PRIORITY LANES ON URBAN FREFWAYS:
SOME OPERATIONAL CONSIDERATIONS
By
THOMAS HAMILTON CULPEPPER
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVfRSI1Y OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1977
To my wife, Maryanne Gillis Culpepper
"Follow your own bent,
no matter what people say."
ACKNOWLEDGMENTS
The author is grateful to all those who provided assistance and
support during the preparation of this dissertation. A special measure
of appreciation is extended to the members of the supervisory committee,
K. G. Courage, J. A. Wattleworth, and D. D. Wackerly, for their indi
vidual contributions. Their comments and constructive criticisms during
this work were of invaluable assistance. A special acknowledgment is
made for the efforts of the committee chairman, Professor Courage. His
willingness to devote his own time and energies to the development of
this material was a source of encouragement for which the author is
extremely grateful.
The research project which provided the motivation for this work
was sponsored by the Federal Highway Administration and the Florida
Department of Transportation. Their sponsorship of this research is
gratefully acknowledged. Along with these agencies, a special thanks
is extended to the various members of the project staff for their con
tributions to this work.
The author is also indebted to Dr. T. J. Hodgson of the Indus
trial Engineering Department for his many suggestions and comments in
the area of optimization techniques and for providing the source code
for the optimization algorithms which were used in the computerized
models.
Last, but by no means least, the author is deeply grateful to
his wife, Maryanne Culpepper, for her many contributions to this work.
In addition to her continuous personal support, she provided invaluable
editorial assistance, typed the preliminary drafts and coordinated the
typing of the final draft.
TABLE OF CONTENTS
Pa.e
ACKNOWLEDGMENTS.................................................. iii
LIST OF TABLES................................................. viii
LIST OF FIGURES.................................................. xi
ABSTRACT....................................................... xiv
CHAPTER 1. INTRODUCTION.................. ... ............ 1
Background and Rationale..................... 1
Statement of the Problem..................... 6
Scope of the Study........................... 6
Organization.................................. 7
CHAPTER 2. SURVEY OF THE LITERATURE ........................... 8
Introduction ................................. 8
Previous Research ............................ 9
Relevance to This Effort..................... 13
Summary.................. ................... 14
CHAPTER 3. DEVELOPMENT OF A CAR POOL DEFINITION MODEL......... 17
Introduction................................. 17
Development of the Basic Model Structure..... 22
Development of the Equilibrium Model......... 30
Development of the Optimization Submodel..... 40
Solution Methodology ........................ 55
Page
CHAPTER 4. VALIDATION AND APPLICATION OF THE CAR POOL
DEFINITION MODEL ................................... 65
Introduction............................... 65
Validation of the Model ..................... 66
Applications of the Model .................... 80
General Guidelines........................... 86
CHAPTER 5. DEVELOPMENT OF A PRIORITY LANE ENTRY/EXIT MODEL.... 95
Introduction ............... ................. 95
Development of the Model................. ... 103
Solution Methodology... ..................... 119
CHAPTER 6. VALIDATION AND APPLICATION OF THE PRIORITY
LANE ENTRY/EXIT MODEL.............................. 128
Introduction................................. 128
Validation of the Model...................... 129
Applications of the Model .................... 142
General Observations. ......................... 146
CHAPTER 7. A CASE STUDY: THE 195 PRIORITY LANE SYSTEM........ 149
Introduction ................................. 149
Car Pool Definition Analysis................. 160
Priority Lane Entry/Exit Analysis............ 165
Summary of Findings .......................... 169
CHAPTER 8. CONCLUSIONS AND RECOMMENDATIONS.................... 171
Conclusions................................. 171
Recommendations .............................. 175
APPENDIX A. DEVELOPMENT OF DEMANDSPEED RELA1IONSHIPS.......... 178
APPENDIX B. SUMMARY OF THE 195 CAR POOL DEFINITION ANALYSIS... 187
Page
BIBLIOGRAPHY ........................... .... .................... 194
BIOGRAPHICAL SKETCH ............................................ 201
vii
LIST OF TABLES
Table Page
1.1 SUMMARY OF THE CHARACTERISTICS OF
VARIOUS FREEWAY PRIORITY TECHNIQUES....................... 5
2.1 SUMMARY OF PREVIOUS AND PROPOSED I:3iDELS................... 15
3.1 OPTIMIZATION SUBMODEL STRUCTURE........................... 56
4.1 SUMMARY OF THE TEST SYSTEM OPERATING CHARACTER
ISTICS FOR THE CAR POOL MODEL SENSITIVITY ANALYSES........ 70
4.2 SUMMARY OF THE DEMAND CHARACTERISTICS FOR THE
CAR POOL MODEL SENSITIVITY ANALYSES....................... 70
4.3 OPTIMAL SYSTEM OPERATION FOR VARYING
PRIORITY SECTION CAPACITIES................................ 73
4.4 OPTIMAL SYSTEM OPERATION FOR VARYING
NONPRIORITY SECTION CAPACITIES ........................... 74
4.5 OPTIMAL SYSTEM OPERATION FOR VARYING
LEVELS OF VEHICULAR DEMAND .. ............. ................ 75
4.6 OPTIMAL SYSTEM OPERATION F r' VARYING
DEMAND DISTRIBUTIONS ..................................... 79
4.7 TOTAL VEHICULAR AND PASSENGER DEMANDS
FOR COMPARISON OF CARPOOL AND PRIFRE ...................... 81
4.8 TOTAL TRAVEL TIMES PREDICTED BY
CARPOOL AND PRIFRE MODELS ................................ 82
5.1 SUMMARY OF AI.TLRNATIVE ENTRY/EXIT
STRATEGY CIIARACTERISTICS .................. ................ 98
6.1 SUMMARY OF THE TEST SYSTEM OPERATING CHARACTERISTICS
FOR THE ENFRY/EXIT MODEL SENSITIVITY ANALYSES............. 133
6.2 TEST SYSTEM ORIGINDESTINATION CHARACTERISTICS ............ 134
Table
6.3 MINIMUM TOTAL TRAVEL TIME FOR
VARYING PRIORITY SECTION CAPACITIES.....................
6.4 MINIMUM TOTAL TRAVEL TIME FOR VARYING
NONPRIORITY SECTION CAPACITIES.................. .........
6.5 MINIMUM TOTAL TRAVEL TIME FOR VARYING
LEVELS OF VEHICULAR DEMAND................................
6.6 RESULTS OF POSTOPTIMALITY ANALYSES
FOR THE ENTRY/EXIT MODEL ..............................
6.7 TOTAL TRAVEL TIMES PREDICTED BY
STRATEGY AND PRIFRE MODELS .................. ......... ..
7.1 195 ANALYSIS SECTIONSPM PEAK PERIOD....................
7.2 195 ORIGINDESTINATION TABLE.............................
7.3 195 ANALYSIS SECTION DEMANDS .............................
7.4 PASSENGER OCCUPANCY DISTRIBUTIONS FOR 195 SYSTEM........
7.5 OPTIMAL PRI 'TTY LANE ENTRY/EXIT STRATEGY FOR
MINIMUM CAR POOL REQUIREMENT OF 2 PERSONS PER VEHICLE.....
7.6 OPTIMAL PRIORITY LANE ENTRY/EXIT STRATEGY FOR
MINIMUM CAR POOL REQUIREMENT OF 3 PERSONS PER VEHICLE.....
A.1 CORRELATION OF BUREAU OF PUBLIC ROADS FUNCTION
AND HIGHWAY CAPACITY MANUAL CURVE .........................
B.1 RESULTS OF TIIE
FOR TIE PERIOD
B.2 RESULTS OF THE
FOR THE PERIOD
B.3 RESULTS OF THE
FOR THE PERIOD
R.4 RESULTS OF THE
FOR IIE PERIOD
B.5 RESULTS OF THE
FOR THE PERIOD
B.6 RESULTS OF THE
FOR THE PERIOD
CAR POOL DEFINITION ANALYSES
PM TO 6:30 PM ........................
CAR POOL DEFINITION ANALYSES
PM TO 4:00 PM.......................
CAR POOL DEFINITION ANALYSES
PM TO 4:30 PM.........................
CAR POOL DEFINITION ANALYSES
PM TO 5:00 PM ........................
CAR POOL DEFINITION ANALYSES
PM r0 5:30 PM........................
CAR POOL DEFINITION ANALYSES
PM TO 6:00 PM.........................
Page
Table Page
B.7 RESULTS OF THE 195 CAR POOL DEFINITION ANALYSES
FOR THE PERIOD 6:00 PM TO 6:30 PM. ... ..................... 193
LIST OF FIGURES
Fijure LPae
3.1 SCHEMATIC MODEL OF A HIGHOCCUPANCY VEHICLE
PRIORITY LANE SYSTEM ................................... 24
3.2 CONCEPTUAL SYSTEM FOR THE EQUILIBRIUM MODEL.............. 33
3.3 STRUCTURE OF THE EQUILIBRIUM MODEL....................... 34
3.4 CONCEPTUAL SYSTEM FOR THE OPTIMIZATION SUBMODEL.......... 39
3.5 EVOLUTION OF A WEIGHTED TOTAL TRAVEL TIME
VS. DEMAND RELATIONSHIP ................................. 44
3.6 PIECEWISE LINEAR APPROXIMATION OF WEIGHTED
TOTAL TRAVEL TIME VS. DEMAND ............................. 45
3.7 EFFECT OF PIECEWISE LINEAR APPROXIMATION................. 45
3.8 CONCEPTUAL STRUCTURE OF THE OPTIMIZATION SUBMODEL........ 47
3.9 LEVEL OF PRIORITY CONSTRAINTS................ ............ 53
3.10 CAR POOL DEFINITION MODEL SOLUTION PROCESS............... 58
3.11 BLOCK DIAGRAN FOR PROGRAM CARPOOL ........................ 63
4.1 SCHEMATIC OF THE TEST SECTION FOR THE CAR POOL
MODEL SENSITIVITY ANALYSES .............................. 69
4.2 ASSUMED DEMANDSPEED RELATIONSHIPS FOR THE
CAR POOL MODEL SENSITIVITY ANALYSES....................... 71
4.3 EFFECT OF VARYING PRIORITY SECTION CAPACITIES
ON MINIMUM TOTAL SYSTEM TRAVEL TIME...................... 73
4.4 EFFECT OF VARYING NONPRIORITY SECTION CAPACITIES
ON MINIMUM TOTAL SYSTEM TRAVEL TIME...................... 74
4.5 EFFECT OF VARYING VEHICULAR DEMANDS ON MINIMUM
TOTAL SYSTEM TRAVEL TIME ................................ 75
F i gute iPaqe
4.6 DISTRIBUTION OF VEHICULAR DEMAND
BY LEVEL OF OCCUPANCY.................................... 78
4.7 EFFECT OF VARYING DEMAND DISTRIBUTION ON
MINIMUM TO1AL SYSTEM TRAVEL TIME......................... 79
4.8 COMPARISON OF TOTAL TRAVEL TIME
PREDICTIONS OF CARPOOL AND PRIFRE MODELS................. 82
4.9 THE EFFECT OF VIOLATION RATE ON TOTAL
TRAVEL TIME............................................ 89
4.10 THE EFFECT OF NONUTILIZATION RATE
ON TOTAL TRAVEL TIME..................................... 89
4.11 THE EFFECT OF A FIXED LEVEL OF PRIORITY
ON TOTAL TRAVEL TIME..................................... 91
4.12 THE EFFECT OF AN INCREASING LEVEL
OF PRIORITY ON TOTAL TRAVEL TIME......................... 93
4.13 THE EFFECT OF A DECREASING LEVEL
OF PRIORITY ON TOTAL TRAVEL TIME......................... 94
5.1 ALTERNATIVE PRIORITY LANE ENTRY/EXIT STRATEGIES.......... 97
5.2 CONCEPTUAL MODEL OF PRIORITY LANE SYSTEM OPERATION....... 106
5.3 NETWORK FLOW STRUCTURE................................... 108
5.4 NETWORK MODEL OF PRIORITY SYSTEM FLOW.................... 111
5.5 EVOLUTION OF A TOTAL TRAVEL TIME VS.
DEMAND RELATIONSHIP.................................... 114
5.6 PIECEWISE LINEAR APPROXIMATION OF
TOTAL TRAVEL TIME VS. DEMAND............................. 116
5.7 EFFECT OF PIECEWISE LINEAR APPROXIMATION................. 116
5.8 FLOW NETWORK FOR TIE PRIORITY LANE
ENTRY/EXIT MODEL......................................... 120
5.9 GENERAL SOLUTION PROCEDURE FOR
THE ENTRY/EXIT MODEL ......... ..................... ....... 122
5.10 BLOCK DIAGRAM FOR PROGRAM STRATEGY....................... 126
6.1 SCHEMATIC OF THE TEST SECTION OF THE
ENTRY/EXIT MODEL SENSITIVITY ANALYSES................... 131
Figure Page
6.2 ASSUMED DEMANDSPEED RELATIONSHIPS FOR
THE ENTRY/EXIT MODEL SENSITIVITY ANALYSES................ 134
6.3 EFFECT OF VARYING PRIORITY SECTION CAPACITIES
ON OPTIMUM SYSTEM PERFORMANCE.. ......................... 136
6.4 EFFECT OF VARYING NONPRIORITY SECTION
CAPACITIES ON OPfIMUM SYSTEM PERFORMANCE................. 137
6.5 EFFECT OF VARYING LEVELS OF VEHICULAR
DEMAND ON OPTIMUM SYSTEM PERFORMANCE..................... 138
6.6 COMPARISON OF TOTAL TRAVEL TIME PREDICTIONS
OF STRATEGY AND PRIFRE MODEL ............................ 143
7.1 THE 195 CORRIDOR.................. ..................... 150
7.2 SCHEMATIC OF 195.................. ..................... 152
7.3 DEMANDSPEED RELATIONSHIPS FOR THE 195 SYSTEM........... 156
7.4 OPTIMUM CAR POOL DEFINITIONS FOR MINIMUM PASSENGER
HOURS DURING THE PERIOD 3:30 TO 6:30 PM.................. 162
7.5 DEGREE OF PRIORITY FOR MINIMUM PASSENGER HOURS
DURING THE PERIOD 3:30 TO 6:30 PM....................... 162
A.1 TYPICAL VOLUMETRAVEL TIME CURVES........................ 179
A.2 ORIGINAL BUREAU OF PUBLIC kOADS FUNCTION................. 182
A.3 COMPARISON OF MODIFIED BUREAU OF PUBLIC
ROADS FUNCTION AND HIGHWAY CAPACITY MANUAL CURVE......... 182
A.4 EXTENDED TRAVEL TIME CURVE............................... 185
A.5 EXTENDED DEMANDSPEED RELATIONSHIPS..................... 185
xiii
Abstract of Dissertation Presented to the
Graduate Council of the University of Florida in Partial
Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
PRIORITY LANES ON URBAN FREEWAYS:
SOME OPERATIONAL CONSIDERATIONS
By
Thomas Hamilton Culpepper
August 1977
Chairman: Kenneth G. Courage
Major Department: Civil Engineering
Since its introduction in the late nineteenth century, the
automobile has played a key role in shaping the current urban form and
way of life in the United States. In recent decades, the emergence of
the major problems of congestion, pollution, energy consumption, and
environmental deterioration have pointed out the need to make more
efficient use of current transportation resources. One approach to
this problem which is now being investigated is encouraging travel
in highoccupancy vehicles by providing positive incentives in the form
of preferential treatment. A primary means of promoting this type
travel is the reservation of freeway lanes for the exclusive use of
highoccupancy vehicles during hours of peak demand. This disserta
tion is concerned with the development of techniques for investigating
the optimal control and operation of these priority lane systems.
One of the major control parameters asset 'ted with this type
priority treatment is the definition of a "highoccupancy" vehicle.
This work develops a methodology for establishing the minimum number
of occupants which should be required for qualification as a priority
vehicle for a given reserved lane system. The proposed technique is
a mathematical optimization model, based on linear programming, which
considers the total system demand, the operating characteristics of
the facility, and the desired degree of priority, ard minimizes the
total travel time for the system. This model is validated, various
applications are discussed, and some general guidelines are developed.
Another important consideration in the operation of these sys
tems is the manner in which reserved lane access and egress is to be
provided. A methodology for investigating the priority lane entry/exit
strategy is proposed, again based on a mathematical optimization tech
nique. This model, a network flow analysis procedure, considers the
system operating characteristics and the priority and nonpriority demands
in the development of an entry/exit strategy which will minimize the
total hours of travel within the system. This model is validated, a
variety of applications are discussed, and some general observations
relating to priority lane entry/exit are presented.
These techniques are applied to an existing priority lane sys
tem operating in Miami, Florida. Reasonable results were obtained,
and the application demonstrated that the models are viable analysis
tools.
CHAPTER 1
INTRODUCTION
This dissertation presents methodologies for determining optimal
control parameters and operating strategies for reserved bus and car
pool lanes on urban freeways.
Background and Rationale
America was introduced to the automobile in the late nineteenth
century. By the year 1900 there were approximately 8,000 privately
owned automobiles in operation on a total of 2,300,000 miles of roadway
[Motor Vehicle Manufacturers Association (MVMA), 1976]. As the country
grew and mass production developed, the dependence of the American
people on the automobile also grew. It is estimated that in 1976, over
106 million automobiles were registered in this country. In the same
year, approximately 1.6 trillion passenger miles were traveled on some
3,816,000 miles of roadway [MVMA, 1976].
During this period, this country also experienced a shift from
an agricultural economy to an industrial economy. This produced a
migration from the sparsely settled farmland areas to the more densely
populated urban areas. It is now estimated that more than 50% of the
U.S. population lives in suburban areas. This suburban dwelling
pattern has increased the dependence on the automobile. In 1976, over
77% of the employed American public were dependent on the private
automobile for the daily travel to and from their place of employment. This
hometowork trip accounted for 31.9% of the total persontrips and 33.7%
of the passengermiles of travel in 1976 [MVMA, 1976]. In raw numbers,
this means that 37,102,997,860 persontrips covering 357,626,566,400
vehiclemiles were required to transport the American work force to and
from their places of employment.
In order to meet this steadily increasing load on the roadways,
a number of methods have been used. Initially, the approach was an
upgrading (surfacing) of the existing roadways. Between the years 1900
and 1976, the total roadway mileage increased only 65% (2.3 million miles
to 3.8 million miles). However, during this same period, the percentage
of paved roadway was increased from 8% to over 80% of the total mileage
[MVMA, 1976]. Recognizing the need for even better highway facilities,
a nationwide system of interstate and defense highways was conceived in
1944. This system of highspeed, limitedaccess roadways was 88%
complete in 1975 with a total designated mileage of 42,500. The concept
of highspeed facilities has been adopted in most metropolitan areas,
and local crosstown expressways or freeways now serve a large proportion
of the urban area travel.
As can be witnessed in any urban area, these methods based on
providing more and more vehicular capacity have been unable to keep pace
with increasing demand. Traffic congestion is now considered a "way of
life" in many areas. Additionally, environmental and energy considera
tions have detracted from these alternatives during the past two decades.
The American public is no longer willing to devote large portions of
the land or energy resources to roadways. As a result of this increased
environmental concern and the recent energy shortages in the world,
alternative methods for meeting the country's transportation require
ments are now being explored.
One of the more obvious alternatives is mass transit. This mode
of transportation has been available, in various forms, since the days
of the stagecoach. Recently, the development or expansion of conven
tional rapid rail and fixedroute bus systems has been utilized in
several localities. Additionally, experimentation with the concept
of demandresponsive systems in several forms has been in progress.
This activity has been successful to varying degrees and future promise
is evident in this approach. However, it should be noted that one of
the primary drawbacks to these systems is the relatively low population
density in this country. In European countries where the average
population densities range from 200 to over 800 persons per square mile,
this approach has been very successful. The average density in the
U.S. is 57 persons per square mile [MVMA, 1976]. In this lower density
situation, iass transit simply cannot be made as convenient for the
American public as for their European counterparts.
A second alternative now being explored is making more
efficient use of the automobile by increasing the average occupancy
level. This approach has the effect of meeting the total person demand
while reducing both the vehicular demand and the overall energy consump
tion. The primary target area for this effort is the peakhour, hometo
work travel. This travel accounts for 33.7% of the yearly person travel
and is made during the combined daily peakperiods, covering only 4 to 6
hours per day. Also, the average automobile occupancy during peak
periods is only 1.4 persons per vehicle, somewhat lower than the overall
average of 1.6 persons per vehicle [MVMA, 1976].
As a means of achieving this increased automobile efficiency
and, at the same time, encouraging the use of public transportation,
consideration is being given to providing preferential treatment for
highoccupancy vehicles in the urban corridors. This is not a new or
revolutionary concept since transit priority in the form of exclusive
rightsofway was first introduced in Chicago in 1939 [United States
Department of Transportation (USDOT), 1975]. More recently, the
provision of preferential or priority treatments on freeways has been
given serious consideration. A summary of the advantages and disadvan
tages of current techniques in this area is presented in Table 1.1.
The use of normalflow reserved lanes, which forms the subject
of this investigation, offers potential benefits in the reduction of
both vehicular demand and energy consumption, while meeting the total
person demand. The person capacity of a single freeway lane is approxi
mately 2,800 persons per hour (2,000 vehicles per hour at an average
occupancy of 1.4 persons per vehicle). The same lane could carry 8,000
persons per hour if used only by car pools of 4 persons, or 25,800
persons per hour as a busonly lane. As to the energy consumption, an
increase in the average peakhour occupancy of 1.4 persons per vehicle
to the overall average of 1.6 persons per vehicle would save an es
timated 2.3 billion gallons of gasoline per year. More realistically,
an increase in the average occupancy to 1.5 persons per vehicle, which
has been achieved in 2 priority freeway lane projects [USDOT, 1975],
would save approximately 1.2 billion gallons of gasoline each year.
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I
Statement of the Problem
In the area of designing and evaluating priority treatments for
highoccupancy vehicles, a substantial amount of work has been done.
This work has provided techniques for designing preferential treatments
for freeway ramps in terms of the priority cutoff level or car pool
definition and control strategies for nonpriority demand, and for
simulating the operation of priority ramp systems, reserved normal or
contraflow freeway lanes or exclusive rightsofway. However, the
current state of the art does not directly address the design and opera
tional criteria for reserved freeway lanes. The intent of this effort
then is directed toward the development of methodologies to investigate
operational design of these priority lanes on urban freeways.
Scope of the Study
Two of the major operational considerations for a reserved
highoccupancy vehicle lane on an urban freeway are (1) the definition
of "highoccupancy" and (2) the provision of entry/exit points for the
lane. The primary focus of this study will be the development of tech
niques which can be utilized to define a highoccupancy vehicle or
car pool and to determine the locations at which priority access and
egress should be provided. Additionally, it will be demonstrated that
these methodologies can be used to investigate other aspects of
priority lane operations, such as the geographical limits and hours
of operation.
Organization
The presentation of the methodology and results of the project
is contained in the following chapters. .A survey of current methodolo
gies for priority treatment investigations is provided in Chapter 2.
This is followed by the development of the proposed techniques for es
tablishing a car pool definition and identifying an optimal entry/exit
strategy in Chapters 3 and 5, respectively. Chapters 4 and 6 contain
demonstrations of the validity of the proposed techniques, as well as
discussion of their areas of application and some general guidelines or
observations.
A case study application of the proposed techniques to an
actual reserved lane system in operation on 195 in Miami, Florida,
is presented in Chapter 7. Conclusions based on this investigation and
suggestions for future research are contained in Chapter 8.
CHAPTER 2
SURVEY OF THE LITERATURE
Introduction
Classification of This Investigation
The work that was carried out in support of this dissertation
can be properly classified as an application of systems analysis tech
niques to an investigation of traffic flo'. Specifically, the work
presented in this report is directed toward the development of tech
niques for determining optimal control and operating strategies for
reserved lanes on urban freeways.
Scope of the Review
Inasmuch as a review of all previous efforts in the area of
trafficflow theory and systems analysis is well beyond the scope of
this work and would contribute little to the final product, this review
will be limited to those previous efforts in the area of modeling
freeway control systems, particularly reserved lane operations. In the
discussion to follow, primary consideration will be given to the nature
and application of previously developed freeway control system models
which are related to this investigation. Presentations of the various
techniques and methodologies adopted in this work will be made as the
models are developed in subsequent chapters.
For additional information related to freeway control systems,
reference can be made to previously compiled stateoftheart documents.
These include a comprehensive survey of current freeway surveillance
and control techniques by Everall [1972], and detailed guidelines for
design and operation of freeway ramp control systems by Masher et al.
[1975].
Previous Research
Early Applications
The idea of applying systems analysis techniques in the area
of traffic engineering is not a recent development. This concept was
first suggested by Lewis [1954] and utilized by Edie [1954] in determin
ing the number of toll booths required on the George Washington Bridge
in New York. The use of optimization techniques in developing a minimal
travel time assignment of vehicles to a traffic network was first demon
strated by Charnes and Cooper [1959]. This model was later used by
Pinnell and Satterly [1962] to determine the optimal operation of a
freeway with a continuous frontage road.
Freeway Models
The use of systems analysis in investigations on freeway on
ramp controls was first demonstrated by Wattleworth [1962]. This model
was based on a linear programming approach and was designed to determine
the optimal metering rates for a series of ramps. This basic methodology
has been adopted in a variety of subsequent efforts, including those of
Goolsby et al. [1969], Messer [1969], Brewer et al. [1969], Wang and
May [1973a], and Ovaici and May [1975].
A general freeway operations evaluation model based on a simu
lation approach was proposed by Makigami et al. [1970]. This model,
known as the FREEQ model, was applicable to investigations of the
operating characteristics of directional freeways and has been applied
successfully by Allen and May [1970] and Stock et al. [1971]. This
model was later refined by Blankenhorn and May [1972] and then again by
Stock et al. [1973]. As a result of these refinements, the FREQ3 model
was developed. This model, still a simulationbased technique, now
evaluated directional freeway operations with consideration given to
implementationof ramp control strategies, such as metering. This final
model formulation was used successfully in an investigation of the
East Shore Freeway in the San Francisco Bay Area [May, 1974].
A methodology for designing freeway ramp control strategies
based on the FREQ3 model was proposed by Eldor and May [1973]. This
model, known as FREQ3D, was, in essence, a search process based on
iterative application of the FREQ3 model. This procedure was subse
quently converted to a deterministic optimization model by Wang and
May [1973a]. The iterative process in FREQ3D was replaced by a linear
programming technique similar to the one proposed by Wattleworth. The
resulting model, called FRLQ3C, used the optimization technique to
determine the optimal metering rates for each freeway ramp, and then
simulated operations before and after control as in previous models.
Subsequent to the initiation of a program to reduce traffic
congestion by providing preferential treatment for buses and multi
passenger vehicles, a number of priority treatment models have been
developed. The first of these was proposed by May [1968]. This
rudimentary model was designed to simulate the operation of an exclu
sive bus lane on a freeway and was based on the assumption of constant
peakperiod demand and a simple Greenshields flow submodel. This
model was later refined by Stock [1969] to include consideration of
timevarying demands and a variety of speedflow submodels. The
name EXBUS was adopted for this refined model.
In order to consider exclusive lanes for both buses and car
pools, Sparks and May [1970] proposed another step in the evolution of
the EXBUS model. This thirdgeneration model, still known as EXBUS,
simulated the operation of a freeway lane reserved for the joint use
of buses and car pools. This final version of the EXBUS model has
been used in priority lane feasibility studies for the San Francisco
Oakland Bay Bridge [Martin, 1970] and for a segment of 190 in Cleve
land, Ohio [Capelle et al., 1972].
As a result of the application of the EXBUS model in these
feasibility studies, some weaknesses in the model structure and appli
cation procedure were identified. Specifically, the simulation model
lacked tie capability to consider temporal or spatial variations in
the demand and/or capacity and manual interfacing of the priority
lane simulation, and a simulation of the normal lanes was required to
determine the operation of the total system. These weaknesses prompted
the development of the PRIFRE model by Minister et al. [1973]. This
model combined the philosophy of the EXBUS model and the more realistic
approach of the FREEQ model, and allowed simulation of a directional
freeway operating with one or more lanes reserved for the use of buses
and car pools. Toe PRIFRE model was developed primarily to simulate
oneway "nonnal" priority lane operations, i.e., the reserved lane(s)
on the same side of the median as the nonreserved lanes. However,
with manual interfacing, it can be applied to contraflow on reversible
lanes, separate priority roadways, freeway design alternatives, and
evaluation of ramp control strategies.
A second consequence of this program to provide preferential
treatment for highoccupancy vehicles was the need to consider priority
access at freeway onramps. In response to this, the FREQ3C model was
modified by Ovaici et al. [1975]. The modified model, known as
FREQ3CP, uses a deterministic optimization technique to determine the
priority cutoff level for ramp priority and the optimum metering
rates for nonpriority vehicles at each onramp. Additionally, the
model simulates the system operation with and without the provision of
priority access. In the simulation process, an onfreeway priority
lane may also be considered.
Corridor Models
It is also possible to evaluate freeway control strategies as a
part of the freeway corridor operations. One model which has been
developed for simulation of corridor operations is the SCOT model pro
posed by Liebenrnan [1971]. This model is based on the prototype Urban
Traffic Control Simulator (UTCS1) model developed by Bruggeman et al.
[1971] for network flows and on another model, DAFT, proposed by
Lieberman [1970] for the dynamic assignment of freeway corridor traffic.
In this assignment and simulation process it is possible to consider
freeway control strategies in the form of ramp metering or closure
[Lieberman, 1971].
Another corridor model, CORQ, has been proposed by Yagar [1975].
This model was designed to predict the selfassignment of timevarying
demand in a freeway corridor. The basic structure of this model is an
iterative simulation process in which the flow assignment is determined.
With appropriate application, this model can he used to evaluate or
design normal ramp control strategies, however, priority treatments
cannot be considered.
Another possibility for considering various freeway control
strategies is in the context of a full network model. The primary
development in this area is the Urban Traffic Control Simulator (UTCS)
model mentioned previously. The prototype for this network simulation
model was proposed by Bruggeman et al. [1971], and additional refine
ments were suggested by Lieberman et al. [1972]. The UTCS model is a
microscopic simulation model designed as an evaluative tool for urban
traffic control policies. As such, the model might be used to evaluate
alternative freeway control strategies, but evaluation or design of
priority treatments would be outside its realm cf applicability.
Relevance to This Effort
As has been demonstrated, previous research in the area of
freeway control systems has produced a number of models for (1) inves
tigating normal freeway operations, (2) simulating and designing free
way ramp control strategies with or without priority considerations,
(3) simulating freeways with reserved lane operations, and (4) consider
ing freeway control strategies in relation to their corridor or network
effects.
However, these efforts have notdirectly addressed the deter
mination of optimal reserved lane control parameters and operating
strategies, which is the subject of this investigation. Therefore, the
contributions of previous research to this particular study are limited
to (1) establishing an overall philosophy for the development of traffic
flow models, (2) identifying acceptable procedures for use in modeling
efforts, and (3) demonstrating the validity of applying systems analy
sis techniques in traffic engineering.
Summary
A summary of the characteristics of these previously developed
freeway control system analysis models is presented in Table 2.1.
The models to be developed in the remainder of this work have been
included in this summary table in order LU demonstrate the manner in
which they extend the range of currently available techniques.
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CHAPTER 3
DEVELOPMENT OF A CAR POOL DEFINITION MODEL
Introduction
Objectives
One of the major control parameters associated with any priority
treatment technique is the definition of a priority vehicle. In the
case of transit priority treatments, this definition is simple and
straightforward. The transit vehicles, i.e., buses, are the only com
ponent of the traffic demand to be given priority status. However, in
the case of a highoccupancy vehicle (HOV) priority system, the defini
tion of a priority vehicle takes on a new dimension. For these treat
ments, the level of passenger occupancy which is to be considered
"highoccupancy" must be determined. Although this decision is often
influenced by convention or social and political considerations, it
should ideally be based on an application of sound engineering princi
ples in each particular situation.
In this chapter, an examination of the engineering considera
tions involved in making this determination will be presented. As a
result of these considerations, a methodology for investigating the
level of occupancy which should constitute "highoccupancy" for a
particular HOV priority treatment will be developed. This development
will address both the framework for the investigation and the mathema
tical tools which will be utilized.
Description of the Problem
As is the case with any proposed methodology, certain limita
tions with respect to the range of applicability must be imposed on the
problem at hand. In this case, the techniques to be developed will be
restricted to investigations of HOV priority lanes on urban freeways.
The motivation for this is twofold. First, it is unlikely that the
need for implementing priority treatments will develop outside the
populous urban areas. Second, previous research [Ovaici et al., 1975]
has yielded acceptable methodologies for defining "highoccupancy" for
priority entry systems.
Additionally, it will be assumed that the decision to implement
reserved lanes treatment rather than a priority entry system has been
made. Contrasts or comparisons of these alternatives will not be
considered as a part of this effort. It will also be assumed that the
number of lanes to be reserved for priority traffic has been predeter
mined. However, as will be discussed in subsequent chapters, this car
pool definition methodology can be readily adapted for making this
determination as well as investigating additional reserved lane concepts.
Therefore, considering these limitations and assumptions, the
methodology to be developed will be a technique for determining the
level of occupancy which should constitute priority status for HOV
priority lanes on urban freeways. This minimum level of passenger oc
cupancy or priority cutoff level will then be the car pool definition
for the reserved lane system.
Method of Analysis
A primary consideration in the development of any analysis
methodology such as the one proposed here should be the work of
previous researchers. As was discussed earlier, there is little in the
current literature which might be used as a basis for this investiga
tion. The work that has been done is limited in scope, oriented toward
other prime objectives, or only generically related to this effort.
However, the research that is documented does provide a general back
ground for this investigation in the form of philosophical guidelines
and acceptable techniques for modeling traffic flow and control systems.
It is then possible to consider the basic structure of a car
pool definition model. When evaluating candidate techniques to be
used as a basis for any mathematical model, certain desirable charac
teristics should be considered. For this particular effort, the follow
ing criteria were used:
1. The model should adequately reflect the character
istics of the physical system.
2. The model should address a reasonably broad range
of alternative situations.
3. The model should be adaptable for investigations
of special conditions.
4. The model should, to the extent possiblP, be
based on currently acceptable technique,
The candidate techniques which were identified for this effort
include: (1) analytical models, (2) simulation models, and (3) optimiza
tion models. Analytical modeling would, in this case, consist of
developing mathematical expressions which describe the various relation
ships of traffic flow and combining them to develop an expression
defining the car pool requirements in terms of the traffic flow
parameters. Development of a simulation model would require formula
tion of a mathematical framework which would describe the traffic flow
interactions encountered in an HOV priority treatment. This model
would then be utilized to evaluate the system operation for different
combinations of the control parameters. Optimization techniques also
require development of mathematical descriptions of the traffic flow
relationships and system characteristics, but would be used to deter
mine the manner in which the system should be controlled to achieve
"optimal" operation.
Detailed examination of the potential value of an analytical
technique in this case quickly demonstrates several weaknesses in the
approach. Primarily, there is a tendency for this technique to become
unwieldy or unsolvable for complex situations. Although an analytical
model can be used successfully with artificially simple representations
of the traffic flow interactions, the degree of simplification required
to achieve a manageable model would be costly in terms of maintaining
a realistic representation of the physical system. Another significant
weakness in this approach is that the resulting expressions are gener
ally more narrow in their range of application than is desirable for
this effort. Finally, the resulting expressions are not readily adapt
able for use under special conditions.
Simulation techniques overcome the limitations of the analytical
techniques to a large degree. These models, if properly developed,
can realistically represent the physical system operations and are
generally quite flexible in application. Some difficulties may be
encountered in adapting a model of this type for investigations of
special conditions, but, as a general rule, this can be accomplished
more readily than would be the case with analytical models. For
the purposes of this work, the simulation approach does have one major
drawback. The basic intent of any simulation model is to provide
information on how a system will operate under fixed conditions
rather than determining the conditions under which optimal performance
will be achieved. This technique can be utilized tu evaluate alterna
tives in an exhaustive search process which will ultimately define the
best or optimal conditions for system operation. It is this lack of
deterministic results that has led many operations research analysts
to refer to simulation as a "method of last resort" [Wagner, 1975,
p. 907]. It should be pointed out, however, that simulation techniques
do iave their rightful place as a method of evaluating system operations
and have been utilized quite successfully in modeling various priority
treatment techniques [Ovaici et al., 1975; Minister et al., 1973].
The last method to be considered is that of optimization m(odls.
This approach is one in which the ultimate goal is the optimization,
i.e., maximization or minimization, of a numerical function of a set
of variables which are subject to a number of constraints [ladley,
1963, p. 1]. This class of techniques exhibits many of the strengths
of simulation, such as realistic representation of the physical system,
formulation for broadrange applications, and adaptability for special
conditions. Additionally, this approach is directed toward a decision
making process as opposed to an evaluation of system operations.
Specifically, these techniques deal with the optimal allocation of
limited resources to meet given objectives. This approach has been
utilized in determining optimum controlparameters for several traffic
control systems since the early 1960s [Wattleworth, 1962].
The preceding considerations indicate than an optimization
technique would be the most direct approach, and is, therefore, the
approach that will be taken in the development of a car pool definition
model.
Development of the Basic Model Structure
Description of the Physical System
The first step in developing any mathematical model must be the
definition of the system which is to be modeled. In the case of a car
pool definition model for an HOV priority lane on an urban freeway, the
basic component of the physical system is the freeway. Specifically,
it is that section of the freeway in which priority treatment is to be
provided. This freeway section is in reality composed of two distinct
subsections, the lane(s) reserved for priority traffic and the lanes
remaining for nonpriority traffic. Although these subsections operate
concurrently in time and space, they do possess distinct operating
characteristics, such as capacity and speedflow relationships. If
the utilization of roadway capacity by individual vehicles is viewed
as a basic supplyanddemand situation, these subsections would
represent the supply portion of the system. This "supply" of roadway
capacity is then the resource which is to be allocated.
The demand portion of this relationship is represented by the
individual vehicles desiring to use this section of freeway. This
"demand" can be viewed as the total number of vehicles which use the
roadway or as a stratified demand consisting of vehicles with a single
occupant, vehicles with two occupants, etc. For the purposes of this
effort, the latter viewpoint will be taken for reasons to be discussed
later.
The physical system to be modeled is then a section of freeway
with both reserved and nonreserved lanes, the available capacity of
which are to be allocated to the vehicular demand in such a manner
as to optimize the system operation. The objective of the model is
to determine the level of occupancy which should be required for prior
ity status such that optimal operation is realized. This system is
shown graphically in Figure 3.1.
General Modeling Considerations
One common element of all optimization models is the objective
function. This is some mathematical function of a set of variables,
known as decision variables, the value of which is the quantity to
be maximized or minimized. This expression is normally considered
to be the "cost" associated with operating a particular system in
a certain manner. In the area of traffic flow analysis, a wide variety
of objective functions or figures of merit have'been proposed and
utilized. Among these are such measures as total travel time, total
system input rate, total system output rate, vehiclemiles of travel,
and passengermiles of travel.
In considering the figure of merit to be utilized as an objec
tive function for this model, it was felt that the selected criterion
24
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should directly reflect the fact that the overall intent of priority
treatment is to reduce the trip time for highoccupancy vehicles. Ad
ditionally, the selected figure of merit must not be independent of
the control parameter under consideration, nor can it be artificially
optimized at the expense of overall system operation.
Looking first at the two latter considerations, such measures
as system input and output rates and total miles of travel would be
independent of the control parameter inasmuch as the total demand is
fixed and no diversion will be considered. Previous investigations
have shown that under certain conditions, specifically ramp control
without diversion, maximization of system output may be equivalent to
minimization of total travel time. However, since ramp controls will
not be considered in this model, the more direct measure, total travel
time, can be utilized. In order to reflect the Fact that this is an
HOV priority treatment, provision will be made for the use of total
travel time on a vehicle or passenger basis.
Having identified the criterion function for this optimization
model, the next step is to outline certain constraints which will be
imposed on the optimization process. The first of these constraints
was implied in the preceding paragraph and requires that the total
demand on the system be satisfied without diversion. Another constraint
is that no system element can be loaded past its maximum capacity.
Next, the model should recognize that not all priority vehicles will
utilize the reserved section and that some nonpriority vehicles will
use the reserved section in violation of its use restrictions. Finally,
the formulation should allow variation in the degree of preferential
treatment that is given to priority vehicles.
Basic Model Structure
With the physical system defined, the objective selected, and
the solution constraints identified, alternative optimization modeling
techniques can now be examined, and a basic model structure formulated.
The underlying supplyanddemand nature of this system immediately
indicates that some type of commodity assignment technique would be
appropriate. Considering the stratification of demand that is neces
sary to treat passenger travel time, the assignment technique must be
capable of recognizing and preserving the distinctions among demands at
the various occupancy levels.
This leads into the area of multicommodity assignment tech
niques. As a general rule, however, this type of optimization model
does not lend itself to the incorporation of the priority variation
constraints mentioned in the preceding section. The most notable ex
ception to this rule is the Charnes multicopy technique [Charnes and
Cooper, 1962]. Although this technique does allow the incorporation
of special optimization constraints, For this application it is felt
that this approach would be somewhat inefficient due to the general
nature of the process required for solution of a multicopy model.
Briefly, this technique requires that two separate optimization models
be solved for each commodity or copy in the system. For consideration
of 5 to 6 levels of passenger occupancy, this would mean solution of
10 to 12 optimization problems.
As a result of the above considerations, examination of an al
ternative methodology is now in order. The multicommodity techniques
originally considered are in fact special cases of the more general
linear programming technique. This linear programming methodology is
quite flexible in nature and, as such, has been utilized in a wide
variety of applications including modeling of traffic control systems.
The primary requirements of this technique are that both the objective
function and constraint equations be linear expressions. This require
ment does present some problems when modeling traffic systems, but
these are relatively minor and can be readily eliminated.
The most significant drawback to this approach is that in the
formulation of the objective function, it is necessary to eliminate
the distinctions among the various levels of passenger occupancy in
determining the optimal system operation. However, this too is a
manageable problem. The linear programming model can be utilized as an
optimization submodel within a more flexible framework that will account
for the multicommodity nature of the physical system .
The specific problem which arises when attempting to use a
linear programming model to optimize (minimize) total travel time
on a passenger basis lies in the formulation of the objective function.
The general form of this function is given by Equation 3.1 [Sivazlian
and Stanfel, 1975, p. 133].
n
Optimize Z = E ci x. (3.1)
i=1
Where: ci = cost coefficient
xi = decision variable.
An expression for the minimization of total travel time on a passenger
basis is given by Equation 3.2.
ill n
Minimize TTTP = Z T. ( 1i x..) (3.2)
j=1 J i=1 i'
Where: TTTP = total passenger travel time
T. = unit travel time in section j
xij = amount of demand at occupancy
level i assigned to section j.
Since the unit travel time (T) is a function of the demand (x),
it is not possible to reduce Equation 3.2 to the required form given
by Equation 3.1 and still retain the assigned vehicular demand as the
decision variable and a scalar cost coefficient. However, this situa
tion may be circumvented by accounting for the passenger occupancy in
the unit travel time cost coefficient. This is accomplished by intro
ducing the average passenger occupancy as a scalar multiplier of the
unit travel time. The average number of occupants (N) in the vehicles
assigned to a given section is
n
T i xij
j n (3.3)
i=l
Where: Nj = average number of occupants in all
vehicles assigned to section j.
Equation 3.2 can then be rewritten as
m n
Minimize TTTP = E Tj N. 7 xij (3.4)
j=1 J i =1i
or
Minimize TTTP = Z T Nj xj (3.5)
j=1
n
Where: xj = xij.
i=l
Equation 3.5 can then be utilized as the basis for developing an objec
tive function for a linear programming model by considering the cost
coefficient (ci) to be the product of the unit travel time (T ) and
the average occupancy (Nj). The complete development of this objective
will be considered in later portions of this chapter.
The preceding demonstration readily identifies the function of
the model which will serve as the general framework for the linear
programming optimization submodel, that is, to determine the average
occupancy of the vehicular flows in both the priority and nonpriority
sections. Obviously, since the average occupancies are functions of
the assigned flows, and the flow assignments are dependent on the
average occupancies, this superstructure model will be an equilibrium
assignment technique. That is to say, the model will assume values
for the average occupancies of the priority and nonpriority sections,
determine the optimal flow assignments, revise the occupancy estimates,
and continue the process until equilibrium has been achieved between
the estimated occupancies and the optimal flow assignments.
In summary, the basic structure of the car pool definition model
will be a deterministic optimization technique. The overall model will
be an equilibrium assignment process with an internal linear programming
optimizaton submodel. The equilibrium model framework will seek a
balance between the estimated average occupancies and the assigned
optimal flows for the priority and nonpriority sections. The linear
programming submodel will determine the optimal flow assignments for
the priority and nonpriority sections based on the estimated average
occupancies, the system operating characteristics and the related
system constraints. The result of the model operation will be a
recommendation as to the minimum number of passengers which should be
required in order to qualify a vehicle for priority status in conjunc
tion with an HOV priority lane system on an urban freeway.
Development of the Equri i briumn Model
Objective
Recalling the statement of the proposed function of the equilib
rium model in the previous section, this structure may be viewed as an
iterative traffic assignment technique. The purpose of this process
is to achieve a balance, or point of equilibrium, between the assigned
flows in the priority and nonpriority sections and the average vehicular
occupancies for those sections. The iterative nature of this process
is necessary since the average section occupancies are functions of the
assigned flows, which are in turn dependent on these average occupan
cies.
At this point, it would be appropriate to note one deviation
from the previous discussion. This is in the definition of the overall
objective function for this model. References have been made to total
travel time on both a vehicle and a passenger basis. In reality,
these measures of effectiveness are not entirely independent. It is
convenient to view the total passenger travel time as simply a "weighted"
total vehicle travel time. The weighting factor, which must be applied
at the stratified flow level, then becomes a method by which the objec
tive function of the model can be defined. If a weighting factor of
1.0 is assumed for all levels of occupancy, the model will use total
vehicular travel time as its objective criterion. On the other hand,
if this weighting factor is the passenger occupancy at each level, the
total passenger travel time will be the figure of merit for the model.
These examples by no means exhaust the possible variations of this
weighting factor approach. In fact, these factors can be viewed as
the relative importance of the flow at each occupancy level, and as
such, reflect the opinions of individual uiers. The two examples that
were presented do result in measures with physical meaning, whereas,
other formulations would not possess this property. For the remaining
discussions pertaining to this model, the selected figure of merit
will simply be referred to as the weighted total travel time (TTTW).
General Structure
The physical system being addressed by this portion of the
model is a section of urban freeway with lane(s) reserved for use by
highoccupancy vehicles. This section has certain operating charac
teristics such as capacity and speedflow relationships which can be
different ior the two subsections. The total demand for use of this
section has a known distribution of occupancy levels and can be
treated as a stratified set of demands. For each of these demand
levels, the relative importance is known via the set of flow weighting
factors which is specified. This system structure is shown in Figure
3.2
The equilibrium model has a basic iterative structure in which
the average subsection weighting factors are balanced with the optimal
flow assignments. The technique is initiated with an estimate of the
average flow weights for the two subsections. This estimate is then
used to determine the optimal flow levels in each subsection through
the optimization submodel. These optimal flow levels are in turn used
in a demand assignment process, which results in a determination of the
actual subsection flow weights. If the estimated and actual flow
weights are the same, the process ceases and final recommendations may
be formulated. Otherwise, the flow weight estimates are revised, and
the process is repeated until equilibrium has been achieved. This
overall process is shown as a block diagram in Figure 3.3.
Mathematical Development
As was previously stated, the objective of this portion of the
model is to achieve a balance between average subsection flow weights
and assigned flows. Assuming that an estimate of the average flow
weights for the two subsections is available (01, 02), the optimal flow
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Start
rFtimate the average flow
weights for the priority
and nonpriority sections
_~I
Determine the optimal flow
levels for the priority
and nonpriority sections
Assign system demand to priority
or nonpriority section considering
the optimal flow levels and the
violation and nonutilization rates
specified for each occupancy level
Compute the average flow
weights for the priority
and nonpriorit sections
No
Has equilibrium been achieved ?
Yes
End
Figure 3.3 STRUCTURE OF THE EQUILIBRIUM MODEL
split between the two sections is known, and the demand assignment has
been made, the equilibrium of the model can be tested. The first
step in this process is to determine the actual flow weights as indi
cated in Equation 3.6.
n
Swi xij
i (3.6)
j n
: x.ij
i=l
Where: 0. = actual average flow weight for
J subsection j
wi = flow weight factor for level of
occupancy i
xij = portion of demand at occupancy
level i assigned to the priority
section.
Equilibrium is then tested against specified tolerance limits as indi
cated in Equation 3.7.
L ix 100% 5 E (3.7)
0.
oj
Where: E = specified maximum error (%) for average
flow weight estimates.
If both inequalities hold, equilibrium will have been achieved. If
not, the flow weight estimates are revised and the process is repeated.
This revision process is indicated in Equation 3.8.
6' = + a (0. 6.)
3 J 3 J
(3.8)
Where: 0' = revised estimate of average flow
S weight in section j
O previous estimate of average flow
weight in section j
a = specified stepsize for the revision of
flow weight estimates.
It should be noted that the revision procedure in Equation 3.8 actually
revised the original estimate by some fraction (a) of the difference
between the estimated and computed weights as opposed to using the
computed values as the new estimates. The rationale here is to reduce
the likelihood of developing oscillations which would preclude the
achievement of equilibrium. This technique can then be described as
a bivariate search technique with a fixed fractional stepsize.
Recalling that the primary motivation for using this equilibrium
model structure is to account for the multicommodity nature of the
system, a discussion of how this is achieved would be in order.
Since this multicommodity nature is a result of the desire to incor
porate weighting factors for each level of passenger occupancy, refer
ence to Equation 3.6 will show that the flow weight factors are in
corporated in the computation of these scalar quantities. It is
these scalars which are used as multipliers for the unit travel time
cost coefficients in the optimization submodel. Through this process,
the stratification of demand is made possible.
The actual determination of the optimal priority and nonpriority
subsection flow levels is made by the optimization submodel. At this
point, it will be sufficient to say that the equilibrium model supplies
this submodel with the physical system parameters, the demand characteristics
and an estimate of the average subsection flow weights. The
submodel uses this information to detennine the optimal system operation
and returns to the equilibrium model information pertaining to the
optimal level of flow in each subsection.
With the optimal flow levels supplied by the optimization
submodel, the assignment state of the equilibrium process can be carried
out. This is basically a direct assignment technique in which flows at
the higher levels of occupancy are assigned to the priority section in
accordance with the optimal flow levels. This assignment process
should, however, account for two observations related to priority
treatment operations. These are the violation of the lane use restric
tions by nonpriority vehicles and the nonutilization of the reserved
lane(s) by qualified priority vehicles.
The violation of the reserved section restrictions by non
qualified vehicles is a phenomenon that has been observed in practically
all implementations of HOV priority lanes. Simply stated, this means
that some portion of those not qualified as highoccupancy vehicles will
use the reserved lane(s) in violation of the use restriction. For the
purposes of this model, this can be stated mathematically as in
Equations 3.9 and 3.10.
xil > cidi (3.9)
or conversely,
Xi2 (1 ai) di
(3.10)
Where: xil = amount of flow at occupancy level i
which uses the reserved section
xi2 = amount of frow at occupancy level i
which uses the nonreserved section
ai = proportion of vehicles at occupancy
level i which will violate the reserved
lane restrictions if that level is not
considered highoccupancy
di = total demand at occupancy level i.
Another characteristic which has been observed is that not all
qualified vehicles will utilize the reserved section. This factor
must also be taken into consideration when assigning system demand.
This consideration is, in essence, the complement of the violation
rate, as is shown in Equations 3.11 and 3.12.
xi2 Bi di (3.11)
or conversely,
xil S (1 Pi' di (3.12)
Where: Bi = proportion of the demand at occupancy
level i which will not utilize the
reserved sections if that level is
considered highoccupancy.
This system is shown pictorially in Figure 3.4.
Summary of the Equilibrium Model
In summary, this model balances the average subsection flow
weights and assigned flows to achieve equilibrium. The actual deter
mination of optimal flow levels is the function of the optimization
40)
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submodel. The flow assignment process takes into account the violation
and nonutilization rates for each occupancy level. The output of the
total model is then the optimal assignment, by level of occupancy, of
the system demand to the priority and nonpriority subsections.
Development of the Optimization Submodel
Objective
As has been previously defined, the objective of the optimiza
tion submodel is the determination of the optimal flow levels for the
priority and nonpriority subsections. This determination is to be
based on the physical system characteristics, total system demand,
and the average subsection flow weights as determined by the equilibrium
structure. The figure of merit to be utilized in developing the ob
jective function for optimization is the weighted total system travel
time (TTT").
General Structure
The physical system which is to be considered at this point is
similar to the original system. It represents a section of urban free
way which is partitioned into subsections designated for use by priority
and nonpriority demand. The section has certain operating character
istics such as capacity and speedflow relationships which can be
different for the subsections. The system demand, however, is not
stratified as in the original system. The individual level of occu
pancy distinction is treated indirectly through the use of the average
flow weight estimates.
The modeling approach to this physical system is a basic linear
programming technique. In this technique, the objective function (TTTW)
is to be minimized within the limits ofcertain constraints which
describe the physical system limitations and the level of priority
which is to be given to the highoccupancy vehicles.
The linear programming process can be thought of as the allo
cation of a resource or resources in such a manner that some function
of this allocation is optimized (maximized or minimized) within the
limitations of a set of constraints which are also functions of the
allocation [Sivazlian and Stanfel, 1975, p. 133]. As implied by the
term "linear" programming, a basic requirement of this technique is
that the set of equations describing the objective and constraints be
linear expressions. In matrixvector notation, this structure can be
expressed as follows:
MAX, MIN Z = cx (3.13)
Subject to: Ax { > = } b (3.14)
x > (3.15)
Mathematical Development
With the basic model structure identified, development of a
linear programming approach which will meet the stated objectives can
proceed. Expressing these required objectives in a format similar to
Equations 3.13 to 3.15 the model becomes
Minimize Z = TTTw
Subject to: (1) Physical system limitations
(2) Level of priority constraints
(3) Nonnegative flows.
Looking first at the objective function for this model, the
weighted total travel time can be expressed initially as follows:
2 m
TTTw = T* E wi x. (3.16)
j=1 i=1
Where: Tj = unit travel time in section j
wi = flow weight factor for level of
occupancy i
xi = amount of demand at occupancy level i
assigned to section j.
However, as has been shown, this is a multicommodity structure and
can be reduced to a singlecommodity format for use in a linear pro
gramming approach. This was done by introducing the average flow weights
(01, 02) for the priority and nonpriority section flows. With this
variable, Equation 3.16 can be reduced to the following.
2
TTTw = E Tj 0. x. (3.17)
j=1 J
Where: 0. = average flow weight in section j
J
x. = total demand assigned to section j.
This formulation is then parallel in structure to Equation 3.13 con
sidering c = (Tj, 0 i and x = {x.l.
A basic relationship of traffic flow that should be considered
at this point is that the unit travel time on a roadway is a nonlinear
function of the vehicular demand. This means that the objective func
tion in Equation 3.17 is nonlinear since the unit travel time portion
(Tj) of the cost coefficient is a function of the decision variable,
xj. This nonlinearity can be removed as follows. First, recognizing
that unit travel time (T) is a function of demand (x) as illustrated
in Figure 3.5a, the total travel time at any demand x is defined as
TTTIx = x TIx. Utilizing this as a transformation procedure, the total
travel time (TTT) can be expressed as a function of demand. This is
illustrated in Figure 3.5b. The weighted total travel time (TTTw) at
any demand x is then equal to wTTTIx, where w is the flow weight
factor. Transforming this TTT function into a TTTw function, an ex
pression for TTTw as a function of demand can be developed as shown in
Figure 3.5c.
At this point the original nonlinearity in the objective func
tion has been removed, however, the resulting cost coefficients are
variable rather than fixed quantities. A relatively simple technique
can be utilized to alleviate this problem. This consists of approxi
mating the curve of Figure 3.5c with a series of straight line segments,
known as a piecewise linear approximation (PLA) [Wagner, 1975, p. 563].
This approximation process, shown in Figure 3.6, allows the cost
function to be treated as a set of linear cost functions. The effec
tive result of this technique is to treat the flow between any two
points (i, j) on a roadway as flow on a series of "branches" between
these same points as shown in Figure 3.7. Each branch has a maximum
capacity, determined by the "break points" in the PLA, and a unit flow
cost which is the slope of the PLA segment represented by the particular
branch.
Travel
Time
Total
Travel
Time
T II
x
Demand
(a) Travel Time vs. Demand
TTTx = x Tix
TTTWx = w TT1T[
x 'X
Demand
(b) Total Travel Time vs. Demand
TTT W
TITWx
Weighted TTT = f(Demand)
Total
Travel
Ti nme
Demand
(c) Weighted Total Travel Time vs. Demand
Figure 3.5 EVOLUTION OF A WEIGHTED TOTAL TRAVEL TIME VS.
DEMAND RELATIONSHIP
TTTWIz
Z        
Piecewise Linear
Weighted Approximation
Total /
Travel 2 2
Time
1
,, Original
S Cur',e
xI x2 z x3
1 X2 Z 3
Demand
Figure 3.6 PIECEWISE LINEAR APPROXIMATION OF
WEIGHTED TOTAL TRAVEL TIME VS. DEMAND
Cost = S3
3
Capacity = xI
TT1 z = S1 4 S2(x2x1) + S3(zx2)
Figure 3.7 EFFECT OF PIECEWISE LINEAR APPROXIMATION
Applying this approximation technique to Equation 3.17, the
objective function can be restated as follows:
2 m
TTT" = Y Sjk xjk (3.18)
j=1 k=1l
Where: Sjk = unit flow cost on branch k of section j
Xjk = amount of demand assigned to branch k
of section j
Although the utilization of this technique has transformed the approach
to the general class of separable programming, the term linear program
ming will still be applied to tlhe model. This separable programming
technique has merely reduced a nonlinear programming problem to a linear
programming format.
In the final objective function formulation, Equation 3.18,
the multicommodity nature of the physical system has been addressed
with the use of the average flow weight variable, which is a derivative
of the individual occupancy level weights and the assigned section
flows. Additionally, the nonlinearities in the original formulation
were eliminated through the use of a piecewise linear approximation
of a weighted total travel time cost coefficient function. The result
ing conceptual system for this formulation is shown in Figure 3.8.
The constraints which are to be imposed on the solution
process can now be considered. The general classes of constraints
which will be developed are system demand constraints, priority and
nonpriority subsection capacity constraints, and the level of priority
J
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CY
Q
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C
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co
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constraints. The speedflow relationships mentioned in previous sec
tions are omitted from this list since they are reflected in the travel
time portion of the cost coefficients for the objective function.
The first type of constraint to be considered is that of
satisfaction of the total demand. This constraint is required as a
result of a previous decision that no traffic diversion was to be
allowed. Thus, the model must accept and consider the total vehicular
demand on the system. In terms of the model variables, this constraint
is expressed mathematically as follows:
2 in n
E xjk = Z di (3.19)
j=1 k=l i=1
Where: xjk = the amount of demand assigned
to branch k of section j
d. = the total system demand at
occupancy level i.
Next, consideration must be given to the system capacity. Prior
to this point, no restriction has been made limiting the system operation
in the realm where demand is less than capacity. It is not the intent
of this type constraint to impose this restriction on the model.
Rather, it is intended to insure that the individual flow branches
resulting from the approximation of the weighted total travel time
curve are not loaded beyond their individual capacities. This series
of constraints can be expressed as follows:
Xjk < Cjk for (j=1,2), (k=1,2,...,m)
(3.20)
Where: xjk = amount of demand assigned to
branch k of section j
Cjk = capacity of branch k of section j.
With these constraints, the total system capacity is indeed restricted,
but not to the physical system capacity. It is intended that the upper
limit on the assigned subsection flows not be the capacity of the sub
section per se, but the maximum level of demand for which the demand
speed characteristics can be determined or estimated. This point will
be addressed more fully in subsequent portions of this chapter.
The final type of constraint to be considered is that of level
of priori y. This constraint class is intended to provide a means
by which the amount of priority that is given to the highoccupancy
vehicles can be controlled. Prior to detailed discussion of these
constraints, some general comments are in order.
The first pertains to the manner in which the level of priority
is to be measured. Several performance characteristics such as speed,
travel, time, and delay are commonly used to evaluate the quality of
traffic flow. These characteristics are strongly related to one another,
as well as to the ratio of system demand to capacity. This demand to
capacity (D/C) ratio is the basic measure of the extent to which a
system is loaded. As this ratio increases as a result of increasing
demand, the operating speeds tend to decrease and system travel time
and delay tend to increase. Recognizing this interdependence and the
fact that the basic parameters that have been utilized in the model
to this point are the system demands and capacities, selection of the
D/C ratio as the measure of priority level wouid be appropriate. Spe
cifically, the level of priority will be reflected by the relative
D/C ratios for the reserved and nonreserved sections.
Attention must now be directed toward the development of a
series of constraints which will provide control over the degree of
priority which is to be afforded highoccupancy vehicles. The basic
constraint is that the priority section should always operate with a
D/C ratio less than that of the nonpriority section. By definition,
priority treatment cannot be provided if this constraint is violated.
In general terms, this minimum level of priority constraint can be
expressed as follows:
D/C, D/Cp > 0 (3.21)
Where: D/Cn demand to capacity ratio For
the nonpriority section
D/C = demand to capacity ratio for
the priority section.
In terms of the model variables, Equation 3.21 becomes
S2k Xlk
k k1  0 (3.22)
Where: Xjk = assigned flow on branch k of section j
c. = capacity of section j.
J
However, in some cases it might be desirable to maintain a
minimum differential between the D/C ratios for the two sections. In
this event, another constraint which requires a fixed level of priority
would be required. This constraint would be of the form
D/Cp D/Cn a
Where:
(3.23)
a = a minimum level of priority differential.
In terms of the model variables, this is expressed as follows:
(3.24)
The third type of priority level constraint addresses the case
where it is desired to provide an increasing level of priority as the
system becomes more heavily loaded. This condition can be expressed
as
6 (D/C ) D/Cp = 0
Where:
(0 < B 5 1)
(3.25)
, = desired ratio of D/C values.
Again, in terms of the model variables,
(3.26)
The final consideration is to allow a decreasing level of
priority as the system loading increases. This can be expressed as
1 (0 < p < 1) (3.27)
P (P"/Cn) DIC p
Where: p = offset of the D/C vs. D/C curve along
the D/C, axis.
Expressing this in terms of the previous model variables,
m m
7 x2, k IY x1k
1 > k_1 P (3.28)
1 p C 1 p
In summary, four priority level constraints have been formu
lated. These are:
Type 0 minimumm Level of Priority)
D/Cn D/C p 0 (3.29)
Type 1 (Fixed Level of Priority)
D/Cn D/Cp a (3.30)
Type 2 (IncrL.ising Level of Priority)
(D/Cn) D/Cp 0 (3.31)
Type 3 (Decreasing Level of Priority)
1 (D/Cn) D/Cp 3.
These priority level constraints are shown graphically in Figure 3.9.
Type
Type 3~. /
1/
0 //
1.0
T //
I /
0 Type /
a0 1
/
ci
C/
4,
0/ /0
n r e/
/ Type 1
'Type 2
Figure 3.9 LEVEL OF PRIORITY CONSTRAINTS
Naturally, these constraints can be employed independently to
define an operating strategy for the priority lane system. A close
examination of Figure 3.9 reveals that certain combinations of these
constraints also define reasonable operating strategies. These combi
nations are a constraint Type 0 or 1 in combination with a constraint
Type 2 or 3. Also, it may be seen from examination of Equations 3.21
through 3.28 that these relationships can be reduced to an equivalent
Type 0 constraint (a=0, B=1, p=0). Thus, it i;, apparent that these
alternative strategies can be expressed with, at most, two constraint
equations. These equations are as follows:
m m
x^k E x1k
k=EI cx2k. kj1 6 (3.33)
c2 L c,__
m "m
Z x2k xlk
S. k__ kc1l > (3.34)
c2 cl
Where: 6 = 0 or a
(,,T)= (8,0) or ( p ) .
This final formulation, Equations 3.33 and 3.34 can then be used as
model constraints to control the level of priority which will be given
to highoccupancy flow under any operating strategy.
Summary of Opt imi nation Submodel
In summary, the linear programming submodel is a deterministic
optimization procedure for identifying the optimal priority and non
priority flows with the objective of minimizing the weighted total
travel time. Constraints have been developed for the model which
require that all system demand be satisfied, that no travel branch
is loaded beyond capacity, and that the level of priority given to the
highoccupancy vehicles meets certain criteria. This model structure
is shown in matrix form in Table 3.1.
Solution Methodolo~y
Underly ing Process
The structure of the model which has been developed for deter
mining an optimal car pool definition for HOV priority lane systems
is based on the solution of an equilibrium model with an internal
linear programming optimization submodel. For the model as developed,
the following steps should be included in the solution technique:
1. Establish system parameters and operating character
istics. This includes determination of the capacities
and speedflow relationships for the priority and
nonpriority sections, as well as ascertaining the
vehicular demand by level of occupancy.
2. Develop optimization constraints. These are both
the demand arn; capacity constraints, and the
operating strategy constraints.
Table 3.1 OPTIMIZATION SUBMODEL STRUCTURE
Decision Variable
Sense RHS
Xl1 X12 x13 x14 x15 x21 x22 x23 x24 x25 n
1 1 1 1 1 1 1 1
            
1 1 1 1 1 1 1 I 1 1
1 __ r2
C C CCC C1 11p
p p p p cn 1
1 1 1/ P 11 1____1
p p p p p n n CC Cn
1 1 1 1 1 1 1 1 1i T di
< C. 11
1< c12
1 < C13
< C14
1 15
I C 21
<__ c___ 22
S
< 23
i < C24
I < C25
S11 S12 S13 514 S15 521 22 '23 524 S25 = Z
3. Estimate average section flow weights. These
estimates will be refined within the equilib
rium model operation.
4. Develop cost coefficients for objective Function
of optimization submodel. These are based on
the speedflow relationships and flow weight
estimates.
5. Find optimal flow levels for priority and non
priority sevi.ions. This requires solution
of the optimization submodel.
6. Determine actual section flow weights. This
requires assignment of the demand to the
sections considering the optimal flow levels
and the violation and nonutilization factors.
7. Check for equilibrium. If the original flow
weight estimates compare favorably with the
actual values, the process can be terminated.
If not, revise the flow weight estimates and
the objective function cost coefficients and
repeat steps 5 through 7 until equilibrium
has been achieved.
8. Formulate recommendations. In this final
step, the results of the optimization process
are examined to develop a recommended car pool
definition.
This solution process is illustrated in Figure 3.10.
Figure 3.10 CAR POOL DEFINITION MODEL SOLUTION PROCESS
Recommended Techniques
Specific techniques which can be utilized in this solution
process have been left to the discretion of model users since they
will vary depending on the particular application. However, the
following considerations should serve as a set of general guidelines.
In the previous discussion, reference has been made to speed
flow characteristics of the system. For operating at levels below the
point where the system demand is equal to the capacity (D/C 5 1.0),
this relationship can be determined by direct field measurements, or a
general relationship can be obtained from any one of several reference
sources. In most cases, however, HOV priority techniques are not
considered until such time as the demand approaches or exceeds the
capacity. Additionally, for a particular application no assurances
can be given that optimal system performance will allow both the
priority and nonpriority sections to operate with demand less than
capacity. For these reasons, the model itself does not require opera
tion with D/C ratios less than 1.0.
In order to permit the model to treat the oversaturated condi
tion, the demandspeed relationships for this range must be provided.
This is by no means a simple task since system demand is difficult to
measure in most cases. It is suggested that consideration be given
to the use of current travel time prediction techniques to extend the
speedflow relationships into the oversaturated region, where they
become speeddemand relationships. Detailed considerations for this
extension are presented in Appendix A of this report. At this point,
it will be sufficient to say that the operational characteristics in
the oversaturated range can be determined by interfacing the speed or
travel time relationships for the two operating regions.
A second point to consider with respect to the speeddemand
relationship pertains to the use of the piecewise linear approximation
(PLA) to the weighted total travel time vs. demand curve as developed
earlier. The use of this technique results in the assumption of
constant speed operation within the range of each PLA segment or branch.
The effect of this can be minimized by the number of PLA branches in
volved and the proper selection of the intersection points for the line
segments. Traditionally, a 3segment PLA has been utilized to describe
the undersaturated portion of this relationship. Considering the ex
tension into the oversaturated region, it is suggested that a 4 or
possibly 5segment PLA would be appropriate for the purposes of this
model. Additionally, selection of the line segments in such a manner
as to concentrate relatively short branches about the expected operating
range will improve the accuracy of the analysis process.
With regard to the equilibrium model, two points should be
considered with respect to the average flow weights. First, for
an initial estimate it is suggested that the overall average flow
weight be used for both sections. This will key the initial estimate
to the occupancy level distribution and, if all flow weights are
equal to 1.0, as would be the case for minimum vehicle travel time,
only one iteration of the equilibrium model will be necessary. Second,
in revising the flow weight estimates based on subsequent computed
values, it is recommended that the stepsize used in Equation 3.8 be in
the range of 0.65 to 0.85. The experience gained during the various
applications of the proposed model indicated that lower values tended to
retard the equilibrium process, while higher values seemed to increase
the likelihood of developing an oscillatory pattern which can prevent
proper convergence of the model.
The solution of the linear programming submodel can be accom
plished with a variety of techniques. These techniques are well
documented in the literature, and several procedures are available
for computer solution at most major installations (Shamblin and
Stevens, 1974, p. 295]. Individual users should consider utilization
of those techniques with which they are most familiar.
This final set of comments deals with the development of a
recommended car pool definition. The information available at the
end of the equilibrium model process is an optimal assignment of flow,
by level of occupancy, to the reserved and nonreserved sections.
As it is unlikely that these optimal flows will break on an occupancy
level boundary, the boundaries on either side of the optimal configura
tion must be investigated. When this is done, the boundary with the
lesser deviation from the optimal condition should be recommended
as the car pool definition provided that the priority level constraints
would not be violated. Under some conditions, the less restrictive
definition (lower boundary) can result in the D/C ratio actually
being higher for the priority section than for the nonpriority section.
If this is the case, a check of the system operation at the less
restrictive definition with user optimization (equal demand/capacity
ratios) should be made and compared with the more restrictive defini
tion as before.
Program CARPOOL
One final note is that the procedures and techniques previously
presented have been used to develop algorithms for application in a
computerized car pool definition model, CARPOOL. This computerized
model will be utilized for the example applications to be presented
in subsequent chapters of this report.
No documentation of the program operation will be presented
here, inasmuch as its algorithmic structure is based directly on
previous considerations. However, it should be noted that the program
was written in the FORTRAN IV programming language and was developed
for operation on an IBM System 370/165. A block diagram of the pro
gram operation is presented in Figure 3.11 for additional reader
information.
I
IL
I
I
I 
I
IL
IF
Figure 3.11 BLOCK DIAGRAM FOR PROGRAM CARPOOL
63
 MAIN
BLOCK DATA
 Subroutine
READIT
.. . Subroutine
CONVRT
I 
Subroutine
MODEL
I
i Subroutine
SIMPLX
I Subroutine
i OPT
I
Subroutines
SUMUP,
ABOUT
J
Figure 3.11 continued
Revis
Weig
Objec
Fu ncd
_ r
Y es
Formulate
Recommended
Car Pool
Definition
Report
Recolimended
Definition
CHAPTER 4
VALIDATION AND APPLICATION OF THE CAR POOL DEFINITION MODEL
Introduction
Objectives
A natural concern associated with the use of any modeling tech
nique, either mathematical or physical, is how well the model reflects
the operation of the fullscale system. If the model is inaccurate
or 'es not consider all system aspects, the results of any investi
gation with the model will be, at best, highly questionable. An
additional concern is whether the scope of the model is sufficient
for application to the particular problem under investigation. As a
general rule, it is more desirable to develop a model in such a manner
that application to a variety of situations or physical variations is
possible, rather than to limit the application of the model to a pre
determined case.
This chapter addresses these concerns with respect to the car
pool definition model which has been proposed. A demonstration of how
well the model represents the physical system operation will be pre
sented in the form of a validation process. The scope of the model
will be defined through a discussion of the potential applications of
the proposed technique. Additionally, some general guidelines for con
sideration with respect to priority lane operations will be presented.
Organization
The first area to be addressed is the validity of the proposed
model. This section will present a number of consideraLions in support
of the overall accuracy of the modeling process. Next, the flexibility
of the model will be demonstrated. This demonstration will consist
of an examination of the basic application of the model and identifi
cation of other considerations which may be investigated with the model.
Finally, the general guidelines will be presented.
Validation of the Model
Validation Methodology
Simply stated, validation of a mathematical model consists of
verifying that known physical system operations are adequately reflec
ted by the model. The validation process, in and of itself, cannot
make a strong positive statement with respect to the accuracy of the
modeling technique. The fact that a model can reproduce known condi
tions does not insure that it can reliably predict operations under
other conditions. However, an absence of the ability to reproduce
these known conditions does make an extremely strong negative statement
about the validity of a model.
Traditionally, the validation process has consisted of applying
the model to an existing system for which the operating characteristics
are known, and determining how well the model reproduces this system
operation. However, a somewhat different approach to the validation
of the car pool definition is required. This is necessitated by the
fact that the required data for an existing system are not available,
and the field implementation and evaluations necessary to produce them
are beyond the scope of this effort. The validation of this model will
then consist of a series of sensitivity analyses, designed to demon
strate the effects of varying system parameters on the predicted system
operation, and a comparison of the results obtained with the proposed
technique and currently accepted simulation models, which will show
that the model accurately reflects the system operation.
Sensi tivity analyses
In this section, the sensitivity of the car pool definition
model to varying system operating parameters will be addressed. Through
this investigation, the degree to which the model predictions conform
to accepted traffic flow relationships will be demonstrated. These
sensitivity analyses will be limited to those parameters which are
scalar quantities or to relationships which may be described by a scalar
quantity. Specifically, this section will present the sensitivity of
the model to variations in (1) the priority section capacity, (2) the
nonpriority section capacity, (3) the total system demand, and (4) the
demand distribution with respect to the level of occupancy.
Prior to beginning the sensitivity analyses, it is necessary
to establish a base condition or basic system for subsequent variation.
For this purpose, a portion of an HOV priority land system currently in
operation on 195 in Miami, Florida, was selected. This test system is
16,910 Feet (3.07 miles) in length and has 1 reserved lane and 4 non
reserved lanes. The estimated capacity of this system is 1,500 vehicles
per hour (vph) in the reserved section and 8,000 vehicles per hour in
the nonreserved section. The free flow operating speed for each
section was assumed to be 60 miles per hour (mph). The total peak
hour demand on this system is 8,463 vehicles, and the violation and
nonutilization rates were assumed to be 10% and 15%, respectively,
for each occupancy level. This test section is shown in Figure 4.1,
and the operating characteristics and deiind description are presented
in Tables 4.1 and 4.2. The speeddemand relationship developed in
Appendix A was adapted for this system and is presented in Figure
4.2.
An initial evaluation of this system indicates that for minimum
passenger travel time, a car pool definition of 3 persons per vehicle
(ppv) is preferable, with the total travel time equal to 914 passenger
hours. For minimum vehicle hours of travel, a car pool definition of
3 or 2 ppv results in 684 vehicle hours of travel, if user optimization
is assumed for the lower definition. At optimality, the minimum total
passenger and vehicular travel times were found to be 913 and 684 hours,
respectively.
A widely accepted traffic flow relationship is that the total
travel time on a roadway has an inverse curvilinear relationship with
the roadway capacity when other factors remain constant [Highway
Research Board (HRB), 1965]. It can be inferred from this that if the
capacity of the priority section of the basic system were decreased,
the optimal total travel time would be increased.
In order to demonstrate that this relationship is reflected in
the car pool definition model predictions, a series of evaluations were
perfonned at various priority capacity levels, ranging from 70% to 130%
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Table 4.1 SUMMARY OF THE TEST SYSTEM OPERATING CHARACTERISTICS
FOR THE CAR POOL MODEL SENSITIVITY ANALYSES
I Capacity Free Flow Speed
SLength Number of Lanes, (P h
Nofeen Non Non
Sect1 (fet) Reserved Reserved rvd Reserved
S reserve_ reserved reserved
1 448 1 4 1500 8000 60 60
2 2577 1 4 1500 8000 60 60
3 20757 1 4 1500 8000 60 60
4 3091 1 4 1500 8000 60 60
5 1644 1 4 1500 8000 60 60
6 1054 1 4 1500 8000 60 60
7 1506 1 4 1500 8000 60 60
8 3795 1 4 1500 8000 60 60
Tota 16,190 NA NA NA NA NA NA
Table 4.2 SUMMARY OF THE DEMAND CHARACTERISTICS FOR
THE CAR POOL MODEL SENSITIVITY ANALYSES
Level of
Occupancy
(ppv)
Demand
Vehicles
1 6287
2
3
4
5
Total 0
6 0
Total 8463
1613
448
76
39
Violation Nonutiliza
Rate tion Rate
(%) (%)
10.00 15.00
10.00 15.00
10.00 15.00
10.00 15.00
10.00 15.00
10.00 15.00
NA NA
71
60
50
Nonpriority_ \ Priority
Section Section
40
ra
Ea
a\
 30
U )
20
10
0 \
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Demand/Capacity Ratio
Figure 4.2 ASSUMED DEMANDSPEED RELATIONSHIPS FOR
THE CAR POOL MODEL SENSITIVITY ANALYSES
of the basic system value. The results of these analyses, presented in
Table 4.3, show that this relationship is maintained within the mathemat
ical model. The expected curvilinear nature of this relationship and
the sensitivity of the model to priority capacity variations are both
shown in Figure 4.3.
This same relationship of total travel time and capacity should
also hold true for variations in the capacity of the nonpriority section.
To ascertain that this is indeed the case, a similar series of analyses
were conducted for varying levels of nonpriority capacity, again ranging
from 70% to 130% of the basic system value. The results of these analy
ses are presented in Table 4.4, and again reflect the expected relation
ship. The sensitivity of the model to nonpriority capacity variations
is shown in Figure 4.4.
A second relationship that should be reflected in a valid
traffic flow model is the effect of demand on total travel time. As the
vehicular demand on a section of roadway increases, the total travel
time also increases, but in a nonlinear fashion. In order to demonstrate
the validity of the model with respect to this relationship, a third
series of analyses were conducted in which the total demand was varied
within the range of the basic system demand i30%, and all other factors
were held constant. The results of these analyses, contained in Table
4.5, again demonstrate that this basic relationship is reflected by the
mathematical model. The nonlinear nature of this relationship and the
sensitivity of the model can be seen in Figure 4.5.
A Final consideration in this series of sensitivity analyses
is the effect of shifts in the distribution of the total demand with
Table 4.3 OPTIMAL SYSTEM OPERATION FOR VARYING
PRIORITY SECTION CAPACITIES
Capa Total Vehicle Hours Total Passenger Hours
city Reseved N
(vph) Reserved NToteservserved Non Total
reserved I reserved
1050 73.5 710.7 784.2 120.4 927.3 1047.7
1200 84.1 658.9 743.0 151.9 839.3 991.2
1350 94.6 607.7 701.7 183.4 753.8 937.2
1500 103.6 580.5 684.1 189.7 724.0 913.7
1650 111.5 568.1 679.6 182.0 724.0 906.0
1800 119.1 556.1 675.2 174.4 724.0 898.4
1950 126.3 544.4 670.7 166.7 724.0 890.7
J 15
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Hours
20 10 0 10 20 30
% Change in Priority Capacity
Figure 4.3 EFFECT OF VARYING PRIORITY SECTION CAPACITIES
ON MINIMUM TOTAL SYSTEM TRAVEL TIME
74
Table 4.4 OPTIMAL SYSTEM OPEPAFION FOR VARYING
NONPRIORITY SECTION CAPACITIES
Capa Total Vehicle Hours Total Passenger Hours
city
(vph) Reserved Non Total Reserved Non Total
i reserved reserved
5600 273.9 1056.7 1330.6 544.7 1240.5 1785.2
6400 105.1 990.1 1095.2 208.6 1207.6 1416.2
7O00 105.1 772.7 877.8 208.6 942.4 1151.0
8000 103.6 580.5 684.1 189.7 724.0 913.7
8800 92.5 570.4 662.9 87.6 801.5 889.1
9600 83.1 558.6 641.7 87.6 772.8 860.4
10400 74.9 545.4 620.3 87.6 744.1 831.7
1 Lp
Passenger Hours
20 10 0 10 20 30
Change in Nonpriority Capacity
Figure 4.4 EFFECT OF VARYING NONPRIORITY SECTION CAPACITIES
ON MINIMUM TOTAL SYSTEM TRAVEL TIME
Table 4.5 OPTIMAL SYSTEM OPERAII FOR VARYING
LEVELS OF VEHICULAR DEMAND
SD n Total Vehicle Hours Total Passenger Hours
en(veh) Reserved Non Non .
Seh Reserved N Total .Reserved Total
reserved reserved
5924 60.9 341.1 402.0 103.8 411.8 515.6
6770 75.1 420.9 496.0 106.0 555.4 661.4
7617 89.4 500.7 590.1 97.9 690.9 788.8
8463 103.6 580.5 684.1 189.7 724.0 913.7
9309 105.1 879.9 985.0 205.0 1089.8 1294.8
10156 105.1 1204.9 1310.0 194.9 1521.9 1716.8
11002 127.4 1509.6 1637.0 230.1 1923.3 2153.4
E
F
100
F
I
0 0
4 5
=
^r
0c 
Vehicle Hours
20 10 0 10 20
% Change in Vehicular Demand
Figure 4.5 EFFECT OF VARYING VEHICULAR DEMANDS
ON MINIMUM FOTAL SYSTEM TRAVEL TIME
respect to the level of occupancy. As the proportion of singleoccupant
vehicles in the traffic stream increases, the relative passenger demand
on the system decreases for constant total vehicular demand. This
shift in distribution should have no effect on the minimum vehicular
travel time, since the total vehicular demand is fixed. However, since
the persondemand is decreasing, the optimal total passenger travel time
shoul also be decreasing. The demand distribution originally assumed
for the basic system, Table 4.2, cannot be directly expressed as a
scalar quantity for this analysis. However, a plot of these data seems
to indicate that they are distributed with an approximate negative
exponential relationship, which may be described with a scalar parameter.
A mathematical expression of this type can then be used to approximate
this demand distribution. The general form of the expression is given
by Equation 4.1.
(% ? n) = 100ep(n1) (4.1)
Where: (% ? n) = percent of the demand with n or
more occupants
p = constant determining the relative
curvature of the relationship.
This shifted negativeexponential relationship approximates the original
demand distribution at a value of p=1.30.
With this expression for the demand distribution, additional
distributions can be developed by varying the parameter p, which is a
scalar quantity. This procedure will permit a sensitivity analysis to
be performed for the effect of shifts in the demand distribution. For
the purposes of this sensitivity analysis, values of p were selected
over the range of 1.1 to 1.6. This distribution will show a shift into
singleoccupant vehicles for increasing values of p. The original
cumulative demand distribution and the variation introduced with this
expression are shown in Figure 4.6.
Applying these occupancy distributions to a fixed total demand,
the effect of occupancy shifts can be examined. As is shown in Table
4.6, the hypothesized relationship between demand distribution and
total travel time is reflected by the optimization process. The sensi
tivity of the model to shifts in the occupancy distribution can be seen
in Figure 4.7.
C'Larison with Simulation Technique
Another technique which can be used to infer the validity of the
car pool definition model is a comparison with a currently accepted
simulation technique. For this purpose, the PRIFRE priority lane
simulation model [Minister et al., 1973] was selected to provide
additional verification of the proposed optimization procedure. If
both models are applied to a system in such a manner as to avoid con
flict between the basic assumptions of the models, the resulting pre
dictions of system operation should be comparable.
In order to compare these models on a common basis and elimi
nate discrepancies which might result from conflicting assumptions of
the two techniques, the test case must be carefully defined. The system
which was developed for this comparison is a priority lane treatment
2 miles in length with 1 reserved lane and 2 nonreserved lanes. The
capacities of the reserved and nonreserved sections were assumed to be
\ e1. 1(n1)
\ \\ /Original Data
3 4
Figure 4.6 DISTRIBUTION OF VEHICULAR DEMAND
BY LEVEL OF PASSENGER OCCUPANCY
Table 4.6 OPTIMAL SYSTEM
VARYING DEMAND
OPERATION FOR
DISTRIBUTIONS
Total Vehicle Hours Total Passenger Hours
R Non Non
Reserved .erved Total Reserved Reserve Total
1.1 103.6 580.5 684.1 214.3 801.9 1016.2
1.2 103.6 580.5 684.1 203.5 768.4 971.9
1.3 103.6 580.5 684.1 195.3 739.2 934.5
1.4 103.6 580.5 684.1 186.6 716.5 903.1
1.5 103.6 580.5 684.1 176.4 700.3 876.7
1.6 103.6 580.5 684.1 168.0 685.6 853.6
(%> n) = 100 e(n1)
Passenger Hours
Vehicle Hours
20 10 0 10 20 3
% Change in Distribution Parameter
Figure 4.7 EFFECT OF VARYING DEMAND DISTRIBUTIONS ON
MINIMUM IOTAL SYSTEM TRAVEL TIME
1,600 vph and 3,700 vph, respectively, and the speeddemand relation
ships used in the previous section were adopted for this comparison.
The total demand was assumed to be constant at 4,500 vph, and the
occupancy distribution was allowed to vary over the range identified
in the last section of the sensitivity analyses. The resulting vehicle
and passenger demands are shown in Table 4.7. Finally, for this com
parison, a car pool definition of 2 ppv was assumed for the simulation
model, and minimum vehicular travel time was used as the objective
criterion in the optimization model.
Both models were applied to this test system for each of the
5 occupancy distributions. As can be seen in Table 4.8 and Figure 4.8,
the results of these analyses show that the predicted system operations
were similar for both models. Comparison of thuse results indicates
that the average discrepancy was 8.4 vehiclehours or 2.7%. The maxi
mum difference of 17 vehiclehours (5.3%) was observed at the apparent
discontinuity in the predictions of the PRIFRE model and is possibly
attributable to that model's treatment of the piecewise linear approxi
mation of the demandspeed curve.
Applications of the Model
Objective
In this section, a number of potential areas of application
for the car pool definition model will be examined. This presentation
will not be a series of "cookbook" procedures; rather, it will serve
to identify various ways in which the model can contribute to investi
gations of priority lane treatments. Although it is quite possible
that additional applications will arise in the considerations of a
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Table 4.8 TOTAL TRAVEL TIMES PREDICTED BY
CARPOOL AND PRIFRE MODELS
PRIFRE
CARPOOL0
Figure 4.8 COMPARISON OF TOTAL TRAVEL TIME PREDICTIONS
OF CARPOOL AND PRIFRE MODELS
400 i
300
200
100
1.1 1.2 1.3 1.4 1.5
particular system, the following discussion will address those areas
which are of more general interest.
Basic Car Pool Definition
The simplest and most obvious application of this model is the
development of a basic car pool definition; that is, a single definition
to be applied throughout the system for the full period of operation.
It is this fixed definition concept that has been adopted for all HUV
priority lane systems implemented to date.
An application of the model for this purpose is reasonably
straightforward, as has been discussed. One primary area of concern,
however, should be the development of the system demandspeed relation
ships for oversaturated operation. If it is likely that the freeway
system will operate with demands near or in excess of the capacity,
as would be true in many cases, estimation of these operating charac
teristics is necessary to allow the total system demand to be considered
in the optimization process. Additionally, it might be beneficial or
necessary in some instances to allow congestion in some portion of
the system in order to achieve overall optimality. A convenient method
of extending the basic speedflow relationships into the oversaturated
region is the utilization of a travel time estimation procedure. These
procedures were developed primarily for use in the transportation
planning process; however, they are readily adaptable for use in this
model, as is discussed in Appendix A.
Another area in which caution would be advised is the segmen
tation of the freeway system into homogeneous subsystems. If the system
is reasonably stable with respect to level of demand and capacity
throughout its length, this subdivision process would not be necessary.
On the other hand, if these parameters vary to a significant degree,
i.e., changes in the number of lanes or substantial changes in the
demand, it would be advisable to conduct the system analysis as a series
of subsystem analyses. Thi.. multiple analysis process would avoid the
situation in which the demand or capacity within any section would he
incorrectly considered.
Spatial Variation of the Car Pool Definition
A second area which merits investigation in an HOV priority
lane system is the concept of a spatially varying car pool definition.
As has been mentioned, it is quite possible that within a given priority
lane system, sections with distinctly different demand/capacity ratios
may be evident as a result of demand variations, capacity variations,
or both. If this situation does exist, it could possibly be used to an
advantage in lowering the overall travel time by developing different
car pool requirements for the various sections.
In considering this approach, a series of analyses with the
car pool definition model would be of use. The potential benefits of
this concept can be determined by performing an independent analysis
for each system section in which the capacities, demands, or operating
characteristics vary. These applications would then define the optimal
car pool definition for each section, which in turn would become a
set of spatially varying car pool definitions for the entire system.
Another application of this concept might also be of interest
in conjunction with priority lane systems. This is the determination
of the appropriate geographical limits for the priority treatment.
Conceptually, this determination can be made by extending the idea of
a spatially varying car pool definition to its logical conclusion. This
would occur at the point where the optimal definition is found to be
1 person per vehicle. With this definition, the priority lane no
longer offers any benefits to highoccupancy vehicles and should be
discontinued. Thus, with this technique, the point at which the lane
should be returned to normal operations can be determined. The point
at which the priority treatment should begin can be similarly defined
as that point at which the optimal car pool definition becomes 2 or
more persons per vehicle.
Temporal Variation of the Car Pool Definition
A third area of consideration with regard to priority 'ine
systems is the use of a temporally varying car pool definition. In
herent in the structure of the analysis model previously developed
is the assumption of constant levels of demand. While this was ex
pedient for the modeling process, it does represent a simplification
of reality, in that the peakperiod demand pattern is normally tri
angular or trapezoidal with respect to time. By investigating this
demai.d fluctuation, it might be possible to improve the system opera
tion by developing a time variant set of car pool definitions.
The analysis procedure required for this investigation can
be described as the development of an independent car pool definition
