Title: Priority lanes on urban freeways
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 Material Information
Title: Priority lanes on urban freeways some operational considerations
Physical Description: xv, 201 leaves : ill., graphs ; 28 cm.
Language: English
Creator: Culpepper, Thomas Hamilton, 1949-
Publication Date: 1977
Copyright Date: 1977
 Subjects
Subject: Express highways -- Mathematical models   ( lcsh )
Traffic engineering -- Mathematical models   ( lcsh )
Civil Engineering thesis Ph. D
Dissertations, Academic -- Civil Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 194-200.
Statement of Responsibility: by Thomas Hamilton Culpepper.
General Note: Typescript.
General Note: Vita.
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Bibliographic ID: UF00099389
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000209977
oclc - 04164130
notis - AAX6796

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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 1-95 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 DEMAND-SPEED RELA1IONSHIPS.......... 178

APPENDIX B. SUMMARY OF THE 1-95 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 ORIGIN-DESTINATION 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 1-95 ANALYSIS SECTIONS--PM PEAK PERIOD....................

7.2 1-95 ORIGIN-DESTINATION TABLE.............................

7.3 1-95 ANALYSIS SECTION DEMANDS .............................

7.4 PASSENGER OCCUPANCY DISTRIBUTIONS FOR 1-95 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 1-95 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 HIGH-OCCUPANCY 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 DEMAND-SPEED 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 DEMAND-SPEED 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 1-95 CORRIDOR.................. ..................... 150

7.2 SCHEMATIC OF 1-95.................. ..................... 152

7.3 DEMAND-SPEED RELATIONSHIPS FOR THE 1-95 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 VOLUME-TRAVEL 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 DEMAND-SPEED 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 high-occupancy 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

high-occupancy 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 "high-occupancy" 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

home-to-work trip accounted for 31.9% of the total person-trips and 33.7%

of the passenger-miles of travel in 1976 [MVMA, 1976]. In raw numbers,

this means that 37,102,997,860 person-trips covering 357,626,566,400

vehicle-miles 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 high-speed, limited-access roadways was 88%

complete in 1975 with a total designated mileage of 42,500. The concept

of high-speed facilities has been adopted in most metropolitan areas,

and local cross-town 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 fixed-route bus systems has been utilized in

several localities. Additionally, experimentation with the concept

of demand-responsive 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 peak-hour, home-to-

work travel. This travel accounts for 33.7% of the yearly person travel

and is made during the combined daily peak-periods, 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

high-occupancy vehicles in the urban corridors. This is not a new or

revolutionary concept since transit priority in the form of exclusive

rights-of-way 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 normal-flow 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 bus-only lane. As to the energy consumption, an

increase in the average peak-hour 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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Statement of the Problem

In the area of designing and evaluating priority treatments for

high-occupancy 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 cut-off level or car pool

definition and control strategies for nonpriority demand, and for

simulating the operation of priority ramp systems, reserved normal or

contra-flow freeway lanes or exclusive rights-of-way. 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

high-occupancy vehicle lane on an urban freeway are (1) the definition

of "high-occupancy" 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 high-occupancy 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 1-95 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

traffic-flow 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 state-of-the-art 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 simulation-based 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

peak-period demand and a simple Greenshields flow submodel. This

model was later refined by Stock [1969] to include consideration of

time-varying demands and a variety of speed-flow 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 third-generation 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 1-90 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

one-way "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 high-occupancy vehicles was the need to consider priority

access at freeway on-ramps. 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 cut-off level for ramp priority and the optimum metering

rates for nonpriority vehicles at each on-ramp. Additionally, the

model simulates the system operation with and without the provision of

priority access. In the simulation process, an on-freeway 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 (UTCS-1) 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 self-assignment of time-varying

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 not-directly 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.







15





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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 high-occupancy 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

"high-occupancy" 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 "high-occupancy" 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 "high-occupancy" 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 broad-range 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 control-parameters 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 speed-flow relationships. If

the utilization of roadway capacity by individual vehicles is viewed

as a basic supply-and-demand 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, vehicle-miles of travel,

and passenger-miles 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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-uu
4 c -

CL




CL 3-
Su -
















00
o I O
























O




S40
E U I
















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/ t-r





-5





should directly reflect the fact that the overall intent of priority

treatment is to reduce the trip time for high-occupancy 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 supply-and-demand 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 multi-commodity 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 multi-copy 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 multi-copy 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 multi-commodity 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 multi-commodity 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











high-occupancy vehicles. This section has certain operating charac-

teristics such as capacity and speed-flow 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





























C r"



S -0 ,- I- O




u U V) u0,
a ) O C iC -
O4-' .r---
> *r LL" .C .-


C +-' Ur 0_ O

S -- 4 -- 0 O












I.
4-
w 3 0 L. t1) LL 0













F-






















F-mF-
: I 0 C II *, I )

0,- tO *r 4 C
CC U CCI- o uC-- -
O O *r- 0 O *i- s

*- 1- C r -- ..





UJ











W
II U O

















_C0 41 0 4 3 0 I 44 <

SCL C +. CLr C ca..- C .


S0 I 0















0 -
r- 4C
I U- 0a
I l -















\ /[-






*O / -tLn +' / ->> +>t^ -- tt 4- 1 "


m f c ) c d; c ( U ) s:a
*'*' t fD U r r 1-no- hi \ g
4- \ t3 C U C U LU -U LU LU S











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 multi-commodity nature of the

system, a discussion of how this is achieved would be in order.

Since this multi-commodity 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 high-occupancy 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 high-occupancy

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 high-occupancy.


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)
-C
cU


i/ 3 3
OO0

4-' 0 Ll.

Cr 4L4
C) r--

.- +>


u r O
C. C-.--
4- *r- 0..
C 0 C
-c) s
J CL i-



*r- ro I-*
SL L
U~ r h



U) CU 0
C) C
C-)
OC) ) --'



C) 0) C)
4-0





4-'




I-

--


4-,-
-c
r
0)






U -
*r- LLi


- 4- -
L *r- O
-1, --

U *r CL.

C- 0. 0

.CO
C: 0

40





(14 > C)











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 speed-flow 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 of-certain constraints which

describe the physical system limitations and the level of priority

which is to be given to the high-occupancy 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 matrix-vector notation, this structure can be

expressed as follows:


MAX, MIN Z = c-x (3.13)


Subject to: A-x { > = -} 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) Non-negative 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 multi-commodity structure and

can be reduced to a single-commodity 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 w-TTTIx, 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(x2-x1) + S3(z-x2)


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 multi-commodity 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
W














LO
CY
Q





,-~ - ,-~ --o CM CM Cm CM CM










C-)


F-:
I-
C












F-
C-)
CMO






CM 2:
C

C-)

co

P1

NJ .-










constraints. The speed-flow 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 high-occupancy

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 high-occupancy 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__ k-c1l- > (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 high-occupancy 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

high-occupancy 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 speed-flow 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 CC-C C1 -11-p
p p p p cn 1


-1 -1 1/ P 11 1____1
p p p p p n n CC C-n
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 speed-flow 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 demand-speed 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

speed-flow relationships into the oversaturated region, where they

become speed-demand 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 speed-demand

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 3-segment 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 5-segment 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 full-scale 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 1-95 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 speed-demand 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%








I II


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0
0
C. U


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LA


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CO

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t"


-4-






LII

.- T


I

III

III










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 DEMAND-SPEED 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
E


w
5-
>

I- 10
C
rO
I-
E
E 5


C


a, 0.
S -
ro


-.
c-3
;-" -3


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 single-occupant

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 person-demand 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) = 100e-p(n-1) (4.1)


Where: (% ? n) = percent of the demand with n or
more occupants

p = constant determining the relative
curvature of the relationship.


This shifted negative-exponential 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

single-occupant 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







































\ -e-1. 1(n-1)


\ \\ /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-(n-1)


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 speed-demand 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 vehicle-hours or 2.7%. The maxi-

mum difference of 17 vehicle-hours (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 demand-speed 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























O CO O CD ) n m Co Co
0 O N O r 0 U3 O CO
L iO LO IT LO LO 0( L r~
o l- LO LD I:Zj D o -zt L) 'T L)




-












-li Ln : D O O C < >
r p











0 rC co Co C C o C i






CD Ci 0 ( Lx) L I 10 C~1i
3



4--
O0






c) co r- : C 'j 't .0 o oC)
a C)i '-j- CY CO I 00 r r!- co I 3









OIi C) LO LO C O O C l 1O

i-i O m-i m 't IJ tj
oo cu (nrn r r m





0 = '. c I t IT
1 Li Co IU () tU ut C)Cu ) 0 1u 0


>- Cl- >. Q- I CL > C Q aC
cj U 0 0 C u 0o)


U) U) CU C a (

L.2 ~ C -..I .C C-.



Co Co1











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 demand-speed 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 speed-flow 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 high-occupancy 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 peak-period 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




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