% 21 70% 197 625_700_650_350 2 6521.02 1110 -9% 3734 -8% 4547 4% 4445 2% 108 145% 19 63% 204 625_700_650_350 3 6681.04 1313 0% 3735 -5% 4553 18% 4252 24% 107 155% 24 80% 281 700_725_625_350 1 6921.07 1917 -11% 5241 -21% 6258 -6% 6158 -5% 117 92% 24 80% 591 700_725_625_350 2 6911.08 2319 -17% 5849 -16% 7463 -15% 7062 -11% 113 61% 27 90% 782 700_725_625_350 3 6851.07 2419 -21% 5942 -29% 7563 -16% 7362 -15% 123 68% 27 90% 797 700_700_700_350 1 697 1.07 2019 -5% 5451 -6% 7265 -10% 6360 -5% 123 95% 25 83% 635 700_700_700_350 2 7071.10 2521 -16% 6559 -9% 8165 -20% 7963 -20% 114 44% 28 93% 854 700_700_700_350 3 7031.09 2521 -16% 6746 -31% 8564 -25% 8363 -24% 120 45% 28 93% 872 725_700_700_350 1 7291.12 3426 -24% 7264 -11% 9768 -30% 8269 -16% 139 70% 29 97% 1308 725_700_700_350 2 7171.12 3225 -22% 7668 -11% 9968 -31% 9468 -28% 125 33% 29 97% 1224 725_700_700_350 3 7191.12 3423 68% 7551 -32% 10363 -39% 9763 -35% 128 32% 29 97% 1334 vphvehvehvehvehvehvehvehveh 30.0 Averages 625_700_650_350 6641.03 1212 3% 35 36 1% 4553 17% 4252 22% 105 147% 21 71% 227 700_725_625_350 6891.07 2218 -17% 5644 -22% 7061 -13% 6861 -11% 118 73% 26 87% 723 700_700_700_350 7021.09 2320 -13% 6252 -16% 7965 -18% 7562 -17% 119 59% 27 90% 787 725_700_700_350 7221.12 3325 -26% 7461 -18% 10066 -33% 9167 -27% 131 44% 29 97% 1289 Maximum Back of Queue PositionBack of Queue Position 98th PercentileA = Actual P = PredictedCycles per Hour: QUEUING PHASE FAILURES Average Queue Length Maximum Queue Length

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125Table 4-4. Stopped delay prediction Sum of Stopped Delay % of Time Stopped Delay Prediction Steps Adjacent File Name Random Volume v/c BuckQ Queue Not BuckQ Blind Period 15 min input volumes Number Set 1st 45 minRatio Actual Predicted Visible Visible Estimated Adjusted ReadjustedCounter 12FOV v v/c dS dSP ABPC 625_700_650_350vph 1 672 1.04 64.7 82.2 127.0% 70% 60% 89% 124% 128% 8131 625_700_650_350vph 2 652 1.02 65.9 67.9 103.0% 63% 57% 80% 101% 104% 2118 625_700_650_350vph 3 668 1.04 77.3 85.5 110.6% 77% 52% 78% 108% 111% 10921 700_725_625_350vph 1 692 1.07 113.9 129.1 113.3% 83% 36% 56% 93% 113% 30098 700_725_625_350vph 2 691 1.08 137.2 128.5 93.7% 88% 31% 49% 82% 93% 38508 700_725_625_350vph 3 685 1.07 141.3 130.0 92.0% 88% 30% 47% 82% 92% 38761 700_700_700_350vph 1 697 1.07 117.0 148.6 127.0% 86% 35% 55% 93% 127% 33220 700_700_700_350vph 2 707 1.10 144.3 146.2 101.3% 92% 29% 47% 79% 101% 41636 700_700_700_350vph 3 703 1.09 145.0 133.3 91.9% 92% 29% 46% 76% 92% 41762 725_700_700_350vph 1 729 1.12 191.8 189.3 98.7% 98% 22% 36% 66% 99% 48604 725_700_700_350vph 2 717 1.12 183.8 159.2 86.6% 96% 23% 37% 65% 87% 48170 725_700_700_350vph 3 719 1.12 193.3 159.5 82.5% 99% 22% 36% 63% 83% 48199 vph secs/veh Averages 625_700_650_350vph 664 1.03 69 79 113% 70% 57% 82% 111% 114% 7057 700_725_625_350vph 689 1.07 131 129 99% 86% 32% 51% 86% 99% 35789 700_700_700_350vph 702 1.09 135 143 105% 90% 31% 49% 83% 107% 38873 725_700_700_350vph 722 1.12 190 169 89% 98% 22% 36% 65% 90% 48324 ALL 107% 82% 40% 61% 93% 107%

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126Table 4-5. Control delay prediction 1.XX Actual Stopped File NameRandomVolumev/cControl Delay/Control Delay/Plus Queue 1.XX 1.30 StoppedStop & QQueueAccel./ 15 min input volumesNumber Set1st 45 minRatioStopped DelayStop+QMU DelayMove-Up Delay Actual DelayMove-UpMove-UpDecel. 12FOV vv/c dS+ dMUdCdCH+dCPXdCP3625_700_650_350vph 1 6721.04 1.28 1.19 69.482.6160.5 194% 104.9 127% 106.9 129% 78%84%6%16% 625_700_650_350vph 2 6521.02 1.28 1.19 71.284.5171.3 203% 87.1 103% 88.3 104% 78%84%6%16% 625_700_650_350vph 3 6681.04 1.27 1.16 84.698.1163.2 166% 108.5 111% 111.2 113% 79%86%7%14% 700_725_625_350vph 1 6921.07 1.26 1.12 128.5143.9205.8 143% 163.1 113% 167.8 117% 79%89%10%11% 700_725_625_350vph 2 6911.08 1.27 1.10 159.2174.4191.7 110% 163.3 94% 167.1 96% 79%91%13%9% 700_725_625_350vph 3 6851.07 1.25 1.09 162.2176.1212.3 121% 162.0 92% 169.0 96% 80%92%12%8% 700_700_700_350vph 1 6971.07 1.27 1.12 132.8148.1199.6 135% 188.1 127% 193.2 130% 79%90%11%10% 700_700_700_350vph 2 7071.10 1.26 1.08 167.6181.3185.5 102% 183.7 101% 190.1 105% 80%92%13%8% 700_700_700_350vph 3 7031.09 1.26 1.08 168.1182.1193.0 106% 167.4 92% 173.3 95% 80%92%13%8% 725_700_700_350vph 1 7291.12 1.24 1.06 225.1237.9214.0 90% 234.8 99% 246.1 103% 81%95%14%5% 725_700_700_350vph 2 7171.12 1.25 1.05 218.8229.5207.4 90% 198.8 87% 207.0 90% 80%95%15%5% 725_700_700_350vph 3 7191.12 1.25 1.05 228.9241.1220.6 91% 198.9 83% 207.4 86% 80%95%15%5% vph secs/vehsecs/vehsecs/vehsecs/vehsecs/veh Averages 625_700_650_350vph 6641.03 1.28 1.18 75.188.4165.0 187% 100.2 113% 102.1 115% 78%85%7%15% 700_725_625_350vph 6891.07 1.26 1.10 150.0164.8203.3 123% 162.8 99% 168.0 102% 79%91%12%9% 700_700_700_350vph 7021.09 1.26 1.09 156.2170.5192.7 113% 179.7 105% 185.5 109% 79%92%12%8% 725_700_700_350vph 7221.12 1.25 1.05 224.3236.2214.0 91% 210.9 89% 220.1 93% 80%95%15%5% ALL 1.26 1.11 142% 107% 110%79%89%10%11% Control Delay BuckQ PredBuckQ Pred HCM Percentage of Control Delay

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127Table 4-6. Comparison of variation in actual and predicted stopped delay Cumulative 1-Hour Random Stopped Delay (sec) Number Set Actual Predicted 1 68622 77325 2 83364 77713 3 85601 78925 4 80081 69056 5 59339 57874 6 95345 91536 7 94206 78308 8 111432 73012 9 66737 67418 10 78859 75952 Mean 82359 74712 Std Deviation 15441 8836 CV 0.19 0.12 Std. Error 4883 2794 95% C.I. 9571 5477 Lower 72788 69235 Upper 91929 80189

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128 Table 4-7. P-value determination fo r difference in median values Fisher SignTest For paired replicates Null hypothesis: Differences between actual and predicted median delay is zero Cumulative 1-Hour Random Stopped Delay (sec) Number Set Actual Predicted Difference Mu RN X Y Z = Y X u 1 68622 77325 8703 1 2 83364 77713 -5651 0 3 85601 78925 -6676 0 4 80081 69056 -11025 0 5 59339 57874 -1465 0 6 95345 91536 -3809 0 7 94206 78308 -15898 0 8 111432 73012 -38420 0 9 66737 67418 681 1 10 78859 75952 -2907 0 B =2 From Reference Table with n = 10 and b = B = 2: p/2 = 0.0547, p = 0.1094

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129 CHAPTER 5 THEORETICAL BOUNDS FO R DE LAY ESTIMATION This chapter describes the development of theo retical limits on the solution space for the empirical delay prediction procedure (Objective 4). The delay estimation procedure presented in th e previous chapter begins by calculating an "estimated arrival rate", which is actually the depa rture rate. Then, if the back end of the queue is not visible, the procedure modifies the estimat ed arrival rate upward us ing a power function to predict the real arrival rate. This power function adjusts the rate in a manner that, in essence, varies with the amount of time dur ing which the back end of the que ue is not visible. A major advantage of this approach is that the resulting estimated queues and associated delay can be immediately calculated on a secondby-second basis, in real time. A major disadvantage of the approach is that there is no relationship between the departure rate and the real arrival rate. Under the right circumstances, errors can accumulate to the point that the delay estimation is no longer reasonable. The potential for this is high est when the length of time that the end of the queue is not visible covers most of the analysis time frame. However, it is possible to calculate a set of theoretical uppe r and lower bounds on the solution space by using information obtained at th e end of the analysis period when the arrival rate does equal the depa rture rate. In order to make any ty pe of reasonable delay estimation, all queues must dissipate prior to the end of the analysis time frame. Once this occurs, a calculation of the arrival rate (which is equal to the depa rture rate) during the fina l portion of the analysis time frame, the last 15 minutes of the hour, can be made. Knowing this final arrival/departure rate and knowing the total number of vehicles that have crosse d the stop bar during the entire hour we can, by assuming a reasonable minimum peak hour factor, work backwards through the period to identify arriva l curves that serve as both lower and upper bou nds. These theoretical

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130 results can be used, in an ex post facto mann er, to bracket the real-time delay estimation procedure presented in the previ ous chapter. These bounds can also be us ed to identify an independent most probable delay pattern by sele cting an intermediate curve between the upper and lower bounds that minimizes the maximum per cent error between the es timate and the actual delay. Chapter 16 of the 2000 Highway Capacity Manu al [4] contains a widely recognized and well-accepted procedure for calculating per-vehicle c ontrol delay at signalized intersections. In the 2000 HCM, this control delay has three components: d1 (uni form delay), d2 (incremental delay) and d3 (initial queue de lay). Component d2 can be further subdivided into an oversaturation element and a random delay element. The random delay element is based on a coordinate transformation technique originally proposed by Whiting and refined for signalized delay applications by Akcelik [47]. In 2007, Courage [48] demonstrated the relationships between overflow delay, determin istic queue delay, incremental de lay and initial queue delay. Courage showed that overflow delay and deterministic queue delay (both of which can be calculated using the area between the cumulative arrival curve and the uniform cumulative departure curve) were each com posed of initial queue delay a nd the over-saturation portion of the incremental delay. The random portion of the c ontrol delay is not reflect ed in the cumulative arrival and departure curves, nor is the portion of the control delay associated with acceleration or deceleration. In addition, qu eue move-up delay is not explic itly depicted in the cumulative arrival and cumulative departure curves although its effect is somewhat implied within the general treatment of delay as the area betw een the curves. Appendix F of the 2000 HCM discusses the relationship between the initial queue de lay and deterministic queue delay. Five

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131 specific arrival cases are discussed and the pr oper way to account for initial queue delay and deterministic delay for each case is explained. The theoretical delay literature is extended in this chapter through the development of a theoretical framework for establishing the upper and lower bounds of the overflow delay given a terminal arrival rate and a mini mum Peak Hour Factor (PHF). The mathematical bracketing of overflow delay using this type of information represents a new aspect of delay estimation. Derivation of the Bounds During a period of over-saturated flow on a signalized intersection approach, the cum ulative number of arrivals at the back of the queue exceeds the cumulative number of departures from the stop bar, with resulting queue formation. Let us assume that over-saturated flow begins immediately at the start of a one -hour observation period a nd that, at some point near the end of the hour, it is replaced by a peri od of under-saturated flow that causes the queue to dissipate before the hour e xpires. Let us also assume that the component 15-minute flow rates follow a reasonable pattern that result in some minimum Peak Hour Factor (PHF). Figure 5-1 graphically depicts the analysis setting. Both the cumulative arrival cu rve and the cumulative depa rture curve are monotonically increasing functions. If we have enough informati on to construct both of these curves, then the delay during the period can be found by simply calculating the area between the curves. However, if we are dependent upon detection de vices located at the intersection then, during periods of over-saturated flow, we will only be ab le to measure the attributes of the departure curve, not the arrival curve, since the end of the queue will be beyond our Field of View (FOV). Under these circumstances we can still obtain, after the one-hour an alysis period ends, a reasonable estimation of the de lay that occurred during the pe riod. We cannot know with certainty the delay that occurred because we have no direct knowledge of the shape of the arrival

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132 curve. However, we can obtain an estimate of the most-likely amount of delay and can put limits on the expected error associated with that estimate. The delay estimation begins by measuring the following values: 1.) the total number of vehicles that arrived during the an alysis period; which also equals the number of vehicles that departed during the analysis period since it is assumed that the overflow queue fully dissipates, 2.) the overflow queue clearance time, or the tim e point at which the cumulative arrival curve and the cumulative departure curves intersect; which is also the time at which the overflow queue is reduced to zero, and 3. ) the total number of vehicles that have arrived when the overflow queue clearance time was reached. Using this information, the arri val rate during the last 15-m inute period (period 4) of the hour can be calculated: AR4 = (CA60-CAC)/(T60-TC) (1) Where: AR4 = Arrival Rate during period 4 (veh/sec) CA60 = Cumulative Arrivals at tim e point 60 (end of the hour) CAC = Cumulative Arrivals at ove rflow queue Clearance time point T60 = Time point 60 (3600 seconds) TC = Time point when overflow queue Clears In the example shown in Figure 5-2, the arrival rate is calculated to be: AR4 = (575 veh 540 veh)/(3600 sec 3240 sec) = 0.0972 veh/sec This can be converted to an hourly flow rate by multiplying by 3600 sec/hour: V4 = (0.0972 veh/sec)(3600 sec/hour) = 350 veh/hr The cumulative number of arriving vehicles at th e beginning of the last 15-minute period is calculated by multiplying this terminal hourly fl ow rate by the duration of the period and then subtracting the resulting value from the cumulativ e number of arriving vehicles at time point 60: CA45 = CA60 (AR4)(t4) or CA45 = CA60 V4 (2)

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133 Where: AR4 = Arrival Rate during period 4 (veh/sec) V4 = Arrival Flow Rate during period 4 (veh/hr) CA60 = Cumulative Vehicles at time point 60 (end of the hour) CA45 = Cumulative Vehicles at time point 45 t4 = Duration of the 4th 15-minute time period (sec) Continuing the Figure 5-2 example, the cumulativ e number of arrivals at the beginning of the last 15-minute time pe riod is calculated as: CA45 = 575 veh (0.0972 veh/sec)(900 sec) = 487.5 veh Given this value, we can now calculate the am ount of overflow delay that occurs during the last 15-minute period (see Figure 5-3): OD4 = Area between Cumulative Arrival Curve and Uniform Departure Curve OD4 = 0.5 (t4S)2(UDR4AR4) = 0.5(Tc T45) 2(UDR4AR4) (3) Where: OD4 = Overflow Delay during period 4 (veh-sec) UDR4 = Uniform Departure Rate during period 4 (veh/sec) AR4 = Arrival Rate during period 4 (veh/sec) t4S = Duration of over-saturated flow during 4th 15-min time period (sec) For our example, the overflow delay during period 4 is calculated to be: OD4 = 0.5 (3240 sec 2700 sec)2(0.1667 veh/sec 0.0972 veh/sec) = 10,133 veh-sec And the arrival rate in vehicles per hour during period 4 (V4) is calculated as: V4 = (575 veh 487.5 veh)(4/hr) = 350 veh/hr Calculating the overflow delay fo r the other three periods is not as straightforward. The arrival rate during each period cannot be definiti vely established since one can only measure the departure rate, not the true arriva l rate, and since the extent of th e queue is only vi sible to the end of the Field of View. However, even with this limited information, one can still develop a best estimate of the overflow delay. This is done by identifying both a maximum reasonable delay arrival curve and a minimum reasonable delay arrival curve. Max imum and minimum delay

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134 curves are then calculated which correspond to each of these arrival curves and a check conducted to ensure that the delay estimated by th e BuckQ analysis procedure falls within these bounds. We can also use the theoretical bounds to establish an independent best estimate of the overflow delay by construction an intermediate delay curve that minimizes the maximum percent error in delay at each time point. Two reasonable assumptions are required in order to bracket the estimated overflow delay on both the low and high side. The first assumpti on is that the arrival ra te observed during the final 15-minute period is the lowe st rate experienced during the hour The second assumption is that the PHF (Peak Hour Facto r) is greater than or equal to some reasonable minimum value (such as 0.75) that is specified in advance. The minimum PH F value can be easily obtained through an examination of historical tr affic counts for the approach under study. A third assumption is also inherent in th e proposed methodology; the assumption that the arrival rate is constant over each 15 minute pe riod. If the rival rate varies during a given 15minute period then the cumulative arrival curve w ill appear curvilinear in nature. This can be problematic when dea ling with the lower bound. Derivation of the Upper Bound Conservation of flow principa ls dictate that the average of the arrival flow rates during each of the four 15-minute periods must equal th e arrival rate over the entire 1-hour period: (V1 + V2 + V3 + V4)/4 = CA60 (4) Where: Vi = Arrival Flow Rate dur ing period i (veh/hr) CA60 = Cumulative Arrivals at time point 60 (veh) Equation (4) constitutes the firs t constraint on the solution space for both the minimum and maximum reasonable delay curves. Conti nuing our example, equation (4) becomes: (V1 + V2 + V3 + 350 veh/hr)/4 = 575 veh/hr

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135 V1 + V2 + V3 = 1950 veh/hr Maximum overall delay is obtained when th e highest 15-minute flow rates occur at the start of the hour. Consequently, when identifyi ng the maximum reasonable delay curve, the PHF is defined as follows: PHF = (V1+V2+V3+V4)/[(4)Max(V1,V2,V3,V4)] PHF = (V1+V2+V3+V4)/4V1 (5) Equation (5) constitutes the second constraint on the so lution space for the maximum reasonable delay curve. Assuming a mini mum PHF of 0.75 and continuing our example, equation (5) becomes: 0.75 = (V1+V2+V3+350 veh/hr)/4V1 3V1 = (V1+V2+V3+350 veh/hr) 2V1 V2 V3 = 350 veh/hr Equations (4) and (5) cannot be uniquely solved since we have only 2 equations to solve for 3 unknown variables (V1, V2 and V3). However, an examination of the solution space for this problem indicates that we can obtain an additional equation by attempting to set V2 as high as possible (in a continued attempt to maximize de lay). In this case, the upper limit for V2 is V1. V2 cannot be greater than V1 or delay would not be maximized. With V1 forming the upper limit for V2 we have the additional equation: V1 = V2 (6) We can now solve for all of the Vis. Subs tituting equation (6) into equation (4) produces: V1 + V1 + V3 + V4 = 4CA60 2V1 + V3 + V4 = 4CA60 V3 = 4CA60 V4 2V1 (7)

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136 And substituting equations (6) and (7) into equation (5) produces: PHF = (V1 +V1 + (4CA60 V4 2V1) + V4)/(4V1) PHF = (V1 +V1 + 4CA60 V4 2V1 +V4)/(4V1) PHF = (4CA60)/(4V1) 4V1PHF = 4CA60 V1 = CA60/PHF (8) Substituting equation (8) in to equation (6) produces: V2 = CA60/PHF (9) And substituting equations (8) and (9) into equation (4) yields: CA60/PHF + CA60/PHF + V3 + V4 = 4CA60 V3 = 4CA60 2CA60/PHF V4 V3 = 2CA60 (2 1/PHF) V4 (10) Continuing our example and utilizin g equations (8), (9), and (10): V1 = 575/0.75 = 766.7 veh/hr V2 = 575/0.75 = 766.7 veh/hr V3 = 2(575 veh/hr) (2 1/(0.75)) 350 veh/hr = 416.7 veh/hr So, for our example, the cumulative arrival curve that produces the maximum reasonable delay has quartile hour ly flow rates of: 766.7 vph, 766.7 vph, 416.7 vph, and 350.0 vph. This upper bound curve is depicted in Figure 5-4. In this example, V1 was a feasible upper limit for V2, which results in maximum delay. However, it is possible that V1 may not be a feasible upper limit for V2. This occurs when the value of V4 is too high to allow V1 to equal V2 without violating the minimum PHF requirement.

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137 To account for this possibility, equation (10) must be restricted so that V3 is greater than or equal to V4. And since maximum delay occurs when V3 is minimized (which, in turn, maximizes V2 subject to the PHF constraint), V3 must equal V4. In other words, If V 1 does not form the upper limit for V 2 then maximum delay will be obtained when V 3 = V 4 which is the minimum V 3 given our initial assumption that V 3 must be greater than V 4 The value of V4 at which this restriction occurs can be found by setting V3 equal to V4 in equation (10): V3 = 2CA60 (2 1/PHF) V3 2V3 = 2CA60 (2 1/PHF) V3 = CA60 (2 1/PHF) = V4 (11) For our example: V3 = 575 (2 1/0.75) = 383.3 veh/hr V4 = V3 = 383.3 veh/hr Consequently, in our example, if V4 is less than 383.3 then V1 = V2 and equation (10) can be used to calculate V3. Otherwise, V3 must be set equal to V4 and the remaining equations solved accordingly. In general, V3 must be set equal to V4 if V4 > CA60 (2 1/PHF). If V 1 does not form the upper limit for V 2 then we have the additional equation : V3 = V4 (12) We can once again solve for all of the Vis. Substituting equation (12) into equation (4) produces: V1 + V2 + V4 + V4 = 4CA60 V1 + V2 + 2V4 = 4CA60 V2 = 4CA60 V1 2V4 (13) And substituting equations (12) and (13) into equation (5) produces:

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138 PHF = (V1+ (4CA60 V1 2V4) + V4 + V4)/(4V1) PHF = (V1 + 4CA60 V1 2V4 + 2V4)/(4V1) PHF = (4CA60 )/(4V1) 4V1PHF = 4CA60 V1 = CA60/PHF (8) This is the same result as before for V1. Substituting equation (8) into equation (13) produces: V2 = 4CA60 CA60/PHF 2V4 V2 = (4 1/PHF)CA60 2V4 (14) If, in our example, V4 was actually 385 instead of 350, then setting V1 = V2 and using equation (10) would result in a value for V3 of: V3 = 2(575 vph) (2 1/0.75) 385 vph = 381.6 vph But this is not acceptable, since V3 = 381.6 would be less than V4 = 385, which violates our original assumption that the la st period must be the period with the lowest flow rate. Rather, if V4= 385 vph, then V3 must be set equal to V4 and equation (13) used to solve for V2 (The value of V1 does not change): V2 = (4 1/0.75)(575 vph) 2(385 vph) = 763.3 So, for this modified example, the cumula tive arrival curve that produces the maximum reasonable delay has quartile hourly flow rates of: 766.7 vph, 763.3 vph, 385.0 vph, and 385.0 vph. Derivation of the Lower Bound We previously discussed how conservation of fl ow princip als dictate that the average of the arrival rates during each of the four 15-minute periods must equal the arrival rate over the entire 1 hour period:

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139 (V1 + V2 + V3 + V4)/4 = CA60 (4) Where: Vi = Arrival Rate during period i (veh/hr) CA60 = Cumulative Arrivals at time point 60 (veh) For our example, equation (4) became: (V1 + V2 + V3 + 350 veh/hr)/4 = 575 veh/hr V1 + V2 + V3 = 1950 veh/hr Minimum delay occurs when the vertical distance between the arrival curve and the departure curve (the nominal queue length) is continually minimized, without the end of the queue becoming visible. This happens when the nominal queue length equals the Field of View (FOV). Under these conditions, the minimum value for V1 is: V1 = [(UDR1)(t1) + FOV] x 4 periods/hr, or V1 = C1 + 4FOV (15) Where: V1 = Arrival Rate during period 1 (veh/hr) UDR1 = Uniform Departure Rate during period 1 (veh/sec) FOV = Field of View (veh) t1 = Duration of 1st 15-min time period (sec/period) = 900 sec/period C1 = Capacity during period 1 (veh/hr) V1 cannot be any lower than this value or the end of the queue would be visible at the end of period 1 and no estimation of th e delay associated with the ove rflow queue would be required. If V1 equals this absolute lower bound, then we can continue to minimize delay by having V2 equal the following: V2 = [(UDR2)(t2)] x 4 periods/hr, or V2 = C2 (16) This produces a cumulative arrival curve for pe riod 2 that parallels the uniform departure curve for period 2. Assuming a FOV of 12, we continue our example as follows:

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140 V1 = [(0.1667 veh/sec)(900 sec/period) + 12 veh] x 4 periods/hr = 600 + 48 = 648 veh/hr V2 = [(0.1667 veh/sec)(900 sec/period)] x 4 periods/hr = 600 veh/hr We can now solve for all of the Vis. Substitu ting equations (15) and (16) into equation (4) produces: C1 + 4FOV + C2 + V3 + V4 = 4CA60 V3 = 4CA60 C1 C2 4FOV V4 (17) For our example: V3 = 4/hr (575 veh) (600 veh/hr) (600 ve h/hr) 4/hr (12 veh) 350 veh/hr = 702 veh/hr So, for our example, the cumulative arrival curve that produces the minimum reasonable delay has quartile hour ly flow rates of: 648.0 vph, 600.0 vph, 702.0 vph, and 350.0 vph. This lower bound curve is depicted in Figure 5-5. When calculating the upper bound arrival curves the minimum PHF is always maintained; it represents a constraint on the so lution space that is always in effect. However, this is not so with the lower bound. Under lower bound conditions the PHF may or may not pose a constraint. Substituting equations (15) and (16) into equation (5), and recognizing that V3 is the highest 15minute volume in this situation, the following is produced: PHF = (V1+V2+V3+V4)/[(4)Max(V1,V2,V3,V4)] (5) PHF = (V1+V2+V3+V4)/4V3 PHF = (C1 + 4FOV + C2 + V3 + V4) / 4V3 PHF = (C1 + C2 + 4FOV + V3 + V4) / 4V3 (5B) Substituting equation (17) into equation (5B) produces: PHF = (C1 + C2 + 4FOV+ 4CA60 C1 C2 FOV V4 +V4) / 4 (4CA60 C1 C2 4FOV V4) PHF = 4CA60 / 4(4CA60 C1 C2 4FOV V4) PHF = CA60 / (4CA60 C1 C2 4FOV V4) (18)

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141 Continuing our example: PHF = (575 veh/hr)/[(4(575 veh/hr) (600 veh/hr) (600 veh/hr) 4/hr(12 veh) 350 veh/hr] PHF = 575 veh/hr / 702 veh/hr = 0.819 The actual peak hour factor is considerably larger than the minimum required value of 0.75. In this example, it was feasible for both V1 and V2 to meet their absolute minimum lower bounds. However, it is possible that V1 may be able to meet its absolute minimum lower bound while V2 cannot, or even that both V1 and V2 cannot meet their absolute minimum lower bounds. This restriction occurs when the value of V4 is too low to allow V1 and/or V2 to meet their absolute minimum lower bounds without either violating the minimum PHF requirement, the conservation of flow equation, or causing the nominal queue length to shrink to a value that is less than the FOV (thus eliminating the need for delay estimation). If V1 and V2 are at their absolute minimum lower bound, then the maximum value for V4 can be calculated by setting V3 equal to its lowest possi ble bound which, as with V2, is parallel to the cumulative departure curve: V3 = C3 (19) Substituting equation (19) into equation (17) yields: C3 = 4CA60 C1 C2 4FOV V4 V4 = 4CA60 C1 C2 C3 4FOV (20) Or, for our example: V4 = 4/hr (575 veh) (600 veh/hr) (600 ve h/hr) (600 veh/hr) 4/hr (12 veh) V4 = 2300 veh/hr 1800 veh/hr 48 veh/hr V4 = 452 veh/hr

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142 The result is graphically depicted in Figure 5-6. This arrival cu rve produces the overall minimum delay and has quartil e hourly flow rates of: 648.0 vph, 600.0 vph, 600.0 vph, and 452.0 vph. Once again, the PHF does not impose a constraint in this situation. Under conditions of overall minimum delay, V1 is always the highest 15minute volume, therefore: PHF = (V1+V2+V3+V4)/[(4)Max(V1,V2,V3,V4)] = (V1+V2+V3+V4)/4V1 (5) PHF = (C1 + 4FOV + C2 + C3 + V4) / 4(C1 + 4FOV) PHF = (C1 + C2 + C3 + 4FOV + V4) / 4(C1 + 4FOV) (21) Continuing our example: PHF=[600veh/hr+600veh/hr+600ve h/hr+4/hr(12 veh)+452veh/hr]/ 4(600 veh/hr+4/hr(12 veh)) PHF = 2300 veh/hr / 2592 veh/hr = 0.887 The actual peak hour factor is once again cons iderably larger than the minimum required value of 0.75 If V1 and V2 are at their absolute minimum lower bound, then the minimum value for V4 can be calculated by setting V3 equal to its highest possible value while maintaining the minimum required peak hour factor and preservi ng conservation of flow. Substituting equations (15) and (16) into equation (4): (C1 + 4FOV + C2 + V3 + V4)/4 = CA60 C1 + C2 + 4FOV + V3 + V4 = 4CA60 V4 = 4(CA60 FOV) C1 C2 V3 (22) Equation (22) is merely a rearrangement of equation (17). Substituting equations (15) and (16) into equation (5) and recognizing that V3 has the highest arrival volume for this situation: PHF = (C1 + 4FOV + C2 + V3 + V4) / 4V3 4PHFV3 = C1 + C2 + 4FOV + V3 + V4 4PHFV3 V3 = C1 + C2 + 4FOV + V4 (23)

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143 Substituting equation (22) into equation (23) yields: 4PHFV3 V3 = C1 + C2 + 4FOV + 4CA60 C1 C2 4FOV V3 4PHFV3 = 4CA60 V3 = CA60 /PHF (24) Now substituting equation (24) back into equatio n (22) gives: V4 = 4CA60 C1 C2 4FOV CA60 /PHF V4 = (4 1/PHF)CA60 C1 C2 4FOV (25) Using the example values we obtain: V3 = 575/0.75 = 766.7 veh/hr and V4 = (4-1/0.75)(575) 600 600 4(12) = 1533.3 1200 48 V4 = 285.3 veh/hr So, V4=285.3 vph is the lowest possible V4 value that will allow both V1 and V2 to meet their absolute minimum lower bounds (see Figure 5-7). We have now examined the case where V1, V2 and V3 are all at their minimum values, and we have examined the case where V1 and V2 are at their minimum values but V3 is not. The next arrangement of interest is when only V1 is at its minimum value. Substituting equation (15) into equation (4) yields: (C1 + 4FOV + V2 + V3 + V4)/4 = CA60 Solving for V2: V2 = 4CA60 C1 4FOV V4 V3 (26) For this situation, minimum delay is obtained when V3 is maximized, subject to the peak hour constraint. Therefore: PHF = (V1+V2+V3+V4) / [4Max(V1,V2,V3,V4)] = (V1+V2+V3+V4)/4V3 (5)

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144 Substituting equations (15) and (26) into equation (5) yields: PHF = (C1 + 4FOV + 4CA60 C1 4FOV V4 V3 +V3 + V4)/4V3 PHF = (4CA60)/4V3 Solving for V3: V3 = (CA60)/PHF (27) Now substituting equation (27) back into equatio n (26) gives: V2 = 4CA60 C1 4FOV V4 (CA60)/PHF V2 = (4 1/PHF)CA60 C1 4FOV V4 (28) We recognize that the highe st possible value for V4 will occur when V2 is as low as possible, which occurs when: V2 = C2 (16) Substituting equation (16) into equation (28) produces: C2 = (4 1/PHF)CA60 C1 4FOV V4 Solving for V4: V4 = (4 1/PHF)CA60 C1 C2 4FOV (25) This formula is consistent with the results ob tained previously. We also recognize that the lowest possible V4 will occur when V2 is as high as possible, which is when V2 = V3: V2 = V3 (29) Substituting equations (27) and (29) into equation (28) produces: (CA60)/PHF = (4 1/PHF)CA60 C1 4FOV V4 Solving for V4: V4 = (4 1/PHF)CA60 C1 4FOV (CA60)/PHF V4 = 4CA60/hr (CA60)/PHF C1 4FOV (CA60)/PHF

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145 V4 = 4CA60/hr 2(CA60)/PHF C1 4FOV V4 = 2CA60(2 1/PHF) C1 4FOV (30) Using our example values we obtain: V4 = 2(575 veh/hr)(2-1/0.75) 600 veh/hr 4/hr(12 veh) = 118.7 veh/hr So, V4 = 118.7 vph is the lowest possible V4 value that will allow V1 to meet its absolute minimum lower bound (see Figure 5-8). If V4 falls below the value give n in equation (30) then V1 (along with V2 and V3) will no longer be at its minimum va lue. For this situation, minimum delay is obtained when V3 is maximized, subject to the peak hour constraint, and when V2 = V3. Therefore: PHF = (V1+V2+V3+V4)/[(4)Max(V1,V2,V3,V4)] = (V1+V2+V3+V4)/4V3 (5) Substituting equation (29) into equation (5) yields: PHF = (V1 + V3 + V3 + V4)/4V3 PHF = (V1 + 2V3 + V4)/4V3 4V3PHF = V1 + 2V3 + V4 4V3PHF 2V3 = V1 + V4 V1 = 4V3PHF 2V3 V4 (31) Substituting equations (29) and (31) into equation (4) produces: (4V3PHF 2V3 V4 + V3 + V3 + V4)/4 = CA60 4V3PHF/4 = CA60 V3 = CA60/PHF (27) Substituting equation (27) into equation (29) yields: V2 = CA60/PHF (32)

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146 The value for V1 can be determined by substituting e quations (27) and (32) into equation (4), which produces: (V1 + CA60/PHF + CA60/PHF + V4)/4 = CA60 V1 + 2CA60/PHF + V4 = 4CA60 V1 = 4CA60 2CA60/PHF V4 V1 = 2CA60(2 1/PHF) V4 (33) Analysis of Bounds Summary The results of the analys is of the bounds can be summarized as follows: UPPER BOUND V1 = CA60/PHF (8) If V4 < CA60 (2 1/PHF) (11) Then: V2 = CA60/PHF (9) V3 = 2CA60 (2 1/PHF) V4 (10) If V4 > CA60 (2 1/PHF) (11) Then: V2 = CA60 (4 1/PHF) 2V4 (14) V3 = V4 (12) LOWER BOUND If V4 = 4CA60 C1 C2 C3 4FOV (20) Then: V1 = C1 + 4FOV (15) V2 = C2 (16) V3 = C3 (19) PHF = CA60 / (4CA60 C1 C2 4FOV V4) (18) If V4 < 4CA60 C1 C2 C3 4FOV (20) And V4 >= (4 1/PHF)CA60 C1 C2 4FOV (25) Then: V1 = C1 + 4FOV (15) V2 = C2 (16) V3 = 4CA60 C1 C2 4FOV V4 (17) PHF = (C1+ C2 + C3 + 4FOV + V4) / 4(C1 + 4FOV) (21)

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147 If V4 < (4 1/PHF)CA60 C1 C2 4FOV (25) And V4 >= 2CA60(21/PHF) C1 4FOV (30) Then: V1 = C1 + 4FOV (15) V2 = (4 1/PHF)CA60 C1 4FOV V4 (28) V3 = CA60/PHF (27) If V4 < 2CA60(21/PHF) C1 4FOV (30) Then: V1 = 2CA60(2 1/PHF) V4 (33) V2 = CA60/PHF (32) V3 = CA60/PHF (27) For our example, the values are: UPPER BOUND V1 = 575 vph/0.75 = 766.7 vph Is V4 = 350 vph < 575 vph (2 1/0.75) = 383.3 vph ? YES Then: V2 = 575 vph/0.75 = 766.7 vph V3 = 2(575 vph)(2 1/0.75) 350 vph = 416.7 vph LOWER BOUND Is V4 = 350 vph > 4CA60 C1 C2 C3 4FOV? Is V4 = 350 vph > 4(575 vph) 3(600 vph) 4/hr (12 veh) = 2300 1800 48 = 452? NO Is V4 = 350 vph > (4 1/PHF)CA60 C1 C24FOV? Is V4 = 350 vph > (4 1/0.75)575 vph 2(600 vph) 4/hr(12 veh)? Is V4 = 350 vph > 1533.3 vph 1200 vph 48 vph = 285.3 vph? YES Then: V1 = C1 + 4FOV = 600 vph + 4/hr(12 veh) = 600 vph + 48 vph = 648 vph V2 = C2 = 600 vph V3 = 4CA60 C1 C2 4FOV V4 V3 = 4(575 vph) 2(600 vph) 4/hr(12 veh) 350 vph V3 = 2300 vph 1200 vph 48 vph 350 vph = 702 vph Is V4 = 350 vph < 2CA60(21/PHF) C1 4FOV? Is V4 = 350 vph < 2(575 vph)(21/0.75) 600 vph 4/hr(12 veh)?

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148 Is V4 = 350 vph < 766.67 vph 600 vph 48 vph = 118.7 vph? NO Derivation of Delay for Upper and Lo wer Bounds Figure 5-9 shows the first two periods of the upper bound curve for our example. The Overflow Delay for period 1 (OD1) is simply the area between th e arrival and departure curves within period 1. On the other hand, the Deterministic Queue Delay for period 1 (DQD1) is composed of two elements: the in-period delay for period 1 (Dp1) and the out-of-period delay for period 1 (Dc1). Both of these elements of the peri od 1 Deterministic Queue Delay are associated with vehicles that arrive at the back of th e queue during period 1, how ever, only the in-period delay actually occurs during period 1, th e out-of-period delay occurs during period 2. For period 1, the in-period DQD equals the Ov erflow Delay, and can be calculated using the following formulas: CA15 = (V1/3600 sec/hr)( T15 T0) (34) UCD15 = (C1/3600 sec/hr)( T15 T0) (35) OD1 = Dp1 = 0.5(T15 T0)(CA15 UCD15) (36) Substituting equations (34) and (35) into equation (36) yields: OD1 = Dp1 = 0.5(T15 T0)[(V1/3600 sec/hr)(T15 T0) (C1/3600 sec/hr)( T15 T0)] OD1 = 0.5(T15 T0)(T15 T0)(V1 C1)/3600 sec/hr OD1 = Dp1 = (T15 T0)2(V1 C1)/7200 sec/hr (37) Where: CA15 = Cumulative Arrivals at time point 15 (veh) UCD15 = Uniform Cumulative Departures at time point 15 (veh) OD1 = Overflow Delay during period 1 (sec) C1 = Capacity during period 1 (veh/sec) V1 = Arrival Rate during period 1 (veh/hr) T0 = Time Point at Beginning of 15 minutes (sec) = 0 sec T15 = Time Point at End of First 15 minutes (sec) = 900 sec

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149 For our example: CA15 = (766.7 veh/hr/3600 sec/hr)(900 sec) = 191.7 veh C1 = 600.0 veh OD1 = Dp1 = (900 sec)2(766.7 veh/hr 600 veh/hr)/7200 sec/hr OD1 = Dp1 = 18,750 veh-sec The out-of-period portion of the DQD for peri od 1, which actually occurs in period 2, is calculated using the following formulas. Accumulating departures: UCDC1 = UCD15 + (C2/3600 sec/hr)(TC1 T15) (38) A critical time point occurs when the last arrivi ng vehicle during period 1 departs. This occurs when: UCDC1 = CA15 (39) Where: UCDCi = Uniform Cumulative Departures at time point Ci (veh) Ci = Capacity during period i (veh/sec) TCi = Critical Time Point (TC1 is the critical time point at which the number of Uniform Cumulative Departures = CA15) CDCi = Cumulative Departures at Critical Time Point TCi (sec) Substituting equation (39) into e quation (38) and solving for TC1 yields: CA15 = UCD15 + (C2/3600 sec/hr)(TC1) (C2/3600 sec/hr)(T15) (CA15 UCD15) + (C2/3600 sec/hr)(T15) = (C2/3600 sec/hr)(TC1) TC1 = 3600 sec/hr (CA15UCD15)/C2 + T15 (40) For period 1, the out-of-period DQD can be ca lculated using the following formula: Dc1 = 0.5(TC1 T15)(CA15 UCD15) (41) For our example: TC1 = 3600 sec/hr (191.7 veh 150.0 ve h)/600 veh/hr + 900 sec = 1150 sec

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150 And: Dc1 = 0.5(1150 sec 900 sec)(191.7 veh 150 veh) = Dc1 = 5208 veh-sec Figure 5-10 shows the second and third periods of the upper bound curve for our example. The Overflow Delay for period 2 (OD2) is still simply the area between the arrival and departure curves within period 2. On the other hand, th e Deterministic Queue Delay for period 2 (DQD2) is now composed of four elements: the in -period over-saturation delay for period 2 (Dp2), the out-of-period over-saturati on delay for period 2 (Dc2), the in-period initia l queue delay for period 2 (DIQA2) and the out-of-period initia l queue delay for period 2 (DIQB2). All four components of the period 2 Deterministic Queue Delay are associated with vehicles that arrive at the back of the queue during period 2, however, only the in-per iod delay and in-peri od initial queue delay actually occur during period 2, the out-of-peri od delay and out-of-period initial queue delay occur during period 3. The in-period DQD for Period 2 can be calculated using the following formulas: Accumulating arrivals: CA30 = (V2/3600 sec/hr)(T30 T15) + CA15 (42) Accumulating departures: UCD30 = (C2/3600 sec/hr)(T30 T15) + UCD15 (43) By inspection we see that the bottom boundary of the area for Dp2 begins at point C15 and is parallel to the departure curve. Defining UCD30A as the cumulative number of vehicles obtained when this parallel boundary line reaches T30 (1800 sec), we have: UCD30A = (C2/3600 sec/hr)(T30 T15) + CA15 (44) The in-period over-saturation de lay is then calculated as:

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151 Dp2 = 0.5(T30 T15)(CA30 UCD30A) (45) For our example: CA30 = (766.7 veh/hr/3600 sec/hr)(1800 sec 900 sec) + 191.7 veh = 383.3 veh UCD30 = (600 veh/hr/3600 sec/hr)(1800 sec 900 sec) + 150 veh = 300.0 veh UCD30A = (600 veh/hr/3600 sec/hr)(1800 sec 900 sec) + 191.7 veh = 341.7 veh Dp2 = 0.5(1800 sec 900 sec)(383.3 veh 341.7 veh) Dp2 = (450 sec)(41.6 veh) Dp2 = 18,750 veh-sec The out-of-period over-saturation delay for peri od 2, which actually occurs in period 3, is calculated using the following formulas. Accumulating departures: UCDC2A = UCD30A + (C3/3600 sec/hr)(TC2AT30) (46) A critical time point occurs when the last arriving vehicle during period 2 would have departed had there not been an initial queue at the beginning of time period 2: UCDC2A = CA30 (47) Substituting equation (47) into e quation (46) and solving for TC2A yields: CA30 = UCD30A + (C3/3600 sec/hr)(TC2A) (C3/3600 sec/hr)(T30) (CA30 UCD30A) + (C3/3600 sec/hr)(T30) = (C3/3600 sec/hr)(TC2A) TC2A = (3600 sec/hr)(CA30 UCD30A)/C3 + T30 (48) For period 2, the out-of-period ov er-saturation delay can be ca lculated using the following formula: Dc2 = 0.5(TC2A T30)(CA30 UCD30A) (49)

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152 For Figure 5-10 to be an accurate representati on of the delay situation, the nominal queue length at T30 must be greater than th e nominal queue length at T15. If it is less, then both DP2 and DC2 are equal to zero. The nominal queue length at T30 is calculated as: Q30 = CA30 UCD30 (50) And the nominal queue length at T15 is: Q15 = CA15 UCD15 (51) Consequently: If Q30 > Q15 then equations (45) a nd (49) hold, otherwise Dp2 = Dc2 = 0 For our example: Q30 = CA30 UCD30 = 383.3 veh 300 veh = 83.3 veh which is greater than: Q15 = CA15 UCD15 = 191.7 veh 150 veh = 41.7 veh Therefore, equations (45) and (49) hold: Dp2 = 0.5(T30 T15)(CA30 UCD30A) = 0.5(1800 sec 900 sec)( 383.3 veh 341.7 veh) Dp2 = 18,750 veh-sec TC2A = (3600 sec/hr)(CA30 UCD30A)/C3 + T30 TC2A = (3600 sec/hr.)(383.3 veh 341.7 veh)/600 veh/hr + 1800 sec TC2A = 2050 sec Dc2 = 0.5(TC2A T30)(CA30 UCD30A) = 0.5(2050 sec 1800 sec)( 383.3 veh 341.7 veh) Dc2 = 5208 veh-sec An inspection of Figure 5-10 reveals that the in -period initial queue delay for period 2 is represented by a trapezoid and a triangl e. The trapezoid has a base of TC1 T15 and a height of UCD30 CA15. The triangle also has a base of TC1 T15 but its height is UCD30A-UCD30. Consequently:

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153 DIQA2 = (TC1 T15)(UCD30 CA15) + 0.5(TC1 T15)(UCD30A UCD30) DIQA2 = (TC1 T15)[(UCD30 CA15) + 0.5(UCD30A UCD30)] (52) The total out-of-period delay fo r period 2, which actually occurs in period 3, is calculated using the following formulas. Accumulating departures: UCDC2 = UCD30 + (C3/3600 sec/hr)(TC2 T30) (53) Another critical time point occu rs when the last vehicle ar riving during period 2 departs: UCDC2 = CA30 (54) Substituting equation (54) into e quation (53) and solving for TC2 yields: CA30 = UCD30 + (C3/3600 sec/hr)(TC2) (C3/3600 sec/hr)(T30) (CA30 UCD30) + (C3/3600 sec/hr)(T30) = (C3/3600 sec/hr)(TC2) TC2 = (3600 sec/hr)(CA30 UCD30)/C3 + T30 (55) For period 2, the total out-of-period delay can be calculated using the following formula: DT2 = 0.5(TC2 T30)(CA30 UCD30) (56) The out-of-period initial queue delay for period 2 is then calculated by simply subtracting the out-of-period over-saturation delay from the total out-of-period delay: DIQB2 = DT2 DC2 DIQB2 = 0.5(TC2 T30)(CA30 UCD30) DC2 (57) For Figure 5-10 to be an accurate representation of the delay situation such that equations (52) and (57) apply, the nominal queue length at T30 must be greater than the nominal queue length at T15. If it is less, then both DIQA2 and DIQB2 are calculated using different equations, as we shall soon see for period 3. For our example the nominal queue length at T30 was previously shown to be greater than th e nominal queue length at T15. Therefore: DIQA2 = (TC1 T15)[(UCD30 CA15) + 0.5(UCD30A UCD30)]

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154 DIQA2 = (1150 sec 900 sec)[(300 veh 191.7 veh) + 0.5(341.7 veh 300 veh)] DIQA2 = (1150 sec 900 sec)[(300 veh 191.7 veh) + 0.5(341.7 veh 300 veh)] DIQA2 = 32,292 vehsec TC2 = (3600 sec/hr)(CA30 UCD30)/C3 + T30 TC2 = (3600 sec/hr)(383.3 veh 300 veh)/600 veh/hr + 1800 sec TC2 = 2300 sec DIQB2 = 0.5(TC2 T30)(CA30 UCD30) DC2 DIQB2 = 0.5(2300 sec 1800 sec)(383.3 veh 300 veh) 5208 veh-sec DIQB2 = 15,625 vehsec Figure 5-11 shows the third and fourth periods of the upper bound curve for our example. The Overflow Delay for period 3 (OD2) is still simply the area between the arrival and departure curves within period 3. On the other hand, since the queue at the end of the period is smaller than the queue at the beginning of the period the Deterministic Queue Delay for period 3 (DQD3) is now composed of the following two eleme