REALTIME 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
RealTime 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 NONVISIBLE DELAY .................................... ................... ............... 68
Data Analysis Programs .............................. ....... .. .. ............. ........ 68
Prediction Algorithm for N onVisible Delay ........................................ ...... ............... 80
NonVisible Queue Estimation Technique........... ............................... ...............80
NonVisible Queue Adjustment Technique: ................................ ...............82
NonVisible Queue ReAdjustment 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 TrajectoryBased 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 OVERSATURATED FLOW.................. ...... .............276
Simplified Example of CyclePeriod Issues in Calculating d3 ..........................................276
R esidual Q ueue D iscrepancy ............... ........ ................ .............................................281
Detailed Example of CyclePeriod 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: RealTime 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 CYCLEPERIOD DELAY EXAMPLE: ...........................................353
R E F E R E N C E S ..................................................................................................................3 7 6
BIOGRAPHICAL SKETCH ...............................................................381
LIST OF TABLES
Table page
41 Example summary volume and capacity ......................... ...... ...............122
42 Example summary queue discharge, delay check and goodnessoffit......................123
43 Queue prediction........ .......... ............ .. ... .... ...... ......... 124
44 Stopped delay prediction .......................................................... ..................................125
45 C control delay prediction ......................................................................... ................... 126
46 Comparison of variation in actual and predicted stopped delay ........... .....................127
47 Pvalue determination for difference in median values.............................128
61 Calculation of cumulative curve delay conversion factors, volume pattern
625_700_650_350vph.............................................................................. ...............249
62 Calculation of cumulative curve delay conversion factors, volume pattern
700_725_625_350vph ............. .............................. ...... ............................... 251
63 Calculation of cumulative curve delay conversion factors, volume pattern
700_700_700_350vph .............. .............................. ..... .... ..................... 253
64 Calculation of cumulative curve delay conversion factors, volume pattern
725_700_700_350vph ............. .............................. ...... ............................... 255
65 Cumulative curve delay for standard 4period case............................................. 257
66 Cumulative curve delay with multiple visible periods ......................................... 258
67 Stopped delay prediction results for 700_725_625_350vph volume pattern ..................259
68 Average stopped delay prediction results for 700_725_625_350vph volume pattern ....262
69 Stopped delay prediction results for 700_700_700_350vph volume pattern ..................263
610 Average stopped delay prediction results for 700_700_700_350vph volume pattern ....266
611 Stopped delay prediction results for 725_700_700_350vph volume pattern ..................267
612 Average stopped delay prediction results for 725_700_700_350vph volume pattern ....270
613 Stopped delay prediction results for 625_700_650_350vph volume pattern ..................271
614 Average stopped delay prediction results for 625_700_650_350vph volume pattern ....274
615 P reduction com prison .......................................................................... .....................275
71 Generalized example of cycleperiod delay discrepancies data ....................................292
72 Generalized example of cycleperiod delay discrepancies summary .........................293
73 Detailed example of cycleperiod delay discrepancies, residual queue determination ...296
74 Detailed example of cycleperiod delay discrepancies, delay comparison....................297
75 Detailed example of cycleperiod delay discrepancies, delay comparison with
m modified d2 term ............. ......... .. .... ............ ......................... 297
76 Detailed example of cycleperiod delay discrepancies, delay comparison with d3
ad ju stm en t ...................................... ................................................... 2 9 7
Bl US 1 machine counts (Southern St. Johns County)...................................................334
B2 US1 M machine counts (northern St. Johns County).........................................................339
B3 Atlantic Boulevard m machine counts...........................................................................342
B4 University Boulevard machine counts (Jacksonville)............................345
B5 SR A1A machine counts (Crescent Beach) ........................................ ............... 348
B6 SR A1A machine counts (Ponte Vedra) PDF 17 KB ............................................... 351
B 7 A ppendix B data sum m ary...................................................................... ...................352
C1 Generalized example of cycleperiod delay discrepancies data. ................................354
LIST OF FIGURES
Figure page
4 1 Q u eu e relation ship s........ ........................................................................ ...... .............94
42 Signalized intersection delay components .................................................. .............. 95
43 M measured versus estim ated delay............................................................ .....................96
44 V isible and nonvisible variables............................................................ .....................97
45 Relationship between v/c ratio and ratio of control delay to stopped delay ....................98
46 Requeuing that results in simultaneous queues ..................................... .................99
47 Requeuing that does not result in simultaneous queues ..............................................100
48 E xam ple of a blind period.. ..................................................... ........................................ 101
49 Exam ple of adjacent blind periods........................................................... ............... 102
410 C ounters and queue statu s........................................................................ .................. 103
411 Base case for P, C and X; stopped delay comparison......................................................104
412 Effect of increasing the power constant on stopped delay comparison........................... 105
413 Q ueue propagation exam ple ................................................. ............................... 106
414 A actual vehicle queues ............................................................................ ....................107
415 A average queue length com parison....................................................................... ...... 108
416 M axim um queue length com parison.......................................... .......................... 109
417 98th percentile back of queue comparison................................................110
418 V vehicle requeuing ......... .. .................................. .. .. .. ............................ ... 111
419 Stopped delay com prison ......... ................................ ...................................... 112
420 Stopped delay prediction, 12 FOV ...... ............................................................ 113
421 Comparison of actual and predicted stopped delay ............. ..................................... 114
422 Adjacent blind period counter v. stopped delay..........................................................115
423 Control delay com prison .............................................................. ............. 116
10
424 Ratio of control delay to stopped delay .................................. .......................... ......... 117
425 Graphical control delay comparison, ......................................................... ...... ......... 118
426 C control delay estim ates.......................................................................... ......... ........... 119
427 C control delay com position ...................................................................... ...................120
428 Ratio of control delay to stopped plus moveup delay ...............................................121
51 Cumulative arrivaldeparture curves and overflow delay.................................... 207
52 Critical time and volume points for period 4..........................................................208
53 O overflow delay in period 4 ........................................... .................. ............... 209
54 M aximum reasonable cumulative arrival curve................................... ..................210
55 M minimum reasonable cum ulative arrival curve ..............................................................211
56 Minimum overall reasonable cumulative arrival curve ....................................... 212
57 Minimum reasonable cumulative arrival curve (minimum V4 for minimum Vi and
V 2) ........................................................................... ..................... 2 13
58 Minimum reasonable cumulative arrival curve (minimum V4 for minimum Vi) ..........214
59 Period 1 delay for the upper bound............................. ......................... ............... 215
510 Period 2 delay for the upper bound............................. ......................... ............... 216
511 Period 3 and period 4 delay for the upper bound..................... .................................217
512 Reasonable overflow delay region for 600 vph capacity and 0.75 minimum PHF .........218
513 Reasonable overflow delay region for 600 vph capacity and 0.80 minimum PHF .........219
514 Reasonable overflow delay region for 600 vph capacity and 0.85 minimum PHF .........220
515 Maximum delay estimation error for 0.75 minimum PHF ..............................................221
516 Maximum delay estimation error for 0.80 minimum PHF ...........................................222
517 Maximum delay estimation error for 0.85 minimum PHF ...........................................223
518 Maximum reasonable cumulative arrival curve with period 1 visible...........................224
519 Minimum reasonable cumulative arrival curve with period 1 visible ...........................225
520 Maximum reasonable cumulative arrival curve with 5 periods ............... .................226
521 Minimum reasonable cumulative arrival curve with 5 periods ......................................227
61 Trajectory example A) Complete chart B) Detailed view of circled area in upper
rig h t c o rn e r ...................................... ................................................... 2 4 4
62 Cumulative arrivaldeparture curve example........................................ ............... 246
63 Trajectory conversion of cumulative curve example........... .............................. 247
64 D elay and travel tim e com ponents........................................................ ............... 248
71 Cycle v. period initial queue delay analysis................................ ....................... 294
72 Cycle v. period "control delay" analysis..................................... ......................... 295
73 Upward bias in HCM residual queue calculation ................................. ..................... 298
Ai Queue discharge headw ay histogram ........................................ ........................... 309
A 2 Startup lost tim e histogram ....... ......... ......... ............................... ............... 310
A3 Comparison of control delay and stopped delay by cycle length (g/C =0.30)...............311
A4 Comparison of control delay and stopped delay (g/C =0.30) ................. ............... 312
A5 Comparison of control delay and stopped plus queue moveup delay by cycle length
(g /C = 0 .3 0 ) ....................................................................... .. 3 1 3
A6 Comparison of control delay and stopped delay plus queue moveup delay (g/C
= 0.3 0) ................... ................ ...........................................3 14
A7 Relationship between v/c ratio and stopped delay..................... ........ ............. ......... 315
A8 Relationship between v/c ratio and stopped delay by cycle length ..............................316
A9 Relationship between v/c ratio and stopped plus queue moveup delay .......................317
A10 Relationship between v/c ratio and stopped plus queue moveup delay by cycle
len g th ................ ......... .................................. ............................... 18
A11 Relationship between v/c ratio and control delay .............................. ...............319
A12 Relationship between v/c ratio and control delay by cycle length................................320
A13 Relationship between vehicle requeues and control delay ..........................................321
A14 Relationship between v/c ratio and vehicle requeues...............................................3.22
A15 Relationship between v/c ratio and vehicle requeues by cycle length .........................323
A16 Relationship between v/c ratio and cycles with phase failure ......................................324
A17 Relationship between v/c ratio and cycles with phase failure by cycle length..............325
A18 Percentage of cycles in 1 hour with phase failure by cycle length...............................326
A19 Percentage of cycles in 1 hour with phase failure................................. ............... 327
A20 Linear relationship between ABPC and stopped delay.................. ............................ 328
A21 Exponential relationship between ABPC and stopped delay........................................329
A22 Relationship between ABPC and control delay................................... ............... 330
Bl US 1 S. PM peak hour factor, southbound (outbound) flow .............. .............. 332
B2 US 1 S. PM peak period factor, southbound (outbound) flow........................................ 333
B3 US 1 N. PM peak hour factor, northbound (outbound) flow..................................... .....337
B4 US 1 N. PM peak period factor, northbound (outbound) flow........................................338
B5 Atlantic Boulevard PM peak hour factor, eastbound (outbound) flow............................ 340
B6 Atlantic Boulevard PM peak period factor, eastbound (outbound) flow......................... 341
B7 University Blvd. PM peak hour factor, northbound (outbound) flow ............................. 343
B8 University Blvd. PM peak period factor, northbound (outbound) flow .......................... 344
B9 SR A1A S. PM peak hour factor, southbound (outbound) flow..............................346
B10 SR A1A S. PM peak period factor, southbound (outbound) flow..............................347
B11 SR A1A N. PM peak hour factor, southbound (outbound) flow ..............................349
B12 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
REALTIME 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 oversaturated
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 realtime procedure for estimating delay during oversaturated
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 secondbysecond 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 oversaturated periods preceded by an initial queue. A
remedial modification to the d2 term is proposed.
Finally, it is demonstrated that the HCM's periodbased queue accumulation procedure
has drawbacks that can produce substantial errors in delay during oversaturated conditions. A
remedial cyclebased 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 realtime 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 nontechnical people." As recognized by
Dowling [3], many MOEs (such as queue length, speed, stops, and density) are relatively
invariant during highly oversaturated 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 retiming 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 laborintensive 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 oversaturated 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 oversaturation
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 nonstationary 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 cyclebycycle 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 oversaturated
conditions. Resulting delay data could then be used for project evaluation or for realtime
modification of controller settings.
Using realtime delay obtained from intersectionbased 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 reidentification 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 intersectionbased 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 realtime manner to obtain realtime 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 workedout 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 wellrecognized analytical
procedure for calculating control delay at signalized intersections, with control delay being
defined as the sum of deceleration delay, stopped delay, queue moveup 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, userfriendly 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, startup lost time, g/C ratio,
arrival type) and that the vehicle arrival rate on the approach remains constant within each of the
four 15minute periods. In reality, conditions change on a cyclebycycle basis depending on
random fluctuations in approach volumes and driver composition. For example, the considerable
variation in cyclebycycle 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 cyclebycycle 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 sitespecific 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 oversaturated conditions. In order to
properly measure delay during oversaturated conditions, multiperiod 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, moveup 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 realtime with limited information, and being able to do so even with
oversaturated 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 costprohibitive 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, realtime
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 realtime procedure that can reasonably
estimate delay associated with vehicles that are beyond the reach of the detection system. The
procedure should function during both undersaturated and oversaturated, 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 (nonvisible 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 realworld
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 oversaturated 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
microsimulation model. CORSIM allows us to quickly simulate a variety of realworld
conditions in a relatively realistic manner and to accumulate important measures of
effectiveness, including delay. CORSIM was used because it is a wellaccepted and well
understood model that has the capability to accommodate a wide range of input variables,
including variable combinations that produce grossly oversaturated 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 microsimulation models can result in a level of variation that
masks causeandeffect 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 backofqueue 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 "nonvisible delay" that
occurred. The procedure developed is capable of handling both undersaturated conditions
(having little or no "blindness") and oversaturated 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 nonvisible 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 secondbysecond 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 realtime 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 oversaturated 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 multiperiod 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 periodbased 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 cyclebased
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 microsimulation 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 realtime
secondbysecond delay estimation procedures for a onehour 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 15minute period for
a onehour 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 nonvisible queue as well as
departures from the front of the nonvisible queue at each point in time, information that would
otherwise be unknown. Knowing the arrivals and departures we can predict the length of the
nonvisible queue at each point in time. This predicted nonvisible 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 oversaturated intersection approach for
a onehour analysis time frame would proceed as follows:
1. Using the vehicle detection equipment for the approach of interest, realtime second
bysecond 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 nonvisible queue at every second
of the onehour analysis time frame. Secondbysecond 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 realtime traffic control.
4. The time during the last 15minute 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 onehour 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 15minute 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 15minute
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 15minute 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 carriedout 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 oversaturated 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 realtime
traffic control and project evaluation. It is envisioned that the eventual end product of this
theoretical research will be a selfcontained delay estimation module that could be attached to
either a closedloop 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 undersaturated conditions and for situations where all
of the information needed to calculate delay is known. The current state of knowledge with
respect to oversaturated conditions is more primitive and the results less tested.
RealTime 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 rightofway (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 TimeinQueue Delay (a.k.a. TimeinQueue) 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 speedbased 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). Freeflow 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 GPSequipped 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 endacceleration distance of 427 feet downstream of the stop bar was
established. An average begindeceleration 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 30degree 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 midblock delay from
segment delay. The authors cautioned that queue spillback from a downstream intersection or
nonrecurring 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 reidentification problem on freeways. A
vehicle waveform pair can be formed by using one downstream waveform and one upstream
waveform. The vehicle reidentification 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 reidentification 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 reidentify
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 reidentification 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 multiobjective optimization problem where each objective
is ordered according to its importance. MultiObjective 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 multiobjective 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 reidentification. 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 loopbased 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 180 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 reidentification 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 online signal control algorithm to find the
optimal green splits. Vehicle reidentification 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 2hour
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 delaybased online control algorithms performed better than the offline case for both
pretimed 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 reidentification 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 4lane 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 reidentification is license plate matching. Other potential
methods of vehicle reidentification 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 turnon 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 reidentification 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 reidentification rate of over 90% was obtained using multidetector 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 reidentification
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 tiein would improve the
accuracy and possibly yield realtime 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 reidentification 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,
cyclelength based average (CBA) and fixed time average (FTA). A fixed interval of 60 seconds
was used for FTA. Kmeans 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 withingroup variance
B = between group variance
A lower Wilk's lambda value indicates better clustering. Kmeans 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(ARDiAADiAADAi) 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 singleloop detectors while all of the tests can
be applied to dualloop 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 (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 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 fieldsofview
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 worstcase 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 userdefined 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:
nonoptimum camera placement, daytonight 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 redgreen 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
overcounted 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.)
overcounted 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=0T=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=01L1)
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=IT1) x P(V=0L=1) + P(L=0IT1) x [1P(V=1IL0)]
The probability of video detection being on when a vehicle is not present = P(V=1ITo) =
P(L=1IT o) x [1P(V=0L=1)] + P(L=0T0) x P(V=IL 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 volumedensity control).
A VIVDS system is sometimes used to provide advance detection on highspeed
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 = distancetoheight 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 42 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: adjacentlane, samelane and crosslane. 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. AdjacentLane Occlusion (Horizontal Occlusion) occurs when
the blocked and blocking vehicles are in adjacent lanes, which can result in false
detections in adjacent lanes. Table 41 of the paper provides minimum required camera
heights to reduce adjacentlane 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. SameLane 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. CrossLane 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 onehalf 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 imageprocessing 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 highcontrast 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 (MAHPT) lv
Iv* = (lvlro) + 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 rightturning vehicles from
the major road whose headlights sweep across the detection zone. It also eliminates false
calls due to crosslane occlusion caused by tall vehicles on the major road.
A lens adjustment module is an essential VIVDSrelated 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 RG59 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
onsite 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 175 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: selfcorrecting, 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)
InputOutput (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 steadystate 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, steadystate models are useful for predicting delays only at lightly loaded
intersections." Hurdle added: "...there is one group of models, the steadystate 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 nonconstant 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 startup 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 atgrade 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(n1)/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 NonConstant 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(ni)/Amax] 0.0086v 0.23AGI
Calibrated Minimum Discharge Headway Model:
Minimum Headway = H = T + d/Vma, 0.0086v 0.23AGI
Calibrated StartUp Lost Time Model:
StartUp 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 atgrade 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 (TRAFSIM) data were used to validate the oversaturation 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 variancetomean 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 moveup 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 = (1P)fpA/[1g/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[(X1) + {(X1)2+8kIX/Tc}1/2]
I = upstream filtering/metering factor obtained from Exhibit 157 of 2000 HCM
k = incremental delay factor obtained from Exhibit 1613 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
oversaturated conditions at signalized intersections. The delay equations calculate delays
consistent with the more accurate pathtrace method of delay measurement rather than the less
accurate (but easier to carryout) queuesampling method. Delays estimated by the proposed
generalized model were in close agreement with those simulated by TRAFNETSIM.
The pathtrace 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 queuesampling 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 pathtrace 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.
TRAFNETSIM calculates delay by subtracting the freeflow 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 oversaturated conditions.
The following TRAFNETSIM input values (representative of oversaturated 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 TRAFNETSIM) 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 60second period with very low flow; the actual analysis period of
duration T; and a final period of duration T, again with very low flow (TRAFNETSIM 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 oversaturation 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. StartUp 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 fivelane 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.
StartUp 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:
StartUp Lost Time = SULT = SRT + 4*(H4h)
Saturation Headway = h = (TNT4)/(N4)
Average Headway = Hi = (Ti Ti4)/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 StartUp 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 offpeak 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 offpeak thru's and 1.3 for offpeak 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 startup 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 carfollowing system to examine the effects of lane
changing and a heterogeneous vehicle mix on queue discharge headways.
In the Pitt carfollowing model, the first vehicle in the queue begins to move across the
stop line after the lost time (startup delay) has expired. The second vehicle in the queue then
responds to the motion of the leader through the carfollowing system with no additional explicit
lost time added. The effect of lost time on subsequent vehicles is modeled through the
sluggishness of the carfollowing 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 startup 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. StartUp 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 carfollowing 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
tofront time headway curve and the reartofront 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 startup 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 shockwave analysis and that highlighted the common errors
that are made with regard to delay estimation using shockwave analysis. The authors point out
that the main difference between shockwave 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 shockwave analysis considers the horizontal extent of the queue.
Maximum queue reach (a.k.a. back of queue) can only be identified using shockwave analysis.
The authors show that the size of the queue obtained from shockwave 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, PerezCartagena and Tarko [34] demonstrated that, based on studies conducted in
Indiana, town size and lateral lane location (rightmost 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 constantvalue 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 oversaturated as well as undersaturated 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
overcapacity 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 undersaturated, 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 workedout 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 freeflow 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 reidentification 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 realworld problem of queues that extend beyond the
detection system for some period of time; either shortlived queues that occur during under
saturated conditions because of spurts in activity or longerlived, recurring queues that occur
during oversaturated 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 oversaturated conditions, represent a deviation from the research conducted to
date.
CHAPTER 4
ESTIMATING NONVISIBLE DELAY
This chapter describes the methodology that was established to predict nonvisible 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 objectoriented, 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 oneway 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 onehour analysis time frame. The intersection was controlled by a 2phase
semiactuated traffic signal and delay data were collected and analyzed only for the actuated side
street approach. Goodnessoffit testing using the chisquare 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 secondby
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 secondbysecond 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 41. It is important to recognize that the back of queue is not itself a length,
but rather a position. As is shown in Figure 41, 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 timedistance 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 realworld
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 secondbysecond basis, estimate the nonvisible 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 zerovelocityzeroacceleration 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 moveup 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 42
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 realtime 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 realtime estimate of stopped delay". By applying an appropriate factor
(such as the commonlyused 1.3 value) or range of factors, we then scaleup the stopped delay
estimate to obtain a "reasonably accurate realtime 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 moveup time that occurs within the limits (or field of view) of the detection
system. Any stopped delay or queue moveup 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 43). 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 nonvisible 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 nonvisible queue (see Figure 44) and the period of
time during which nonvisible 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 45 summarizes the relationship between this ratio and both the v/c ratio and
cycle length for oversaturated conditions.
For each CORSIM run, a certain Field of View (FOV) was assumed. Measured visible
lockedwheel 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 lockedwheel stopped delay assuming an infinite FOV.
Finally, the predicted stopped delay was factoredup 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 presignal 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 postsignal 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 freeflow speed (with freeflow 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 freeflow 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 overcapacity 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 shockwave theory and when traffic volumes become very high in
relation to the capacity of the approach in question, vehicle requeuing 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 46. 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.
Requeuing 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. Requeuing is a necessary condition for the formation of simultaneous queues; however,
it is not a sufficient one. As shown in Figure 47, requeuing may not result in the formation of
simultaneous queues.
Unusual or atypical events can also result in phase failures and associated requeuing. 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
gapout prematurely, forcing this vehicle and all vehicles 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 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 requeue 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 requeue 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 requeues 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 fillsup 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 nonvisible queue. Figure 48 provides an
example of a blind period. During this blind period, vehicles attach themselves to the end of the
nonvisible 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 nonvisible queue,
but cannot leave the front of the nonvisible 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 T7 to time T34 in Figure 48).
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 nonvisible queue from the front while they continue to attach to the back
of the nonvisible 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 nonvisible queue and leave the front of
the nonvisible queue, as the falling queue blind period (which occurs from time T34 to time T
72 in Figure 48). The length of the nonvisible 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 T72 in Figure 48).
A review of the Figure 48 example reveals that the nonvisible queue actually shrinks to
zero well before the end of the falling queue portion of the blind period (somewhere abound time
T50). However, because of the limited FOV, we cannot be certain that the nonvisible queue
has dissipated until time T72.
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 48 example, 5 seconds), and if another queue does not fill the FOV prior to this 5second
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 nonvisible queue (i.e., the nonvisible queue has been "flushed out"). However, if this
5second 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 49). As one might expect, the
problem of estimating the length of nonvisible 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 nonvisible queue and its associated stopped delay.
The nonvisible 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 410.
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 carryingout 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 onehour analysis time frame having four
15minute periods, a secondbysecond tabulation of items such as queue length, back of queue
position, stopped delay, moveup delay and control delay. It also provides a host of ancillary
capabilities, including automated calculation of: startuplosttime, saturation flow, and capacity
by cycle; HCM queuing and delay information by 15minute period; and arrival type by 15
minute period. In addition, BuckQ allows evaluation of arrival patterns using a chisquared
goodnessoffit test and provides extensive graphing capabilities. However, the most important
feature of BuckQ is its ability to accommodate secondbysecond 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 secondbysecond basis for the
entire 3600second (60minute) 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 nearsaturated and oversaturated
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 oversaturated conditions.
Prediction Algorithm for NonVisible 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 nonvisible 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 nonvisible queue.
NonVisible 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 nonvisible 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 nonvisible 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 nonvisible 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 nonvisible queue passed some time ago or,
conversely, it may be that there are more vehicles in the nonvisible 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 fivesecond 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 nonvisible 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 5second 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 fillsup 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
nonvisible queue length estimations to be made that depend on previous nonvisible 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 nonvisible queue during the blind period. A second "adjustment technique" is needed to
augment the initial "estimation technique" when this occurs.
NonVisible 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
oversaturated 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 411 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 412. 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.
NonVisible Queue ReAdjustment Technique:
ReAdjusted 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 413 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 overcapacity
conditions at some point during the onehour 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 B29.
Tables 41 and 42 summarize the characteristics of these examples while Tables 43
through 45 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 15minute volumes
that were input into CORSIM. Considering the first row, random number set 1 was used and the
15minute input volumes were 625 vph, 700 vph, 650 vph and 350 vph. The input volume for
the last 15minute period was always set at a relatively low value so that all residual queues
would clear by the end of the onehour 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 41. 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 41 are the approach capacity values for each 15minute 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 15minute period, the
needed intermediate variables such as queue discharge Headway (H), StartUp Lost Time
(SULT), and effective green time (g). The Extension of Effective Green (EEG) is determined for
the first 45minutes of the hour by minimizing the sum of the squared deviations between the
cyclebycycle capacity values calculated using the Highway Capacity Manual procedure and
actual cyclebycycle thruput. A review of Table 41 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.
Volumetocapacity ratios are calculated for each 15minute period and for the first 45
minutes of the hour. These values are also provided in Table 41. For individual 15minute
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 45minutes, 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 42 summarizes various values used for capacity analysis,
including cycle length, green time, queue discharge headway, saturation flow rate, and startup
lost time. Our analysis procedure calculates these values on both a cyclebycycle basis and a
15minute 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, startup 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 startup lost time
value of 2.7 seconds is significantly greater than the 2.0 second mean startup delay input into
CORSIM. The difference is due to a definition inconsistency. CORSIM only applies the mean
startup delay to the first vehicle, adding additional delay (of about 0.7 seconds) to subsequent
vehicles. In other words, CORSIM's mean startup delay is not the same as startup lost time.
The next section of Table 42 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 42 summarizes, for the Poisson distribution, the chisquare
goodnessoffit test results based on 20second arrival intervals. During only one of the forty
eight 15minute 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 20second 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 43 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 414
depicts actual queue length as a function of v/c ratio while Figures 415 through 417 compare
actual and predicted queue results for the average queue length, the maximum queue length, and
the 98th percentile back of queue, respectively. Figures 414 through 417 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 415 through 417 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 417 indicates that the HCM procedures grossly
overestimate the 98th percentile back of queue.
Also provided in Table 43 is information on the number of (liberal) phase failures, the
percentage of cycles experiencing a phase failure, and the number of vehicle requeues. 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
requeue, or if 100 vehicles are forced to requeue. 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 418 demonstrates that the number of vehicle requeues tends to increase linearly as a
function of v/c ratio.
Stopped Delay Prediction
Table 44 summarizes the stopped delay prediction results for our analysis procedure as
compared to actual stopped delay. Figure 419 indicates that the procedure does a pretty good
job of predicting stopped delay over all v/c ratios.
Figure 420 shows the relative contribution of each segment of the prediction
methodology. For the examples under consideration, visible delay makesup 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 nonvisible 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 oversaturated 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 oversaturated 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 nonvisible 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 421 provides another
way of visualizing the final predictive results.
The maximum individual overestimation of delay is 27% and the maximum individual
underestimation is 17.5%. If the results are averaged over the three random number replicates,
as is documented at the bottom of 44, the maximum overestimation is 13% and the maximum
underestimation 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 422, 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 45 summarizes the control delay prediction results as compared to actual control
delay. Figure 423 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 424. (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 423 is control delay as predicted by HCM
procedures. The HCM procedures tend to overpredict control delay for the lower oversaturated
v/c ratios.
Figures 425 and 426 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
moveup delay. As shown in Figure 427, 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 moveup 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 requeuing associated with high v/c
ratios produces this steady and dramatic increase in queue moveup delay. Figure 428 provides
factors that convert "stopped delay plus queue moveup 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 B30.
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 startup 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 46 provides a comparison between the actual 1hour 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 46, 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 nonparametric 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 47
contains the test, which produces a pvalue of about 0.11 The pvalue 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 oversaturated 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 nonvisible queue. A "sleeper" is defined as
a motorist that does not exhibit normal carfollowing 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 invehicle 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 nearsaturated conditions. For oversaturated 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 microsimulation 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.
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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 43. Measured versus estimated delay
BIGGER FOV IS BETTER.
BUT ACHIEVABLE FOV DEPENDS ON
THE DETECTION SYSTEMS USED
0
J U
I = VISIBLE VARIABLES
Q= NONVISIBLE VARIABLES
QUEUE WITHIN FOV
(VISIBLE QUEUE)
12 VEHICLE
FIELD OF VIEW
(FOV)
QUEUE BEYOND THE FOV
(NONVISIBLE QUEUE)
VEHICLES LEAVING 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 44. Visible and nonvisible 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 45. 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
r4 M
CD rnrn rnrn rnJ 03
ii__ IIL.L _
v~CLE C
i rnWB r
a2 CnJu
TADCKAoW5'
S 01  =5
I sT0LLr
Ca, bJ~
Q0 =3
02=2
r7 0
I EZWLI W W W
MT CUMLE
1EIGEND
X SIGNAL ~IICATKIN
R = RED
G GRHCLEEN
SVEHKILE #X
Figure 46. Requeuing that results in simultaneous queues
01 4
 Twi I ^ R
r4 * 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 47. Requeuing that does not result in simultaneous queues
EIKl
ww wI r 3
^
i*T QUUE
fE .BTbPmM
avana
14
S~mrnwrrnnE~E
a 4k IIaI r. I ~
P~tdDP ~Uc? I"~"I
BL"DPMM;A
T25 M1
1 s1 Ia 14 rb I1I TI II11110I 1 I r
EEN WA1
ho LemrD FLuP,
QUSM POWWW~~ P~llw
BUNDPERICOUW
r14
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~kLLIIIu ~ d
I.Ll t
QyfLC r,4,,Ro,,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 UuLk J
AN F.J" QIWA IJC ~~~PB~ CFbU~I O 
Figure 48. Example of a blind period
79, Mr11M =7 IM
Mala
Tq r17
r, wQw
H BUMD3E pZL
T47 7
!aLb fo'gl] r1i ew
IgT GREu EN I PERM
wwwmmMIrj ini^j
 Urn.I
'.0It'v
'rNFtit
r<>
H ',bv ,
r1i pn II u y in 11 j innnn
'" b4I ~~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 49. Example of adjacent blind periods
iwa.1t^
~rl ~ ~ICCILh ~I~
I
B

an nlmr
rnrrnB rr'
Fr15 a
rf r"
rr 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(iCatu
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 410. Counters and queue status
I EE OFR IF Jh EimP l]
nEet8ie
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 411. 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 412. Effect of increasing the power constant on stopped delay comparison
I v I ___. r
 l TOPH 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 413. 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= 226.9x221.9
 R2 = 0.928
1.12
1.14
Figure 414. 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 415. 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 416. 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 417. 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 418. Vehicle requeuing
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 419. 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 420. 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 421. 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 422. 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 423. 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 424. 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
16H
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 425. 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 426. 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
420%
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 427. 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 428. Ratio of control delay to stopped plus moveup delay
Table 41. Example summary volume and capacity
File Name Random Arrival at Back of Queue Volumes Calculated Capacity and VolumetoCapacity 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 42. Example summary queue discharge, delay check and goodnessoffit
___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 GoodnessofFit 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+EEGSULT
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
(20sec arrival intervals)
95% Ref. Statistic= 9.49
ChiSquare 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 43. 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 ReQ'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 44. 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 45. 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 MoveUp MoveUp 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 46. Comparison of variation in actual and predicted stopped delay
Cumulative 1Hour
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 47. Pvalue determination for difference in median values
Fisher SignTest
For paired replicates
Null hypothesis: Differences between actual and predicted median delay is zero
Cumulative 1Hour
Random Stopped Delay (sec)
Number Set Actual Predicted Difference Mu
RN X Y Z=YX 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 secondbysecond 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 realtime 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
wellaccepted procedure for calculating pervehicle 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 oversaturation 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 moveup 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 oversaturated 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 oversaturated
flow begins immediately at the start of a onehour observation period and that, at some point
near the end of the hour, it is replaced by a period of undersaturated flow that causes the queue
to dissipate before the hour expires. Let us also assume that the component 15minute flow
rates follow a reasonable pattern that result in some minimum Peak Hour Factor (PHF). Figure
51 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 oversaturated 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 onehour 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 mostlikely 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 15minute period (period 4) of the
hour can be calculated:
AR4 = (CA6oCAc)/(T60Tc) (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 52, 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 15minute 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 15minute time period (sec)
Continuing the Figure 52 example, the cumulative number of arrivals at the beginning of
the last 15minute 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 15minute period (see Figure 53):
OD4 = Area between Cumulative Arrival Curve and Uniform Departure Curve
OD4 = 0.5 (t4s)2(UDR4AR4) = 0.5(T T45) 2(UDR4AR4) (3)
Where: OD4= Overflow Delay during period 4 (vehsec)
UDR4 = Uniform Departure Rate during period 4 (veh/sec)
AR4 = Arrival Rate during period 4 (veh/sec)
t4s = Duration of oversaturated flow during 4th 15min 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 vehsec
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 15minute 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 15minute periods must equal the arrival rate over the entire 1hour 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 15minute 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 54.
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 15minute 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 15min 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 55.
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 56. 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 15minute 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= (41/0.75)(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 57).
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)(21/0.75) 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 58). 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 59 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 inperiod delay for period 1 (Dpi) and the outofperiod 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 inperiod
delay actually occurs during period 1, the outofperiod delay occurs during period 2.
For period 1, the inperiod 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 vehsec
The outofperiod 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 outofperiod 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 vehsec
Figure 510 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 inperiod oversaturation delay for period 2 (Dp2), the
outofperiod oversaturation delay for period 2 (Dc2), the inperiod initial queue delay for period
2 (DIQA2) and the outofperiod 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 inperiod delay and inperiod initial queue delay
actually occur during period 2, the outofperiod delay and outofperiod initial queue delay
occur during period 3. The inperiod 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 inperiod oversaturation 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 vehsec
The outofperiod oversaturation delay for period 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 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 outofperiod oversaturation delay can be calculated using the following
formula:
Dc2 = 0.5(Tc2A T30)(CA3o UCD30A) (49)
For Figure 510 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 vehsec
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 vehsec
An inspection of Figure 510 reveals that the inperiod 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 UCD30AUCD30.
Consequently:
DIQA2 = (Tc T15)(UCD30 CA15) + 0.5(Tc T15)(UCD30A UCD30)
DIQA2 = (Tc1 T15)[(UCD30 CA15) + 0.5(UCD30A UCD3o)] (52)
The total outofperiod 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 outofperiod delay can be calculated using the following formula:
DT2 = 0.5(T2 T30)(CA3o UCD3o) (56)
The outofperiod initial queue delay for period 2 is then calculated by simply subtracting
the outofperiod oversaturation delay from the total outofperiod delay:
DIQB2 = DT2 Dc2
DIQB2 = 0.5(Tc2 T30)(CA3o UCD3o) Dc2 (57)
For Figure 510 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 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 vehsec
DIQB2 = 15,625 vehsec
Figure 511 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 inperiod initial queue delay for
period 3 (DIQA3) and the outofperiod 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 inperiod initial queue delay actually occurs during
period 3, the outofperiod initial queue delay occurs during period 4. The inperiod 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 511 reveals that the inperiod 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 outofperiod 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 outofperiod 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[(2700sec1800sec)(487.5veh383.3veh)(2700 sec1800 sec)(450 veh383.3 veh)]
DIQA3=0.5[(2700sec1800sec)(487.5veh383.3veh)(2700 sec2300 sec)(450 veh383.3 veh)]
DIQA3 = 33,542 vehsec
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 511 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 511 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 511 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 vehsec
It should be pointed out that the period 4 delay situation that is represented in Figure 511
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 511 corresponds to CASE IV.
In addition, the period 2 situation represented in Figure 510 corresponds to Case V and the
period 1 situation represented in Figure 59 corresponds to Case II of Appendix F.
The total overflow delay for the onehour analysis period is obtained by simply summing
the individual 15minute period overflow delays. Inspection of Figures 59 through 511
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 vehsec
OD2 = Dc1 + DP2 + DIQA2 = 5208 vehsec + 18,750 vehsec + 32,292 vehsec
OD2 = 56,250 vehsec
OD3 = Dc2 + DIQB2 + Dp3 + DIQA3 = 5208 vehsec + 15,625 vehsec + 0 vehsec + 33,542 vehsec
OD3 = 54,375 vehsec
OD4 = Dc3 + DIQB3 + DIQ4= 0 vehsec + 4219 vehsec + 5906 vehsec
OD4 = 10,125 vehsec
ODT = ODi+ OD2 + OD3+ OD4
ODT = 18,750 vehsec + 56,250 vehsec + 54,375 vehsec + 10,125 vehsec
ODT = 139,500 vehsec
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 vehsec + 5208 vehsec
DQD1 = 23,958 vehsec
DQD2 = DP2 + Dc2 + DIQA2 + DIQB2
DQD2 = 18,750 vehsec + 5208 vehsec + 32,292 vehsec + 15,625 vehsec
DQD2= 71,875 vehsec
DQD3 = DP3 + DC3 + DIQA3 + DIQB3
DQD3 = 0 vehsec + 0 vehsec + 33,542 vehsec + 4219 vehsec
DQD3 = 37,760 vehsec
DQD4 = DIQ4 = 5906 vehsec
DQDT = DQD1 + DQD2 + DQD3 + DQD4
DQDT = 23,958 vehsec+ 71,875 vehsec+ 37,760 vehsec+ 5906 vehsec
DQDT = 139,500 vehsec
The deterministic delay values can be changed to a "pervehicle" 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 "pervehicle" 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]=18,750vehsec/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 vehsec/(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)[(UCD3oCA15) + 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 512 shows an example set of
feasible delay regions for a minimum PHF of 0.75, while Figures 513 and 514 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 bulletshaped 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 15minute 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 = (400.030.1727X0.0007X2) (404.051.2365X+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 515. Figures 516 and 517 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 15minute 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 15minute 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(41/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 518.
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 15minute periods must equal the arrival rate over the entire 1hour
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 15min 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 519.
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 15minute 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 15minute 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 15minute periods, with a
newly defined "Peak Period Factor" that is consistent with the actual number of 15minute
periods under consideration. For example, the Peak Period Factor for a 5period 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 15minute 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 rework 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 520.
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 rework 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 521. 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 (51/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 C25FOV 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
(5P5F1)[CA5(5 1/ P5F)Cl C2 5FOV] = C, + 5FOV +C2 +CA5 /P5F + V5 + V5 (5P5F 1)
V5 (1+5P5F1) = (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 P5Fconstrained 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 C15 FOV 2CA5/P5F
5P5F V5 = 25 P5F CA5 15CA5 5P5F C1 25 P5F FOV
V5 = 5CA5 3CA5 /P5F C 5FOV
V5 = CA5(53/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 = 1341.7660 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 P5Fconstrained 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 nonnegative 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 nonvisible, 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 (51/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 =5CA5Ci 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(51/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)600600 600 5(12) 350
V4= 2875 180060350
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 180060?
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< 57560060?
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 /(N3))(N (N2)/PNF)
Then: VN3 = CAN/PNF
VN2 = CAN/PNF
VN1 = CAN(N (N2)/PNF) (N4)VN
If VN > (CAN /(N3))(N (N2)/PNF)
And VN < (CAN/(N2))(N (N3)/PNF)
Then: VN3 = CAN/PNF
VN2 = CAN(N (N3)/PNF) (N3)VN
VN1 = VN
If VN > (CAN /(N2))(N (N3)/PNF)
Then: VN3 = CAN(N (N4)/PNF) (N2)VN
VN2 = VN
VN1 = VN
LOWER BOUND
If VN = NCAN (EN1Ci) NFOV
Then: VN4 = CN4 + NFOV
VN3 = CN3
VN2 = CN2
VN1 = CN1
PNF = CAN / (CN4 + NFOV)
If VN < CAN(N (N5)/PNF) (ENCi) NFOV
And VN >= CAN (N (N4)/PNF) (EN2Ci) NFOV
Then: VN4 = CN4 + NFOV
VN3 = CN3
VN2 = CN2
VN1 = CAN (N (N5)/P5F) (EN2Ci) NFOV VN
PNF = CAN / VN1
If VN < CAN (N (N4)/PNF) (EN2Ci) NFOV
And VN >= CAN(N (N3)/PNF) (EN3Ci) NFOV
Then: VN4 = CN4 + NFOV
VN3 = CN3
VN2 = CAN (N (N4)/P5F) (EN3Ci) NFOV VN
VN1 = CAN /PNF
PNF = CAN / VNI
If VN < CAN(N (N3)/PNF) (EN3Ci) NFOV
And VN >= CAN(N (N2)/PHF) (EN4Ci) NFOV
Then: VN4 = CN4 + NFOV
VN3 = CAN (N (N3)/PNF) (EN4Ci) NFOV VN
VN2 = CAN / PNF
VN = CAN / PNF
PNF = CAN / VN1
If VN < CAN(N (N2)/PHF) (EN4Ci ) NFOV
Then: VN4 = CAN (N (N2)/PNF) (ENCi) VN
VN3 = CAN / PNF
VN2 = CAN / PNF
VN = CAN / PNF
PNF = CAN / VN1
C1 <= CAN(N (N2)/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 (N2)/PNF]
Then:
For 1 < k
For k = Nl: Vk = 2VNUL VN
For k = N: Vk = VN
Forj = 2 to N3:
When VN> VNLL = (CAN/j)(N (N
And VN < VNUL = (CAN/(j +1))[N 
Then:
For 1 < k
For k= Nj:
For N >= k > Nj:
Forj = N2:
When VN > VNLL = (CAN /(N2))(N
Then:
Fork = 1:
For k = 2:
For N >= k > 2:
j)/PNF)
(Nj1)/PNF]
CAN /PNF
VNUL (j +1) jVN
VN
2/PNF)
Vk = CAN/PNF
Vk = (N1)VNUL (N2)VN
Vk=VN
(141)
(142)
(143)
(144)
(145)
(146)
(147)
(143)
(148)
(145)
(149)
(143)
(150)
(145)
LOWER BOUND
Forj = 1 to N2:
Ifj 1
And VN = VNUL = NCAN (EN Ci)
Then:
Fork= 1:
For N1 > k > 1:
NFOV
PNF = CAN / (Cl + NFOV)
Vk = Ck+ NFOV
Vk= Ck
Otherwise:
When VN < VNUL = CAN (N (j1)/PNF) (EN Ci) NFOV
And VN > VNLL = CAN (N (j)/PNF) (EN1 Ci) NFOV
Then: PNF = CAN VNI
Fork = 1: Vk = Ck+ NFOV
For
For k= Nj: Vk= VNUL + Ck VN
For N1 > k > Nj: Vk = CAN /PNF
(151)
(152)
(153)
(154)
(155)
(156)
(157)
(153)
(154)
(158)
(159)
Forj = Nl:
IfVN < VNUL = CAN (2/PNF) Ci NFOV (160)
And Ci <= CAN(N (N2)/P5F) NFOV (161)
Then: PNF = CAN / VN1 (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 oversaturated. Appendix B contains a sample of
historical PHF information for various locations in Jacksonville, Florida.
Tarko and PerezCartagena [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 10location 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 15minute
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 15minute
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 15minute
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 timeofday 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 weatherrelated incident.
600
500
. 400
a
S300
200
100
0
300 600 900 1200 1500 1800 2100 2400 2700 3000 3300
Time (sec)
Figure 51. Cumulative arrivaldeparture curves and overflow delay
3000 3300
Time (sec)
Figure 52. 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 53. 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 54. 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 55. 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 56. 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 57.
600
500
a,
S400
E 300
200
100
0
Time (sec)
Figure 58. 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 59. 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 510. 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 511. 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 512. 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 513. 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 514. 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 515. 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 516. 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 517. Maximum delay estimation error for 0.85 minimum PHF
0 300 600 900 1200 1500 1800 2100
Time (sec)
Figure 518. 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 519. 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 520. 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 521. 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 poststop bar acceleration. In addition, queue moveup delay and prestop bar acceleration
delay are overrepresented 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
PreStop 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
PreStop 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
MoveUp Time = Time that it takes the vehicle to travel between queues
Free Speed MoveUp 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
MoveUp Delay = Time that the vehicle is delayed while traveling between queues = MoveUp
Time Free Speed MoveUp 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 + MoveUp 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 61 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 moveup time can be decomposed in 40 seconds of
moveup delay and 10 seconds of time traveling at the free flow speed. The average speed
during moveup 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 prestop 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 poststop 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 moveup 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 freeflow conditions).
Trajectory analysis gives a true picture of vehicular delay. The only component of delay
that is not represented by this singlevehicle 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 62 tracks the vehicle previously shown in Figure 61, but this time using a typical
set of cumulative arrival and cumulative departure curves. As in Figure 61, 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 MoveUp 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 movingup 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 moveup time as well as the free flow speed portion of
the prestop 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 madeup of the following 5 components:
1.) Stopped delay
2.) Moveup delay
3.) Free Speed MoveUp Time
4.) PreStop Bar Acceleration Delay
5.) PreStop 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 poststop bar acceleration delay is accounted for
in the area between the curves.
Reconciling the Difference Between Cumulative Curves and Trajectories
Continuing our example, Figure 63 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 freeflow
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 moveup
time (10 seconds) and the free flow speed portion of the prestop 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 moveup time and free flow speed prestop 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 oversaturated conditions, the free flow speed moveup time and
free flow speed prestop bar acceleration time make up about 10% of the control delay.
Coincidentally, the acceleration delay and poststop 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 counterbalancing 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 poststop bar acceleration delay, and by
subtracting out the free flow speed portion of both the moveup time and the prestop bar
acceleration time.
Figure 64 illustrates the various delayrelated 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 64, 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=NonStopped 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 nonstopped 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 TrajectoryBased Delay Components for the BuckQ Examples
Trajectorybased 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 nonstopped delay portion of the cumulative arrival area
based on equations 162 and 163. Tables 61 through 64 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 61 through 64 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 4period 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 65. The cumulative curve delay for the 4
period Lower Bound was also calculated and the resulting minimum cumulative overflow delay
values for each 15minute period are provided on the left side of Table 65. The middle of
Table 65 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 66. 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 15minute period are
provided on the left side of Table 66. The middle of Table 66 provides similar values for the
estimated actual delay.
Also included in Table 66 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 underutilization of some cycles, as the green time is "starved" due to episodes of
infrequent arrivals, and to the overutilization 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 15minute period whenever a residual
queue does not exist at the beginning of that 15minute 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 15minute period by dividing the number of signals cycles
that occur during the period into the 15minute capacity of that period. For example, if the
hourly capacity for the first 15minute period is 600 vph and the average cycle length is 120
seconds, then the average thruput for the first 15minute 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(Tmin(t,T))/T. The modified d2 term
thus becomes:
d2 = 900T[(X1)+sqrt[(X 1)2 + 8kIX(Tmin(t,T)/cT2] (165)
It is interesting to note that, during oversaturated conditions, variations in cycletocycle
vehicle arrival patterns have much less of an effect on delay than variations in cycletocycle
capacity stemming from relative driver aggressiveness. The amount of startup 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 oversaturated
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 61 through 64 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 67 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
limitedinformation secondbysecond 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 68 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 69 and 610 address volume pattern 700_700_700_350vph. A review of the
embedded figures in Table 69 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 610 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 611 and 612 address the highest volume pattern: 725_700_700_350vph. A
review of the embedded figures in Table 611 shows that the prediction continues to falls within
the theoretical envelope for all three runs. Table 612 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 613 and 614 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 613
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 614 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 615 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 oversaturated
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 63. Trajectory conversion of cumulative curve example
VYENC
peLtL4rAtn
I i
VEMaM
JOleS BOO
teiTs
9rr.
VEYI4CA
prrta
ai, [i.i t D
#1 ww* TTM
VERM
IsA
I
I I
. II
f
I . '"
FREE
I I
SI V
^^UH **Tu; 6f*P
I 0b D DaiDu+ Dv * Lu
TI T E T
TIME
WHERE
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 MCEUP DELAY
Cma FESTP AFB ACCEIRWTI DELAY
Du w PGSTSTP BAR ACCERTION DELAY
Tw FREiE 8EED MOWFI' TIME
To FPRESTPS B AHREE SEED SPEED ACCELERATION ME
Ta POSTTOP BAR FREE S EEDATCEA'NJ tEM
Figure 64. Delay and travel time components.
FREE
Ff.0#
I9L Hg
.::
_m
F
r"
Table 61. Calculation of cumulative curve delay conversion factors, volume pattern 625 700 650 350vph
Period
Random Number Set 1
Stopped Delay
Queue MoveUp Delay
Free Speed Queue MoveUp Time
Accel/Decel Delay
PreStop Bar Accel Delay as % of Accel/Decel Delay
PreStop Bar Acceleration (Al) Delay
Free Speed PreStop Bar Accel Time as % ofA1Delay
Free Speed PreStop Bar Acceleration Time
NonStopped Delay Portion of Cumulative Curve Area
NonStopped 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 MoveUp Delay DMu BuckQ 35 533 2731 3105
Free Speed Queue MoveUp Time TMu BuckQ 28 398 1591 1870
Accel/Decel Delay BuckQ 1821 4479 6973 7924
PreStop Bar Accel Delay as % of Accel/Decel Delay BuckTRAJ 46% 46% 46% 47%
PreStop Bar Acceleration (Al) Delay DA1 Calculated 838 2060 3208 3724
Free Speed PreStop Bar Accel Time as % of A1Delay BuckTRAJ 70% 73% 73% 72%
Free Speed PreStop Bar Acceleration Time TAl Calculated 586 1504 2342 2681
NonStopped Delay Portion of Cumulative Curve Area Ecc Calculated 1487 4495 9871 11381
NonStopped 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 61. Continued
Period
Random Number Set 3
Stopped Delay
Queue MoveUp Delay
Free Speed Queue MoveUp Time
Accel/Decel Delay
PreStop Bar Accel Delay as % of Accel/Decel Delay
PreStop Bar Acceleration (Al) Delay
Free Speed PreStop Bar Accel Time as % ofA1Delay
Free Speed PreStop Bar Acceleration Time
NonStopped Delay Portion of Cumulative Curve Area
NonStopped 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 62. Calculation of cumulative curve delay conversion factors, volume pattern 700 725 625 350vph
Period
Random Number Set 1
Stopped Delay
Queue MoveUp Delay
Free Speed Queue MoveUp Time
Accel/Decel Delay
PreStop Bar Accel Delay as % of Accel/Decel Delay
PreStop Bar Acceleration (Al) Delay
Free Speed PreStop Bar Accel Time as % ofA1Delay
Free Speed PreStop Bar Acceleration Time
NonStopped Delay Portion of Cumulative Curve Area
NonStopped Delay Portion as % of Stopped Delay
Factor That Converts Overflow Delay to Stopped Delay
Random Number Set 2
Stopped Delay
Queue MoveUp Delay
Free Speed Queue MoveUp Time
Accel/Decel Delay
PreStop Bar Accel Delay as % of Accel/Decel Delay
PreStop Bar Acceleration (Al) Delay
Free Speed PreStop Bar Accel Time as % ofA1Delay
Free Speed PreStop Bar Acceleration Time
NonStopped Delay Portion of Cumulative Curve Area
NonStopped 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 62. Continued
Period
Random Number Set 3
Stopped Delay
Queue MoveUp Delay
Free Speed Queue MoveUp Time
Accel/Decel Delay
PreStop Bar Accel Delay as % of Accel/Decel Delay
PreStop Bar Acceleration (Al) Delay
Free Speed PreStop Bar Accel Time as % ofA1Delay
Free Speed PreStop Bar Acceleration Time
NonStopped Delay Portion of Cumulative Curve Area
NonStopped 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 63. Calculation of cumulative curve delay conversion factors, volume pattern 700 700 700 350vph
Period
Random Number Set 1
Stopped Delay
Queue MoveUp Delay
Free Speed Queue MoveUp Time
Accel/Decel Delay
PreStop Bar Accel Delay as % of Accel/Decel Delay
PreStop Bar Acceleration (Al) Delay
Free Speed PreStop Bar Accel Time as % ofA1Delay
Free Speed PreStop Bar Acceleration Time
NonStopped Delay Portion of Cumulative Curve Area
NonStopped 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 MoveUp Delay DMu BuckQ 1185 4090 11561 14343
Free Speed Queue MoveUp Time TMu BuckQ 902 2781 6196 7938
Accel/Decel Delay BuckQ 3092 6138 9307 10171
PreStop Bar Accel Delay as % of Accel/Decel Delay BuckTRAJ 37% 39% 39% 41%
PreStop Bar Acceleration (Al) Delay DA1 Calculated 1144 2394 3630 4170
Free Speed PreStop Bar Accel Time as % ofA1Delay BuckTRAJ 78% 80% 76% 76%
Free Speed PreStop Bar Acceleration Time TAl Calculated 892 1915 2759 3169
NonStopped Delay Portion of Cumulative Curve Area Ecc Calculated 4123 11180 24145 29620
NonStopped 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 63. Continued
Period
Random Number Set 3
Stopped Delay
Queue MoveUp Delay
Free Speed Queue MoveUp Time
Accel/Decel Delay
PreStop Bar Accel Delay as % of Accel/Decel Delay
PreStop Bar Acceleration (Al) Delay
Free Speed PreStop Bar Accel Time as % ofA1Delay
Free Speed PreStop Bar Acceleration Time
NonStopped Delay Portion of Cumulative Curve Area
NonStopped 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 64. Calculation of cumulative curve delay conversion factors, volume pattern 725 700 700 350vph
Period
Random Number Set 1
Stopped Delay
Queue MoveUp Delay
Free Speed Queue MoveUp Time
Accel/Decel Delay
PreStop Bar Accel Delay as % of Accel/Decel Delay
PreStop Bar Acceleration (Al) Delay
Free Speed PreStop Bar Accel Time as % ofA1Delay
Free Speed PreStop Bar Acceleration Time
NonStopped Delay Portion of Cumulative Curve Area
NonStopped 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 MoveUp Delay DMu BuckQ 2117 7524 17590 22139
Free Speed Queue MoveUp Time TMu BuckQ 1515 4341 8819 11428
Accel/Decel Delay BuckQ 2837 5827 8079 8568
PreStop Bar Accel Delay as % of Accel/Decel Delay BuckTRAJ 38% 39% 40% 44%
PreStop Bar Acceleration (Al) Delay DA1 Calculated 1078 2273 3232 3770
Free Speed PreStop Bar Accel Time as % ofA1Delay BuckTRAJ 88% 80% 77% 77%
Free Speed PreStop Bar Acceleration Time TAl Calculated 949 1818 2488 2903
NonStopped Delay Portion of Cumulative Curve Area Ecc Calculated 5659 15956 32129 40240
NonStopped 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 64. Continued
Random Number Set 3
Stopped Delay
Queue MoveUp Delay
Free Speed Queue MoveUp Time
Accel/Decel Delay
PreStop Bar Accel Delay as % of Accel/Decel Delay
PreStop Bar Acceleration (Al) Delay
Free Speed PreStop Bar Accel Time as % ofA1Delay
Free Speed PreStop Bar Acceleration Time
NonStopped Delay Portion of Cumulative Curve Area
NonStopped 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 64. Continued
Table 65. Cumulative curve delay for standard 4period 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 66. 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 67. 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 67. 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 67. 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 68. 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 69. 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 69. 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 69. 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 610. 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
41
40000
20000
0
0 1 Period 2 3 4
Table 611 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 611.
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 611.
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 612. 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 613 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 613. 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 613. 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 614. 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 615. 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 OVERSATURATED FLOW
This chapter describes deficiencies in the Highway Capacity Manual that can lead to
incorrect delay values during oversaturated 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 oversaturated conditions. If the size of the residual queue is not
correctly identified at the start of each 15minute 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 cyclebycycle 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 CyclePeriod Issues in Calculating d3
Formula F166 in the Highway Capacity Manual is touted as yielding the residual queue.
The formula is:
Qb,i+1=max[0,Qb,i+ciT(Xi1)]
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(X1)
Recognizing that X = v/c, we can further simplify this equation to:
Qb2=cT(v/c1)
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 15minute
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 (15minute volume) and the departing vehicles (15minute capacity) for period 1.
However, this is not the correct procedure for determining the residual queue and Tables 7
1 and 72 show why. The simple example illustrated in these tables has just two 15minute
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 15minute
period and a value for d3 is calculated for the second 15minute 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 71 provides the second
bysecond cumulative arrival, cumulative departure and queue length information while the
resulting residual queue, thruput, and associated d3 values are provided in Table 72. (The full
data set associated with Table 71 is provided in Appendix C.)
This example will demonstrate that the HCM formula consistently overpredicts 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 15minute period is V=180 vehicles (or v=720 vph)
and the uniform arrival rate for the second 15minute period is V=130 vehicles (or v=520 vph).
Again, to keep the example simple, vehicles are assumed to depart at constant 2second
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 71 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 overpredicted by 4
vehicles and the resulting d3 is too high by about 15%.
There are 15 cycles during each 15minute period (900/60 = 15) and 150 vehicles (C = 10
x 15 =150) will pass the stop bar during each 15minute 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 endofred (startofgreen)
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 36vehicle "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 startofred 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 periodbased 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 162second 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 endofred 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 cyclebased approach and a
periodbased approach can be readily seen in Figure 71. The true cyclebased delay is always
less than the periodbased delay.
A further demonstration of the loss of accuracy associated with periodbased analysis can
be made by comparing the "actual control delay" (as obtained by summing up the actual queue
lengths on a secondbysecond basis) with the control delay obtained by adding the di term to the
previously calculated d3 term. The results are presented in Figure 72. The cyclebased analysis
is much closer to the actual delay than the periodbased analysis in every case. Also shown in
Figure 72 is the control delay value provided by CORSIM. As with the actual control delay, the
CORSIM delay results are much closer to the cyclebased results than the periodbased 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 periodcycle problem is to always start and stop the counts at the end
ofred (startofgreen), and to keep track of how much time transpires for each count "period".
For the 162second example just discussed, the traffic technician would, for the first period, start
counting at endofred time 108 and stop counting at endofred time 756. This makes period 1
of length 648 seconds (756108). 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 oversaturated 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 startofgreen 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 15minute 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 (endofgreen) 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 73 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 oversaturated volumetocapacity 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 423.
Detailed Example of CyclePeriod 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
realtime 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
onehour (3600 second) example has been developed that is summarized in Tables 73 through
76.
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 oversaturated flow, we have accumulated a residual queue of 80
vehicles. The next 15minute 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 ([400600]/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 15minute 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 ([400480]/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 73 through 76 summarize the comprehensive example. A 90second cycle is used
in this example with the start of the green offset by about 15 seconds from the 15minute 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 startofgreen (or endofred).
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 15minute period, in an HCM multiperiod
analysis. If we also allow capacity to vary on a 15minute basis, as reflected by actual thruput,
then we would enter a value of 156 (which equals 624 vph) and, using equation F166 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 161 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 startofgreen and the thruput has not been
deducted. The closest startofgreen 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 (176149) and, since it occurs at the startof
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 endof
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 (183165). 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 startofgreen (time 885) from the 165 cumulative departing vehicles
from the endofgreen (time 916) yields a thruput of 16 vehicles. Subtracting this thruput (16)
from the startofgreen queue (27) produces the true residual queue of 11 vehicles at the end of
period 1.
The capacity for the 900second 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:
306149 = 157. Since this thruput occurs over a 900second interval, the equivalent hourly
capacity is calculated as: 157 x 3600/900 = 628 vph. Using equation F161 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 endofgreen is actually the endofyellow. 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 endof
green is improved by using the departures that occur 1 second after the endof 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 (352178) and 157
departures (313156) in period 2, with a resulting "queue" of 39 vehicles (352313). 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 15minute intervals, and it is the value that would be
entered into an HCM multiperiod analysis for period 2, the second 15minute 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 F166 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 F161 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 startofgreen and the
thruput has not been deducted. The closest startofgreen 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 (351306) and, since it
occurs at the startofgreen, 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 endofgreen 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 startofgreen (time 1785) from
the 320 cumulative departing vehicles from the endofgreen (time 1816) yields a thruput of 14
vehicles. Subtracting this thruput (14) from the startofgreen queue (45) produces the true
residual queue of 31 vehicles at the end of period 2.
The capacity for the 900second 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 900second interval, the equivalent hourly capacity
is calculated as: 153 x 3600/900 = 612 vph. Using equation F161 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 (502352) and 152
departures (465313) in period 3, with a resulting "queue" of 37 vehicles (502465). 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 15minute intervals, and it is the value that would be
entered into the multiperiod HCM analysis for period 3, the third 15minute 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 F166 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 F161 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 startofgreen and the thruput has not been deducted.
The closest startofgreen 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 (502459) and, since it occurs at the startofgreen, 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 endofgreen 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 (506474). 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 startofgreen (time 2685) from the 474 cumulative
departing vehicles from the endofgreen (time 2716) yields a thruput of 15 vehicles.
Subtracting this thruput (15) from the startofgreen queue (43) produces the true residual queue
of 28 vehicles at the end of period 3.
The capacity for the 900second 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: 588459 = 129. Since this thruput occurs over an
869second interval, the equivalent hourly capacity is calculated as: 129 x 3600/869 = 534 vph.
Using equation F161 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 periodbased
arrivals and capacities instead of cyclebased arrivals and capacities can be quite large.
The periodbased method simply does not produce the correct residual queue.
The situation becomes even worse when we use the periodbased 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 oversaturated periods. If this is done for our example, the d3 delay results are close to
those obtained for the variable capacity periodbased scenario: 134 sec/veh for period 2, 229
sec/veh for period 3, and 65 sec/veh for period 4.
Table 73 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 74 provides the dl, d2 and d3 results for both a periodbased and cyclebased
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 periodbased
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 periodbased variable capacity approach. However, to approach the
CORSIM results, a cyclebased 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
75, 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 cyclebased 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 76 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 cyclebased approach with proper
definition of residual queue, varying capacity by period, and a d2 term that eliminates the
effect of randomness during oversaturated conditions.
Previously, in Tables 67 through 615, 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 periodbased approach
instead of a cyclebased approach may be the cause of these discrepancies.
Table 71. Generalized example of cycleperiod 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 72. Generalized example of cycleperiod 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
PERIODBASED 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(X1) 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
CYCLEBASED 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 (17614010) 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[1g/Cf/[1g/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
PERIODBASED 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(X1) based on theoretical capacity 30 180 30 180 30 180
Calc d3=1800Qb(1+u)t/CT & Qb based on actual thruput
CYCLEBASED 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[1g/C l[1g/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 ycleB a sed (Thruput= C capacity)
7
78
Cycle L e n g th
Figure 71. Cycle v. period initial queue delay analysis
m
I
I
. I
* PeriodBased Using Capacity
Perio dBased, Using Thruput
O CycleBased (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 72. Cycle v. period "control delay" analysis
3i0 r
300 
200
1 0
1 0
c n
60
162
i
::::
::::
::::
::::
::::
::::
::::
::::
::::
::::
::::
::::
::::
::::
::::
::::
::::
::::
::::
::::
::::
::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::::
:::
:::::
:::::
:::::
:::::
:::::
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Table 73. Detailed example of cycleperiod 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 74. Detailed example of cycleperiod 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 75. Detailed example of cycleperiod 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 76. Detailed example of cycleperiod 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 73. Upward bias in HCM residual T"l:queue calcul :' ation
Figure 73. 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 realtime
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 oversaturated conditions.
2. The research indicates that there are two important variables for predicted nonvisible
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 oversaturated volume
tocapacity ratio increases. At lower oversaturated 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 15minute 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 realtime 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 nondelay 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 oversaturated 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 oversaturated periods having v/c ratios slightly above 1.0
13. The 2000 Highway Capacity Manual's multiperiod signalized intersection analysis
procedure uses a periodbased technique for queue accumulation. This technique has
certain drawbacks that can produce substantial errors when calculating control delay
during oversaturated conditions. The degree of error increases with increasing cycle
length. A cyclebased 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 oversaturated
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 oversaturated v/c ratios close to 1.0 A modification
to the d2 delay term is proposed to remedy this situation.
15. During oversaturated conditions, variability in capacity due to cycletocycle 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 hourlong 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 oversaturated intersection approach for a onehour
analysis time frame would proceed as follows:
10. Using the vehicle detection equipment for the approach of interest, realtime second
bysecond 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 nonvisible queue at every second of the onehour analysis time frame. Second
bysecond 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 realtime traffic control.
13. The time during the last 15minute 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 onehour 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 15minute 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 15minute 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 carriedout 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 realtime
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 closedloop
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 finetuned. 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 addon to
the closedloop 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: RealTime Traffic Signal Control
A realtime 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 oversaturated approaches. The appropriate 15minute count data are
collected and, for the undersaturated, nearsaturated, and slightly oversaturated movements, a
multiperiod analysis based on HCM principals is conducted to determine approach control
delay. However, for the grossly oversaturated 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 resetting 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 simulationbased research.
5. Accounting for the effect of multiple approach lanes The queue arrival and
discharge situation is complicated when lanechanges 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 highvolume or oversaturated 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
multilane main street approach in Gainesville, Florida and on a multilane 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 5period and
nperiod 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 backofqueue position, number of phase failures, number of vehicle requeues,
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 websurvey 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 oversaturated conditions
prevail:
"When saturation exists, different measures of effectiveness should be usedfor evaluating
system performance. During undersaturated 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 buildup 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 oversaturated 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 closedloop or traffic signal
controller module that would automate the delay estimation procedure. The module
would provide realtime delay estimation, even during oversaturated 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 A1. 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
StartUp Lost Time, 1 Hour Average (seconds)
Figure A2. Startup 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 A3. 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 A4. 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 moveup"
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 A5. Comparison of control delay and stopped plus queue moveup 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 A6. Comparison of control delay and stopped delay plus queue moveup 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 A7. 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 A8. 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 A9. Relationship between v/c ratio and stopped plus queue moveup 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 A10. Relationship between v/c ratio and stopped plus queue moveup 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 A11. 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 A12. 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 A13. Relationship between vehicle requeues 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 requeues 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 ReQueues
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 requeues
increases as the cycle length decreases
n 1500 If cycle length is accounted for, the / a ch
Q 1400 number of vehicle requeues 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____II
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 A14. Relationship between v/c ratio and vehicle requeues
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 requeues 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 A15. Relationship between v/c ratio and vehicle requeues by cycle length
01
0.95 1.00 1.05 1.10 1.15
v/c Ratio During 1st 45 Minutes
Figure A16. 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 A17. 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 A18. 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 IfAAA
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 A19. 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 A20. 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 A21. 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 :
~ 180I( ^ Iry = =51.186e2E05<
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 A22. Relationship between ABPC and control delay
APPENDIX B
TYPICAL PEAK HOUR FACTORS
1.000
I
0.800
C 9 0              
0.800   
4Tues, 3/28/06
Wed. 3/29/06
 Thur, 3/30/06
0.700
4:005:00 4:155:15 4:305:30 4:455:45 5:006:00 5:156:15 5:306:30 5:456:45 6:007:00
Time Period
Figure B1. US 1 S. PM peak hour factor, southbound (outbound) flow
1.000
0.900
0.800
0.700
4:006:00
4:156:15
Figure B2. 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
4:306:30
Time Period
4:456:45
5:007:00
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
1Hour
PHF
Both Period SB NB
479 1 4:005:00 0.894 0.924
438 2 4:155:15 0.942 0.917
446 3 4:305:30 0.893 0.940
449 4 4:455:45 0.933 0.935
489 5 5:006:00 0.895 0.926
523 6 5:156:15 0.837 0.842
511 7 5:306:30 0.784 0.754
393 8 5:456:45 0.859 0.910
345 9 6:007:00 0.871 0.852
328
317
256
1972
2Hour
Period
4:006:00
4:156:15
4:306:30
4:456:45
5:007:00
PPF
SB NB
0.876 0.892
0.839 0.869
0.818 0.835
0.784 0.809
0.735 0.768
Table B1. 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
1Hour
PHF
Both Period SB NB
403 1 4:005:00 0.973 0.836
477 2 4:155:15 0.888 0.877
415 3 4:305:30 0.880 0.874
444 4 4:455:45 0.906 0.890
487 5 5:006:00 0.887 0.896
536 6 5:156:15 0.816 0.894
508 7 5:306:30 0.815 0.810
425 8 5:456:45 0.854 0.853
395 9 6:007:00 0.833 0.800
323
308
266
1975
2Hour
Period
4:006:00
4:156:15
4:306:30
4:456:45
5:007:00
PPF
SB NB
0.842 0.841
0.826 0.859
0.795 0.817
0.767 0.799
0.718 0.770
Table B1. 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
1Hour
PHF
Both Period SB
459 1 4:005:00 0.948
433 2 4:155:15 0.918
456 3 4:305:30 0.960
438 4 4:455:45 0.948
483 5 5:006:00 0.923
493 6 5:156:15 0.823
461 7 5:306:30 0.840
423 8 5:456:45 0.794
336 9 6:007:00 0.982
383
320
292
1875
2Hour
NB
0.901
0.963
0.930
0.913
0.932
0.896
0.906
0.835
0.770
PPF
Period SB NB
4:006:00 0.902 0.916
4:156:15 0.870 0.887
4:306:30 0.842 0.898
4:456:45 0.803 0.872
5:007:00 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
Ul
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
4:005:00 4:155:15 4:305:30 4:455:45 5:006:00 5:156:15 5:306:30 5:456:45 6:007:00
Time Period
Figure B3. 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
4:006:00 4:156:15 4:306:30 4:456:45 5:007:00
Time Period
Figure B4. US 1 N. PM peak period factor, northbound (outbound) flow
Table B2. 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
600 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
1Hour PHF
Period NB SB
4:00500 0933 0.795
4:15615 0889 0.803
4:30530 0907 0.829
4:455.45 0.922 0.957
5:006.00 0.913 0.929
5156'15 0845 0.849
5 306'30 0783 0.778
5456'45 0 775 0.870
6 00700 0830 0.868
4:0005 0 96
4:15515 0868
430530 0892
4.455.45 0.923
5.006.00 0.876
5:15615 0810
5:30630 0 743
5:45645 0930
6:00700 0912
2Hour PPF
Period NB SB
4:00600 0.874 0777
4:15615 0.847 0744
4:306:30 0.826 0728
4:456:45 0.781 0.815
5:007:00 0.722 0.757
4:006:00 0.841 0852
4:156:15 0.826 0832
4:306:30 0.802 0803
4:456:45 0.761 0.754
5.007:00 0.724 0.703
South of Lewis Speedway
Tuesday, 31/2806 1Hour PHF 2Hour PPF
Start Time End Time Northbound Southbound Both Period NB SB Period NB SB
400 415 251 291 542 1 4.005:00 0.960 0.798 4:006:0D 0901 0.754
4:15 4:30 264 288 552 2 4:155:15 0.967 0.829 4:156:15 0 859 0.721
4:30 4:45 252 378 630 3 4:305:30 0.959 0.813 4:306:30 0 823 0.689
4:45 5:00 270 250 520 4 4:455:45 0.931 0.823 4;456:45 0.705 0.692
5:00 5:15 258 338 596 5 5.008:00 0.902 0.794 5:007:00 0.724 0.652
5'15 5:30 256 263 519 6 5:156:15 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 6007:00 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 4.005:00 0.947 0.B4B 4:006:00 0872 0.802
415 430 266 273 539 2 4:1155:1 0.964 0.864 4:156:15 843 0.756
4:30 4:45 258 364 622 3 4305:30 0.911 0.861 4:306:30 0 13 0.726
4:45 5:00 239 276 515 4 4455:45 0.913 0.841 4:456:45 0.776 0.699
5:00 5:15 267 345 612 5 5.000:00 0.871 0.798 5:007:00 0.741 0.651
5:15 5:30 289 269 558 6 5:156:15 0.796 0.871
5'30 5,45 261 271 532 7 5:306:30 0.792 0.794
545 600 190 216 406 B 5:456:45 0.941 0.B90
600 615 160 188 368 9 6.00700 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
4 005 00
415515
4.305.30
4:455:45
5:006:00
5:156:15
5:306:30
5:45645
6:007:00
4:006:00 0.835 0873
4:156:15 0.817 0857
4.300:30 0.791 0.026
4:456.45 0.744 0.848
5:007'00 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
500 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 4005:00 0.958 0.833 4:006:00 0904 0.769
613 2 4:155:15 0.972 0.880 4:156:1 0878 0.736
709 3 4305:30 0.939 0.044 4:306:30 0.841 0.691
599 4 4.455:45 0.912 0.792 4:456.45 0.787 0.642
678 5 5106:00 0.894 0731 5:007'00 0737 0.597
567 6 5:156:15 0.31 0871
542 7 5306:30 0.877 0791
476 8 5:456:45 0.798 0879
412 9 6.007:00 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
AW of SSide, Tues, 10/5/04 Hur c
W of SSide, Wed, 10/6/04
0.7C W of SSide, Thur, 10/7/04 
4:005:00 4:155:15 4:305:30 4:455:45 5:006:00 5:156:15 5:306:30 5:456:45 6:007:00
Time Period
Figure B5. Atlantic Boulevard PM peak hour factor, eastbound (outbound) flow
1.000
0.900
0.800
0.700
Figure B6. Atlantic Boulevard PM peak period factor, eastbound (outbound) flow
4:006:00 4:156:15 4:306:30 4:456:45 5:007:00
Time Period
Table B3. Atlantic Boulevard machine counts
East f Unversit Boulevard
Tuesday, 10/504 1Hour PHF 2Hour PPF
StartTime End Time Eastbound Westbound Both Period EB WB Period EB W E
400 415 406 290 606 1 400500 0935 065 400600 0932 0856
415 430 363 317 680 2 415515 0967 0967 415615 0956 0864
430 445 366 307 673 3 43305 0964 0866 430630 0944 0851
4:45 5:00 384 009 693 4 445545 0976 0893 445645 0914 0825
5:00 5:15 388 293 681 5 500600 0971 0883 500700 0877 0810
5:15 5:30 358 369 727 6 515615 0954 0898
5:30 5:45 384 347 731 7 530630 0934 0889
545 600 377 295 672 8 545645 0877 0889
600 615 347 314 661 9 600700 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 400500 0960 0922 4006 0940 0 97
415 430 382 259 641 2 415515 0977 0894 415615 0948 0896
4:30 4:45 364 300 664 3 430530 0971 0939 430630 0937 0892
4:45 5:00 389 314 703 4 445545 0962 0946 445645 0924 0857
5:00 5:15 385 339 724 5 500600 0929 0940 500700 0889 0825
5:15 5:30 373 320 693 6 515615 0933 0951
530 545 350 310 660 7 530630 0916 0923
545 600 323 306 629 8 545645 0899 0850
600 615 383 281 664 9 60070 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 400500 0968 0932 400600 0879 0863
2 415515 0968 0843 415615 0878 0861
3 430530 0942 0692 430630 0883 0 86
4 445545 0904 0911 445645 0888 0863
5 500600 0815 0932 500700 0856 0846
6 5 156 15 0821 0937
7 530630 0839 0900
8 545645 0884 0868
9 600700 0907 0965
West ofArlington Road
Tuesday, 10/504 1Hour PHF 2Hour PPF
:artTime EndTime Eastbound Westbound Both Period EB WB Period EB WB StartTime
400 4 15 482 298 780 1 400500 0 846 0 957 4 006 00 0 761 938 4:00
415 430 441 296 737 2 415515 0815 0 945 415615 0 750 0 934 4:15
430 445 407 311 718 3 4305 30 0 849 0 945 4 306 30 0 739 0 928 4:30
445 500 302 286 588 4 445545 0 708 0 937 445645 729 0 896 4:45
500 515 287 282 569 5 5006 0 778 0 934 5007 00 0734 0894 500
5:15 5:30 416 316 732 6 515615 0 845 0 938 515
5:30 5:45 549 300 849 7 530630 0835 0958 530
5:45 6:00 457 283 740 8 545645 902 0 945 545
6:00 6:15 434 286 720 9 600700 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 4005 00 0956 0921 400600 0945 0 912 4:00
415 430 428 242 670 2 4 155 15 971 0 945 4 156 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 445545 0 967 0 943 445645 0 888 0 879 4:45
5:00 5:15 450 305 755 5 5006 O 951 0 929 500700 857 845 500
5:15 5:30 468 313 781 6 515615 0 912 0 910 515
530 545 452 261 713 7 5306 30 0899 0939 530
545 600 411 284 695 8 545645 877 0898 545
600 6 15 503 281 784 9 6007 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 00500 0 949 0 958 4 006 00 0 B48 0 876 4:00
415 430 460 262 722 2 4 155 15 927 0 900 4156 15 0 803 0 859 4:15
4:30 4:45 484 287 771 3 430530 0 893 0 907 4 306 0 799 0 857 4:30
4:45 5:00 448 281 729 4 445545 917 0 900 445645 0 789 0 844 4:45
5:00 5:15 403 319 722 5 500600 896 0 872 5 00700 0 768 0 829 500
5:15 5:30 393 270 663 6 515615 0 748 938 515
530 545 399 278 677 7 5 36 30 772 0 925 530
545 600 250 246 496 8 545645 0 793 0982 545
600 6 15 523 249 772 9 6007 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 1Hour PHF 2Hour PPF
EndTime Eastound Westbound Boh Penod EB WB Penod EB WE
4:15 375 325 700 1 400500 0869 0937 400600 0 67 0889
4:30 378 320 698 2 415515 0825 0919 4 15615 840 865
4:45 267 341 608 3 4 305 30 832 0905 4 30630 837 0848
5:00 294 292 586 4 445545 0864 0957 445645 0859 0924
515 308 300 608 5 500600 0877 0949 5 00700 865 906
530 373 302 675 6 515615 0867 0917
545 314 262 576 7 530630 0937 0956
600 313 282 595 8 545645 0951 0955
615 293 262 555 9 6007 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 400500 0891 1 1400600 04 0901
4:30 329 287 616 24 15515 0960 0915 4 156 15 934889
4:45 338 346 684 3 430530 0938 092 430630 0929 0 78
5:00 315 326 641 4 445545 0986 0939 445645 0911 0876
515 316 307 623 5 50600 0 975 0922 5 007 O 0 904 856
530 299 305 604 6 515615 0920 0894
545 316 286 602 7 530630 0932 0859
600 301 334 635 8 545645 0911 0840
615 342 270 612 9 600700 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 00500 954 0958 4 006 00 0907 0927
4:30 355 312 667 2 4 1555 0 983 0976 4 156 15 937 0939
4:45 343 321 664 3 4 305 30 0 957 0956 4 306 30 0 928 0 929
5:00 352 299 651 4 445545 0952 0942 445645 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 15615 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
4:005:00 4:155:15 4:305:30 4:455:45 5:006:00 5:156:15 5:306:30 5:456:45 6:007:00
Time Period
Figure B7. University Blvd. PM peak hour factor, northbound (outbound) flow
1.000
 
Peak Period Factor
0.8( E of Univ, Wed. 10/6/04
EE 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
       
AW 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
4:006:00 4:156:15 4:306:30 4:456:45 5:007:00
Time Period
Figure B8. University Blvd. PM peak period factor, northbound (outbound) flow
Table B4. 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
1Hour PHF
Period NB SE
1 400500 0931 0900
2 415515 0972 0957
3 430530 0984 0986
4 445545 0947 0959
5 500600 0935 0941
6 515615 0870 0909
7 53030 0817 0956
8 545645 0823 0922
9 60 700 0887 0883
1 400500 0929
2 415515 0953
3 430530 0963
4 445545 0976
5 500600 0966
6 515615 0907
7 530630 0836
8 545645 0818
9 600700 0900
1 400500 0935
2 415515 0944
3 430530 0961
4 445545 0966
5 5 000 0956
6 515615 0911
7 530630 0804
8 545645 0874
9 600700 0852
2Hour PPF
Period NB SE
400600 0899 0917
415615 0886 0924
430630 0873 0925
445645 0031 0096
500700 0791 0858
400600 0925 0913
4 156 0913 0913
430630 0895 0918
445645 0861 0929
500700 0 11 0909
400600 0927 0908
415615 0926 0923
430630 0902 0919
445645 0864 0919
500700 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
1Hour PHF 2Hour PPF
thbound Both Period NB SB Period NB E
316 1 400500 0864
393 2 415515 0935
398 3 4 30530 0959
451 4 445545 0945
444 5 5 0600 0924
456 6 515615 0867
486 7 530630 0811
410 8 545645 0850
334 9 6 007 00 0919
347
303
291
1837
327 1 4 00500 076
363 2 455515 0926
370 3 4 30530 0929
423 4 445545 0960
411 5 00600 0960
443 6 515615 0903
425 7 530630 0862
422 8 545645 0825
310 9 6 00700 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 400500 0925
366 2 415515 0939
406 3 4 3 30530 0962
400 4445545 0964
425 5 5 00600 0970
432 6 515615 0930
440 7 5 30630 0865
410 8 545645 0851
3559 6 00700 0865
317
314
242
0 1707
Lowest 0811
Highest 0970
Average 096
4 00600 063
4 15615 0867
43030 0855
445645 0031
5 007 0 00790
400600 0 89
4 156 0894
4 306 30 0878
445645 0873
5 007 00 0023
400600 0912
4 15615 0919
4 306 30 0905
45645 0879
5 007 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
1Hour PHF 2Hour PPF
Penod NB SB Period NB S
k500 0853 0923 400600
5 15 0912 0960 4 156 15
530 0922 0886 430630 C
545 0920 0094 456 45
60 0895 0864 5 00700
6 15 0833 0874
630 0781 0931
5645 0860 0944
700 0896 0956
5 00 0829 0930 4 00600 871 0 917
515 0 881 0955 415615 0873 0934
5 30 0945 0952 4306 30 0864 0918
545 0953 0955 4 456 45 857 0930
600 0946 09555 00700 0 804 0931
6 15 0900 0965
630 0821 0930
645 0816 0950
70 0873 0953
500 0841 0978 4 00600 0869 0921
5 15 0914 0924 4 156 15 885 0925
530 0955 0916 4 30630 0884 0936
545 0957 0939 4456 45 0862 0939
60 00 923 0946 5 007 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
4:005:00 4:155:15 4:305:30 4:455:45 5:006:00 5:156:15 5:306:30 5:456:45 6:007:00
Time Period
Figure B9. SR A1A S. PM peak hour factor, southbound (outbound) flow
   
Tues, 4/4/06
&Wed, 4/5/06
Thur, 4/6/06
4:006:00
4:156:15
Peak Period Factor
4:306:30
Time Period
Figure B10. SR A1A S. PM peak period factor, southbound (outbound) flow
1.000
0.900
0.800
0.700
4:456:45
5:007:00
Table B5. 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
1Hour
Period
1 4:005:00
2 4:155:15
3 4:305:30
4 4:455:45
5 5:006:00
6 5:156:15
7 5:306:30
8 5:456:45
9 6:007:00
1 4:005:00
2 4:155:15
3 4:305:30
4 4:455:45
5 5:006:00
6 5:156:15
7 5:306:30
8 5:456:45
9 6:007:00
1 4:005:00
2 4:155:15
3 4:305:30
4 4:455:45
5 5:006:00
6 5:156:15
7 5:306:30
8 5:456:45
9 6:007:00
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
2Hour PPF
Period SB NB
4.006:00 0.853 0.933
4:156:15 0.827 0.913
4,306:30 0 807 0.905
4:456:45 0.764 0.892
5 007:00 0 733 0.900
4'006:00 0 878 0.874
4.156:15 0876 0.931
4'306:30 0 863 0.931
4:456:45 0845 0.916
5.007:00 0813 0.893
4006:00 0847 0.873
4:156:15 0852 0.882
4:306:30 0 942 0.848
4'456:45 0816 0.834
5:007:00 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
ATues, 4/24/01, PGA
Wed, 4/25/01, PGA
0Thur, 4/26/01, PGA
0.700~. 
4:005:00 4:155:15 4:305:30 4:455:45 5:006:00 5:156:15 5:306:30 5:456:45 6:007:00
Time Period
     
Tues, 4/24/01, C orona       
T hu r, 4/2 6/0 1 P G A        
Time Period
Figure B11 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
ATues, 4/24/01, PGA
Wed, 4/25/01, PGA
0Thur, 4/26/01, PGA
0.700
Figure B12. SR A1A N. PM peak period factor, southbound (outbound) flow
4:006:00 4:156:15 4:306:30 4:456:45 5:007:00
Time Period
Table B6. 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
1Hour PHF
Period SB NB
1 400500 0933 0922
2 415515 0972 0847
3 430530 0903 0089
4 445545 0992 0926
5 500600 0994 0927
6 515615 0976 0 90
7 530630 0946 0B7
8 545645 0934 0872
9 600700 0937 0919
1 400500 0959
2 415515 0968
3 430530 0968
4 445545 0986
5 500600 0976
6 515615 0960
7 30630 0957
8 545645 0964
9 600700 0927
1 400500 0909
2 415515 0944
3 430530 0956
4 445545 0944
5 500600 0953
6 515615 0952
7 530630 0923
8 545645 0931
9 600700 0919
2Hour PPF
Period SB NB
400600 0955 0873
415615 0966 0855
430630 0953 0543
445645 0959 0829
500700 0937 0805
400600 0968 0905
415615 0961 0097
430630 0956 0906
44564 50966 0883
500700 0943 0644
400600 0910 0002
415615 0935 0866
430630 0918 0865
445645 0914 0050
500700 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
1Hour PHF
rthbound Both Period SR N
362 743 1 400500 0942 08
340 776 2 415515 0945 09
369 704 3 4 30530 0935 9
424 869 4 4455 45 0930 9
408 794 5 500600 0928 0 9
379 798 6 515615 0918 08
398 803 7 5 306 30 074 00
340 786 8 545645 0842 08
314 682 9 6 00700 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 400500 0986 08
S4 15515 0944 08
3 430l530 0951 9
1 445545 0938 0 9
5 500600 0940 03
6 515615 0928 0
7 5 306 30 0905 8
8 545645 0887 08
9 6 007 00 089 07
1 400500 0913 068
2 415515 0931 08
3 430530 0937 08
4 445545 0945 09
5 500600 0975 09
6 515615 0942 09
7 530630 0911 0 8
8 545645 0879 09
9 600700 0931 08
2Hour PPF
3 Period SB NB
81 400600 0934 0890
09 415615 0930 0 876
32 430630 0904 0063
49 4456 45 0885 834
34 500700 0841 0816
99
45
98
61 400600 0 933 03867
97 415615 030 0 84
32 4 30630 0 922 867
43 4 456 45 0906 0 838
99 500700 0801 0793
96
30
50
96
42 400600 0939 0042
40 415615 0931 0839
54 4 306 30 0 919 0821
00 4456 45 0901 0 08
23 500700 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
1Hour PHF 2Hour
id Both Period SB NB Penod
669 1 400500 0964 0911 400600 1
614 2 4 155 15 0920 0925 4 156 15 [
661 3 430530 0924 0970 430630 [
684 4 445545 0930 0949 4 456 45 1
724 5 5006 00 0934 0913 500700 1
683 6 5 15615 0970 0868
701 7 5 30630 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 0061
35 15 0959 0936 4 156
5 30 0962 0972 4306:
545 0966 0935 4456.
600 0962 0943 5007
6 15 0954 0917
630 0907 0866
645 0889 0837
7 00 0875 0904
500 0954 0941 4 006
5 15 0 950 0874 4 156
5 30 0963 0878 4306
545 0974 0076 4456.
5600 0952 0914 5007
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 B7. 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
3Day
Minimum
PHF
U. 81
0.784
0.811
0.815
0.817
0.820
0.825
0.842
0.842
0.909
3Day 3Day
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 CYCLEPERIOD DELAY EXAMPLE:
Table C1. Generalized example of cycleperiod 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 & DXparrenhrupuF 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
rdT 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
Sr 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 ?TH
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 C1. 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
IFL 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 C1. 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 C1. 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 C1. 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 C1. 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 C1. 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 C1. 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 C1. 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 C1. 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 C1. 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 C1. 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 C1. 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 C1. 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
01 741
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 C1. 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 C1. 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 C1. 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 C1. 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 C1. 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 26
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 C1. 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 C1. 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 C1. 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
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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 statecertified unlimited electrical contractor in Florida and Georgia
Mr. Buckholz is a senior level traffic engineer with 26 years of wideranging 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.
PAGE 1
1 REALTIME 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 RealTime 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 NONVISIBLE DELAY............................................................................... 68 Data Analysis Programs......................................................................................................... 68 Prediction Algorithm fo r NonVisible Delay ......................................................................... 80 NonVisible Queue Estimation Technique...................................................................... 80 NonVisible Queue Adjustment Technique:................................................................... 82 NonVisible Queue ReAdjustment 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 TrajectoryBased 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 OVERSATURATED FLOW................................................ 276 Simplified Example of CyclePe riod Issues in Calculating d3.............................................276 Residual Queue Discrepancy................................................................................................281 Detailed Example of CyclePe 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: RealTime 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 CYCLEPE RIOD DELAY EXAMPLE: ................................................. 353
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7 REFERENCES............................................................................................................................376 BIOGRAPHICAL SKETCH.......................................................................................................381
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8 LIST OF TABLES Table page 41 Example summary volum e a nd capacity....................................................................... 122 42 Example summary queue discha rge, delay check and goodnessoffit .........................123 43 Queue prediction........................................................................................................... ...124 44 Stopped delay prediction..................................................................................................125 45 Control delay prediction.................................................................................................. 126 46 Comparison of variation in act ual and predicted stopped delay ...................................... 127 47 Pvalue determination for difference in median values................................................... 128 61 Calculation of cumula tive curve delay conversion factors, volum e pattern 625_700_650_350vph......................................................................................................249 62 Calculation of cumula tive curve delay conversion factors, volum e pattern 700_725_625_350vph......................................................................................................251 63 Calculation of cumula tive curve delay conversion factors, volum e pattern 700_700_700_350vph......................................................................................................253 64 Calculation of cumula tive curve delay conversion factors, volum e pattern 725_700_700_350vph......................................................................................................255 65 Cumulative curve delay fo r standard 4period case ......................................................... 257 66 Cumulative curve delay w ith m ultiple visible periods.................................................... 258 67 Stopped delay prediction re sults for 700_725_625_350vph volum e pattern..................259 68 Average stopped delay predicti on results for 700_725_625_350vph volume pattern .... 262 69 Stopped delay prediction re sults for 700_700_700_350vph volum e pattern..................263 610 Average stopped delay predicti on results for 700_700_700_350vph volume pattern .... 266 611 Stopped delay prediction re sults for 725_700_700_350vph volum e pattern..................267 612 Average stopped delay predicti on results for 725_700_700_350vph volume pattern .... 270 613 Stopped delay prediction re sults for 625_700_650_350vph volum e pattern..................271 614 Average stopped delay predicti on results for 625_700_650_350vph volume pattern .... 274
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9 615 Prediction comparison.....................................................................................................275 71 Generalized example of cyclepe riod delay discrepancies data .................................... 292 72 Generalized example of cyclepe riod delay discrepancies summary ...........................293 73 Detailed example of cycleperiod delay di screpancies, residual queue determ ination... 296 74 Detailed example of cycleperiod de lay discrepancies, delay com parison...................... 297 75 Detailed example of cycleperiod delay discrepancies, delay com parison with modified d2 term.............................................................................................................. 297 76 Detailed example of cycleperiod delay discrepancies, delay com parison with d3 adjustment..................................................................................................................... ...297 B1 US 1 machine counts (Southern St. Johns County)......................................................... 334 B2 US1 Machine counts (northern St. Johns County)........................................................... 339 B3 Atlantic Boulevard machine counts................................................................................. 342 B4 University Boulevard mach ine counts (Jacksonville)...................................................... 345 B5 SR A1A machine counts (Crescent Beach)..................................................................... 348 B6 SR A1A machine counts (Ponte Vedra) PDF 17 KB ...................................................... 351 B7 Appendix B data summary...............................................................................................352 C1 Generalized example of cyclepe riod delay discrepancies data. ..................................354
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10 LIST OF FIGURES Figure page 41 Queue relationships........................................................................................................ ....94 42 Signalized intersection delay components.........................................................................95 43 Measured versus estimated delay.......................................................................................96 44 Visible and nonvi sible variables ....................................................................................... 97 45 Relationship between v/c ratio and rati o of control delay to stopped delay ...................... 98 46 Requeuing that result s in sim ultaneous queues................................................................ 99 47 Requeuing that does not re sult in simultaneous queues ................................................. 100 48 Example of a blind period................................................................................................ 101 49 Example of adjacent blind periods................................................................................... 102 410 Counters and queue status................................................................................................103 411 Base case for P, C and X; stopped delay comparison...................................................... 104 412 Effect of increasing the power constant on stopped delay com parison........................... 105 413 Queue propagation example............................................................................................ 106 414 Actual vehicle queues..................................................................................................... .107 415 Average queue length comparison................................................................................... 108 416 Maximum queue length comparison................................................................................ 109 417 98th percentile back of queue comparison....................................................................... 110 418 Vehicle requeuing........................................................................................................ ...111 419 Stopped delay comparison............................................................................................... 112 420 Stopped delay prediction, 12 FOV...................................................................................113 421 Comparison of actual and predicted stopped delay......................................................... 114 422 Adjacent blind period counter v. stopped delay............................................................... 115 423 Control delay comparison................................................................................................ 116
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11 424 Ratio of control delay to stopped delay........................................................................... 117 425 Graphical control delay comparison,............................................................................... 118 426 Control delay estimates................................................................................................... .119 427 Control delay composition............................................................................................... 120 428 Ratio of control delay to stopped plus m oveup delay.................................................... 121 51 Cumulative arrivaldepartur e curves and overflow delay ................................................ 207 52 Critical time and volume points for period 4................................................................... 208 53 Overflow delay in period 4..............................................................................................209 54 Maximum reasonable cumulative arrival curve............................................................... 210 55 Minimum reasonable cu mulative arrival curve ............................................................... 211 56 Minimum overall reasonable cum ulative arrival curve................................................... 212 57 Minimum reasonable cumulativ e arrival cu rve (minimum V4 for minimum V1 and V2)....................................................................................................................................213 58 Minimum reasonable cumulativ e arrival cu rve (minimum V4 for minimum V1)...........214 59 Period 1 delay for the upper bound.................................................................................. 215 510 Period 2 delay for the upper bound.................................................................................. 216 511 Period 3 and period 4 delay for the upper bound.............................................................217 512 Reasonable overflow delay region for 600 vph capacity and 0.75 m inimum PHF......... 218 513 Reasonable overflow delay region for 600 vph capacity and 0.80 m inimum PHF......... 219 514 Reasonable overflow delay region for 600 vph capacity and 0.85 m inimum PHF......... 220 515 Maximum delay estimation error for 0.75 minimum PHF.............................................. 221 516 Maximum delay estimation error for 0.80 minimum PHF.............................................. 222 517 Maximum delay estimation error for 0.85 minimum PHF.............................................. 223 518 Maximum reasonable cumulative arri val curve with period 1 visible ............................. 224 519 Minimum reasonable cumulative arri val curve w ith period 1 visible............................. 225 520 Maximum reasonable cumulative arrival curve with 5 periods ....................................... 226
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12 521 Minimum reasonable cumulative arrival curve w ith 5 periods....................................... 227 61 Trajectory example A) Complete chart B) Detailed view of circled area in upper right corner. .................................................................................................................. ....244 62 Cumulative arrivalde parture curve exam ple................................................................... 246 63 Trajectory conversion of cumulative curve example....................................................... 247 64 Delay and travel time components................................................................................... 248 71 Cycle v. period initia l queue delay analysis..................................................................... 294 72 Cycle v. period "control delay" analysis.......................................................................... 295 73 Upward bias in HCM residual queue calculation ............................................................ 298 A1 Queue discharge headway histogram............................................................................... 309 A2 Startup lost time histogram............................................................................................. 310 A3 Comparison of control delay and sto pped delay by cycle length (g/C =0.30) ................. 311 A4 Comparison of control delay and stopped delay (g/C =0.30) .......................................... 312 A5 Comparison of control delay and stopped plus queue m oveup delay by cycle length (g/C = 0.30)......................................................................................................................313 A6 Comparison of control delay and stopped delay plus queue m oveup delay (g/C =0.30)...............................................................................................................................314 A7 Relationship between v/c ratio and stopped delay...........................................................315 A8 Relationship between v/c ratio and stopped delay by cycle length ................................. 316 A9 Relationship between v/c ratio a nd stopped plus queue m oveup delay.........................317 A10 Relationship between v/c ratio and st opped plus queue m oveup delay by cycle length................................................................................................................................318 A11 Relationship between v/c ratio and control delay............................................................319 A12 Relationship between v/c ratio a nd control delay by cycle length................................... 320 A13 Relationship between vehicle requeues and control delay ............................................. 321 A14 Relationship between v/c ra tio and vehicle requeues ..................................................... 322 A15 Relationship between v/c ratio a nd vehicle requeues by cycle length ...........................323
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13 A16 Relationship between v/c ratio and cycles with phase failure .........................................324 A17 Relationship between v/c ratio and cycl es with phase failure by cycle length ................ 325 A18 Percentage of cycles in 1 hour with phase failure by cycle length .................................. 326 A19 Percentage of cycles in 1 hour with phase failure ............................................................ 327 A20 Linear relationship betw een ABPC a nd stopped delay....................................................328 A21 Exponential relationship betw een ABPC and stopped delay ........................................... 329 A22 Relationship between ABPC and control delay............................................................... 330 B1 US 1 S. PM peak hour fa ctor, southbound (outbound) flow ...........................................332 B2 US 1 S. PM peak period f actor, southbound (outbound) flow......................................... 333 B3 US 1 N. PM peak hour factor, northbound (outbound) flow........................................... 337 B4 US 1 N. PM peak period factor, northbound (outbound) flow........................................ 338 B5 Atlantic Boulevard PM peak hour factor, eastbound (outbound) flow ............................ 340 B6 Atlantic Boulevard PM peak pe riod factor, eastbound (outbound) flow .........................341 B7 University Blvd. PM peak hour factor, northbound (outbound) flow .............................343 B8 University Blvd. PM peak peri od factor, northbound (outbound) flow .......................... 344 B9 SR A1A S. PM peak hour factor, southbound (outbound) flow ...................................... 346 B10 SR A1A S. PM peak period factor, southbound (outbound) flow ...................................347 B11 SR A1A N. PM peak hour factor, southbound (outbound) flow ..................................... 349 B12 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 REALTIME 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 oversaturated 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 realtime procedur e for estimating delay during oversaturated 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 secondbysecond 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
PAGE 15
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 oversatur ated periods preceded by an initial queue. A remedial modification to the d2 term is proposed. Finally, it is demonstrated that the HCM s periodbased queue accumulation procedure has drawbacks that can produce su bstantial errors in delay duri ng oversaturated conditions. A remedial cyclebased 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 realtim 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 nontechnica l people. As recognized by Dowling [3], many MOEs (such as queue lengt h, speed, stops, and density) are relatively invariant during highly oversaturat 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 retiming 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 laborintensive 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 oversaturated 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 oversaturation 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 nonstationary 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 ebycycle 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 oversaturated conditions. Resulting delay data could then be used for proj ect evaluation or for realtime modification of controller settings. Using realtime delay obtained from intersec tionbased 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 reidentification 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 tersectionbased 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 ealtime manner to obtain r ealtime 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 workedout 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 wellrecognized 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 moveup 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 15minute periods. In reality, conditions change on a cyclebycy cle basis depending on random fluctuations in approach volumes and driver composition. For example, the considerable variation in cyclebycycle 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 cyclebycycle 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 sitespecific 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 oversaturated conditions. In order to properly measure delay during oversaturated conditions, multiper 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, moveup 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 realtime with limited information, and being able to do so even with oversaturated 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 costprohibitive 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, realtime 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 altime 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 undersaturated and oversat 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 (nonvisible 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 realworld 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 oversaturated 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 microsimulation model. CORSIM allows us to quickly simulate a va riety of realworld 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 llaccepted and wellunderstood model that has the cap ability to accommodate a wide range of input variables, including variable combinations that produce grossly oversaturated 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 microsimulation m odels can result in a le vel of variation that masks causeandeffect 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 backofqueue 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 nonvisible delay that
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24 occurred. The procedure deve loped is capable of handling both undersaturated conditions (having little or no blin dness) and oversaturated 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 nonvisi 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 secondbysecond 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 realtime 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 oversaturated 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 multiperiod 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 periodbased 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 cyclebased 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 microsimulation 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 realtime secondbysecond delay estimation procedures for a onehour 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 15minute period for a onehour 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 nonvi sible queue as well as departures from the front of the nonvisible 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 nonvisible queue at each point in time. This pr edicted nonvisible 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 oversaturated intersection approach for a onehour analysis time frame would proceed as follows: 1. Using the vehicle detection equipment for th e approach of interest, realtime secondbysecond 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 nonvisible que ue at every second of the onehour analysis time frame. S econdbysecond 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 realtime traffic control. 4. The time during the last 15minute 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 15minute 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 15minute 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 15minute 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 carriedout 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 oversaturated 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 realtime traffic control and project evaluation. It is e nvisioned that the eventual end product of this theoretical research will be a se lfcontained delay estimation module that could be attached to
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29 either a closedloop 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 undersaturated conditions and for situations where all of the information needed to calculate delay is known. The current state of knowledge with respect to oversaturated conditions is more primitive and the results less tested. RealTime 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 rightofway (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 TimeinQueue Delay (a.k.a. TimeinQueue) 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 speedbased 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). Freeflow 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 GPSe 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 endacceleration distance of 427 feet downstream of the stop bar was established. An average begindeceleration 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 30degree 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 midblock delay from segment delay. The authors cautioned that queue spillback from a downstream intersection or nonrecurring 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 reidentification pr oblem on freeways. A vehicle waveform pair can be formed by us ing one downstream waveform and one upstream waveform. The vehicle reidentif 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 reidentifica 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 reidentify 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 reidentification 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 multiobj ective optimization problem where each objective is ordered according to its importance. MultiObjective 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 multiobjective 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 reidentific 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 loopbased 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 I80 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 reidentification 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 online signal control algorithm to find the optimal green splits. Vehicle reidentificatio 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 2hour 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 delaybased online control algorithms pe rformed better than the offline case for both pretimed 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 reidentification 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 4lane 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 reidentification is license pl ate matching. Other potential methods of vehicle reidentificati 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 turnon 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 rei 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 reidentification rate of over 90% was obtained using multidetector 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 reidentification 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 tiein would improve the accuracy and possibly yield realtim 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, cyclelength based average (CBA) and fixed time av erage (FTA). A fixed interval of 60 seconds was used for FTA. Kmeans 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 withingroup variance B = between group variance A lower Wilks lambda value indicates better clustering. Kmeans 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(ARDiAADi/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 singleloop detectors while all of the tests can be applied to dualloop 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 fieldsofview 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 worstcase 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 userdefined 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: nonoptimum camera placement, daytonight 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 redgreen 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 overcounted 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.) overcounted 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=0T=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=1T=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=1T=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=0L=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=1L=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=0T=1) = P(L=1T=1) x P(V=0L=1) + P(L=0T=1) x [1P(V=1L=0)] The probability of video detection being on when a vehicle is not present = P(V=1T=0) = P(L=1T=0) x [1P(V=0L=1)] + P(L=0T=0) x P(V=1L=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 volumedensity control). A VIVDS system is sometimes used to provide advance detection on highspeed 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 = distancetoheight 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 42 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: adjacentlane, samelane and crosslane. 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. AdjacentLane Occlusion (Horizontal Occlusion) occurs when the blocked and blocking vehicles are in adjacent lanes, which can result in false detections in adjacent lanes. Table 41 of the paper provides minimum required camera heights to reduce adjacentlan 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. SameLane 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. CrossLane 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 onehalf 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 imageproces 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 highcontrast 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 (MAHPT) lv lv* = (lvlro) + 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 rightt urning vehicles from the major road whose headlights sweep across the detection zone. It also eliminates false calls due to crosslane occlusion caused by tall vehi cles on the major road. A lens adjustment module is an essential VIVDSrelated 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 RG59 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 onsite 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 I75 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: selfcorrecti 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) InputOutput (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 steadystate 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, steadystate models are usef ul for predicting delays only at lightly loaded intersections. Hurdle added: there is one group of models the steadystate 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 nonconstant 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 startup 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 atgrade 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(n1)/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 NonConstant Acceleration Calibrated Stop Line Speed Model: Stop Line Speed for vehicle n = Vsl(n) = Vmax (1 enk) 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(n1)/Amax] 0.0086v 0.23AGI Calibrated Minimum Discharge Headway Model: Minimum Headway = H = T + d/Vmax 0.0086v 0.23AGI Calibrated StartUp Lost Time Model: StartUp 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 atgrade 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 (TRAFSIM) data were used to validate the oversaturation 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 variancetomean 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 moveup 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 = (1P)fPA/[1g/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[(X1) + {(X1)2+8kIX/Tc}1/2] I = upstream filtering/metering factor obtained from Exhibit 157 of 2000 HCM k = incremental delay factor obta ined from Exhibit 1613 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 oversaturated conditions at signalized intersections. The delay equations calculate delays consistent with the more accurate pathtrace met hod of delay measurement rather than the less accurate (but easier to carryout) queuesamp ling method. Delays estimated by the proposed generalized model were in close agreem ent with those simulated by TRAFNETSIM. The pathtrace 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 queuesampling 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 pathtrace 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. TRAFNETSIM calculates delay by subtracting the freeflow 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 oversaturated conditions. The following TRAFNETSIM input values (rep resentative of oversaturated 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 TRAFNETSIM) 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 60second 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 (TRAFNETSIM 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 oversaturation 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 fivelane 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. StartUp 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: StartUp Lost Time = SULT = SRT + 4*(H4h) Saturation Headway = h = (TNT4)/(N4) Average Headway = Hi = (Ti Ti4)/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 StartUp 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 offpeak 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 offpeak thrus and 1.3 for offpeak 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 startup 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 carfollowing system to examine the effects of lane changing and a heterogeneous vehicl e mix on queue discharge headways. In the Pitt carfollowing model, the first ve hicle in the queue begi ns to move across the stop line after the lost time (startup delay) has expired. The second vehi cle in the queue then responds to the motion of the leader through the carfollowing system with no additional explicit lost time added. The effect of lost time on subsequent vehicles is modeled through the sluggishness of the carfollowing 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 startup 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 tUp 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 carfollowing 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 fronttofront time headway curve and th e reartofront 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 startup 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 shockwave an alysis and that highlighted the common errors that are made with regard to delay estimation us ing shockwave analysis. The authors point out that the main difference between shockwave 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 shockwave 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 shockwave analysis. The authors show that the size of the queue obtained from shockwave 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, PerezCartagena and Tar ko [34] demonstrated that, ba sed on studie s conducted in Indiana, town size and lateral la ne location (rightmost 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 constantvalue 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 oversaturat ed as well as undersaturated 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 overcapacity 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 undersaturated, 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 workedout 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 freeflow 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 reidentif 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 realworld probl em of queues that extend beyond the detection system for some period of time; ei ther shortlived queues that occur during undersaturated conditions because of spurts in activity or longerlived, recurring queues that occur during oversaturated 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 oversaturated c onditions, represent a deviation from the research conducted to date.
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68 CHAPTER 4 ESTIMATING NONVISIBLE DELAY This chapter describes the m ethodology that wa s established to pred ict nonvisible 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 oneway 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 onehour analysis time frame. The intersection was controlled by a 2phase semiactuated traffic signal and delay data were co llected and analyzed only for the actuated side street approach. Goodne ssoffit testing usi ng the chisquare 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 secondby
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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 secondbysecond 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 41. It is important to recognize that the back of queue is not itself a length, but rather a position. As is shown in Figure 41, 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 timedistance 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 realworld 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 secondbysecond basis, es timate the nonvisible 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 zerovelocityzeroacceleration 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 moveup 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 42 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 realtime es timate of stopped delay. By applying an appropriate factor (such as the commonlyused 1.3 value) or range of factors, we then scaleup the stopped delay estimate to obtain a reasonably a ccurate realtime 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 moveup time that occurs within the limits (or field of view ) of the detection system. Any stopped delay or queue moveu 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 43). 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 nonvisible 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 nonvisible queue (see Figure 44) and the period of time during which nonvisible 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 45 summarizes the relationship between this ratio and both the v/c ratio and cycle length for oversaturated conditions. For each CORSIM run, a certain Field of View (FOV) was assumed. Measured visible lockedwheel 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 lockedw heel stopped delay assuming an infinite FOV. Finally, the predicted stopped delay was factoredup 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 presigna 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 postsignal 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 freeflow speed (wit h freeflow 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 freeflow 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 overcapacity 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 shockwave 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 requeuing 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 46. 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. Requeuing 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. Requeuing is a necessary condition for the formation of simultaneous queues; however, it is not a sufficient one. As show n in Figure 47, requeuing may not result in the formation of simultaneous queues. Unusual or atypical events can also result in phase failures and associated requeuing. 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 gapout 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 requeue 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 requeue 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 requeues 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 fillsup 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 nonvisible queue. Figure 48 provides an example of a blind period. During this blind peri od, vehicles attach themse lves to the end of the nonvisible 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 nonvisible queue, but cannot leave the front of the nonvisible 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 T7 to time T34 in Figure 48). 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 nonvisible queue from the front while they continue to attach to the back of the nonvisible 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 nonvisibl e queue and leave the front of the nonvisible queue, as the falling queue blind pe riod (which occurs from time T34 to time T72 in Figure 48). The length of the nonvisible 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 T72 in Figure 48). A review of the Figure 48 example reveals that the nonvisible 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 T50). However, because of th e limited FOV, we cannot be cer tain that the nonvisible queue has dissipated until time T72. 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 48 example, 5 seconds), and if another queue does not fill the FOV prior to this 5second 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 nonvisible queue (i.e., the nonvisible queue has been flu shed out). However, if this 5second 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 49). As one might expect, the problem of estimating the length of nonvisible 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 nonvisible queue a nd its associated stopped delay. The nonvisible 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 410. 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 carryingout 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 onehour analysis time frame having four 15minute periods, a secondbyseco nd tabulation of items such as queue length, back of queue position, stopped delay, moveup delay and control delay. It also provides a host of ancillary capabilities, including automated calculation of: startuplosttime, saturation flow, and capacity by cycle; HCM queuing and delay information by 15minute period; and arrival type by 15minute period. In addition, BuckQ allows evalua tion of arrival patterns using a chisquared goodnessoffit test and provides ex tensive graphing capabilities. However, the most important feature of BuckQ is its abil ity to accommodate secondbys 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 secondbysecond basis for the entire 3600second (60minute) 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 nearsaturated and oversaturated 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 oversaturated conditions. Prediction Algorithm for NonVisible 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 nonvisible 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 nonvisible queue. NonVisible 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 nonvis 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 nonvisible 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 nonvisible 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 nonvisible que ue passed some time ago or, conversely, it may be that there are more vehi cles in the nonvisible 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 fivesecond 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 nonvisible 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 5second 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 fillsup 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 nonvisible queue length estimations to be ma de that depend on previous nonvisible 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 nonvisible queue during the blind period. A second adjustment technique is needed to augment the initial estimation technique when this occurs. NonVisible 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 oversaturated 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 411 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 412. 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. NonVisible Queue ReAdjustment Technique: ReAdjusted 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 413 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 overcapacity conditions at some point during the onehour 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 B29. Tables 41 and 42 summarize the characteris tics of these examples while Tables 43 through 45 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 15minute volumes that were input into CORSIM. Considering the first row, random number set 1 was used and the 15minute input volumes were 625 vph, 700 vph, 650 vph and 350 vph. The input volume for the last 15minute period was always set at a re latively low value so that all residual queues would clear by the end of the onehour 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 41. 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 41 are the approach capacity values for each 15minute 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 15minute period, the needed intermediate variables such as queue discharge Headway (H), StartUp Lost Time (SULT), and effective green time (g). The Extens ion of Effective Green (EEG) is determined for the first 45minutes of the hour by minimizing the sum of the squared deviations between the cyclebycycle capacity values calculated using the Highway Capacity Manual procedure and actual cyclebycycle thruput. A review of Tabl e 41 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. Volumetocapacity ratios are calculated for each 15minute period and for the first 45 minutes of the hour. These values are also provided in Table 41. For individual 15minute 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 45minutes, 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 42 summarizes va rious values used for capacity analysis, including cycle length, green time queue discharge headway, satu ration flow rate, and startup lost time. Our analysis procedure calculates th ese values on both a cyclebycycle basis and a 15minute 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 artup 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 startup lost time value of 2.7 seconds is significan tly greater than the 2.0 second m ean startup dela y input into CORSIM. The difference is due to a definition inconsistency. CORSIM only applies the mean startup 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 startup lost time. The next section of Table 42 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 42 summarizes, fo r the Poisson distribution, the chisquare goodnessoffit test results based on 20second arrival intervals. During only one of the fortyeight 15minute 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 20second 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 43 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 414 depicts actual queue length as a function of v/ c ratio while Figures 415 through 417 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 414 through 417 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 415 through 417 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 417 indicates that the HCM procedures grossly overestimate the 98th per centile back of queue. Also provided in Table 43 is information on the number of (liberal) phase failures, the percentage of cycles experiencing a phase failure and the number of vehicle requeues. 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 requeue, or if 100 vehicles are forced to requeue. 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 418 demonstrates that th e number of vehicle requeues te nds to increase linearly as a function of v/c ratio. Stopped Delay Prediction Table 44 summ arizes the stoppe d delay prediction results for our analysis procedure as compared to actual stopped delay. Figure 419 i ndicates that the proced ure does a pretty good job of predicting stopped de lay over all v/c ratios. Figure 420 shows the relative contribution of each segment of the prediction methodology. For the examples under considerati on, visible delay makesup 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 nonvisible 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 oversaturated 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 rsaturated 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 nonvisible 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 421 provides another way of visualizing the final predictive results. The maximum individual overestimation of delay is 27% and the maximum individual underestimation is 17.5%. If the results are aver aged over the three random number replicates, as is documented at the bottom of 44, the ma ximum overestimation is 13% and the maximum underestimation 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 422, 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 45 s ummarizes the control delay prediction results as compared to actual control delay. Figure 423 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 424. (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 423 is control delay as predicted by HCM procedures. The HCM procedures tend to overpred ict control delay for th e lower oversaturated v/c ratios. Figures 425 and 426 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 moveup delay. As shown in Figure 427, 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 moveup 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 requeuing associated with high v/c ratios produces this steady and dramatic increase in queue move up delay. Figure 428 provides factors that convert stopped delay plus queue moveup 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 B30. 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 startup 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 46 provides a comparison between the actual 1hour 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 46, 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 nonpa 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 47 contains the test, which produces a pvalue of about 0.11 The pvalue 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 rsaturated 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 nonvisible queue. A sleeper is defined as a motorist that does not exhibit normal carf 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 invehicle 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 nearsaturated conditions. For oversaturated 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 microsimulation 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 41. Queue relationships
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95 Figure 42. Signalized inte rsection delay components
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96 Figure 43. Measured versus estimated delay
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97 Figure 44. Visible a nd nonvisible 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 45. Relationship between v/c ratio and ratio of cont rol delay to stopped delay
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99 Figure 46. Requeuing that re sults in simultaneous queues
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100 Figure 47. Requeuing that does not result in simultaneous queues
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101 Figure 48. Example of a blind period
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102 Figure 49. Example of adjacent blind periods
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103 Figure 410. Counters and queue status
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104 Figure 411. Base case for P, C and X; stopped delay comparison
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105 Figure 412. Effect of increasing the power constant on stopped delay comparison
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106 Figure 413. Queue propagation example
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107 Figure 414. Actual vehicle queues
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108 Figure 415. Average queue length comparison
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109 Figure 416. Maximum queue length comparison
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110 Figure 417. 98th percentile back of queue comparison
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111 Figure 418. Vehicle requeuing
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112 Figure 419. Stopped delay comparison
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113 Figure 420. Stopped delay prediction, 12 FOV
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114 Figure 421. Comparison of act ual and predicted stopped delay
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115 Figure 422. Adjacent blind period counter v. stopped delay
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116 Figure 423. Control delay comparison
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117 Figure 424. Ratio of control delay to stopped delay
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118 Figure 425. Graphical c ontrol delay comparison,
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119 Figure 426. Control delay estimates
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120 Figure 427. Control delay composition
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121 Figure 428. Ratio of control de lay to stopped plus moveup delay
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122Table 41. 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 VolumetoCapacity 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 42. Example summary queue di scharge, delay check and goodnessoffit ActualCORSIMCORSIMCORSIMActualCORSIM File NameRandomCycleGreen DischargeSat FlowLo stExt. ofStoppedStopQueueControlControlDelay 15 min input volumesNumber SetLengthTime g/C ratio HeadwayRateTimeGreenDelayTimeDelayDelayDelayTime CGg=G+EEGSULTHSSULTEEG 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 ChiSquare Test Statistic Queue Discharge Data, Average Over All Cycles 95% Ref. Statistic = 9.49 BuckQ Delay Check GoodnessofFit Test (20sec arrival intervals) sec/veh
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124Table 43. Queue prediction Actual% ofActual File NameRNVol PhaseCyclesVehicle 15 min input volSet45minv/cAP%ErrAP%ErrAP%ErrAP%ErrHCM%ErrFailw/ PFReQ'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 44. 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 45. 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 DelayMoveUp Delay Actual DelayMoveUpMoveUpDecel. 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 46. Comparison of variation in actual and predicted stopped delay Cumulative 1Hour 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 47. Pvalue determination fo r difference in median values Fisher SignTest For paired replicates Null hypothesis: Differences between actual and predicted median delay is zero Cumulative 1Hour 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 secondbysecond 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 realtime 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 wellaccepted procedure for calculating pervehicle 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 oversaturation 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 moveup 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 oversaturated 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 oversaturated 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 undersaturated flow that causes the queue to dissipate before the hour e xpires. Let us also assume that the component 15minute flow rates follow a reasonable pattern that result in some minimum Peak Hour Factor (PHF). Figure 51 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 oversaturated 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 onehour 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 mostlikely 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 15m inute period (period 4) of the hour can be calculated: AR4 = (CA60CAC)/(T60TC) (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 52, 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 15minute 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 15minute time period (sec) Continuing the Figure 52 example, the cumulativ e number of arrivals at the beginning of the last 15minute 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 15minute period (see Figure 53): 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 (vehsec) UDR4 = Uniform Departure Rate during period 4 (veh/sec) AR4 = Arrival Rate during period 4 (veh/sec) t4S = Duration of oversaturated flow during 4th 15min 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 vehsec 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 15minute 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 15minute periods must equal th e arrival rate over the entire 1hour 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 15minute 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 54. 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 15minute 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 15min 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 55. 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 56. 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 = (41/0.75)(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 57). 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)(21/0.75) 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 58). 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 59 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 inperiod delay for period 1 (Dp1) and the outofperiod 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 inperiod delay actually occurs during period 1, th e outofperiod delay occurs during period 2. For period 1, the inperiod 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 vehsec The outofperiod 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 outofperiod 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 vehsec Figure 510 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 oversaturation delay for period 2 (Dp2), the outofperiod oversaturati on delay for period 2 (Dc2), the inperiod initia l queue delay for period 2 (DIQA2) and the outofperiod 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 inper iod delay and inperi od initial queue delay actually occur during period 2, the outofperi od delay and outofperiod initial queue delay occur during period 3. The inperiod 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 inperiod oversaturation 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 vehsec The outofperiod oversaturation 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 outofperiod ov ersaturation delay can be ca lculated using the following formula: Dc2 = 0.5(TC2A T30)(CA30 UCD30A) (49)
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152 For Figure 510 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 vehsec 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 vehsec An inspection of Figure 510 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 UCD30AUCD30. Consequ