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Evaluating traveler costs of alternative traffic management plans at highway work zones using a dynamic traffic assignment model

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Evaluating traveler costs of alternative traffic management plans at highway work zones using a dynamic traffic assignment model
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Robles, Juan
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English
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viii, 120 leaves : illustrations ; 29 cm

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Roads -- Maintenance and repair -- United States ( lcsh )
Traffic flow -- United States ( lcsh )
Roads -- Maintenance and repair ( fast )
Traffic flow ( fast )
United States ( fast )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Bibliography:
Includes bibliographical references (leaves 116-120).
General Note:
Submitted in partial fulfillment of the requirements for the degree, Doctor of Philosophy, Civil Engineering.
General Note:
Department of Civil Engineering
Statement of Responsibility:
by Juan Robles.

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University of Colorado Denver
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Auraria Library
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38277129 ( OCLC )
ocm38277129
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LD1190.E53 1997d .R63 ( lcc )

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Full Text
t
EVALUATING TRAVELER COSTS OF ALTERNATIVE
TRAFFIC MANAGEMENT PLANS AT HIGHWAY WORK ZONES USING A
DYNAMIC TRAFFIC ASSIGNMENT MODEL
by
Juan Robles
B.S., Universidad Autonoma de Guadalajara, Mexico, 1986
M.S., University of Colorado at Denver, 1994
A dissertation submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Civil Engineering
1997
i
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This dissertation for the Doctor of Philosophy
degree by
Juan Robles
has been approved
by
Hyman Brown
Date


Robles, Juan (Ph.D., Civil Engineering)
Evaluating Traveler Costs of Alternative Traffic Management Plans at Highway
Work Zones Using a Dynamic Traffic Assignment Model
Dissertation directed by Associate Professor Bruce N. Janson
ABSTRACT
This dissertation formalizes a procedure to evaluate the impacts of traffic
management plans on traveler costs during major highway work. The proposed
procedure is based upon the use of DYMOD, a dynamic traffic assignment program.
D YMOD is used to calculate differences in travel times for alternative traffic
management scenarios, and the implications of these travel times on construction
costs and traffic safety are discussed. In DYMOD, queuing effects are approximated
through capacity losses on links upstream of a bottleneck or a congested link. This
dissertation explains the approximation of queuing effects currently used in DYMOD
and proposes a better queuing approximation technique to account for how queues
develop in traffic networks.
In addition to the base case, forty different work zone scenarios representing different
traffic management plans were analyzed. These scenarios included three different
work schedules, two levels of freeway capacity reduction, four different work zone
segments, and three traffic control measures. The results suggest the following
important observations: (1) the location and length of a work zone within two
interchanges has a significant influence in the total travel times for all users affected
by the freeway work zone; (2) closing one or two on-ramps upstream of a freeway
hi


work zone can improve or worsen total travel time depending on the severity of the
work zone closure and the characteristics of streets adjacent to the work zone; and (3)
the variation in travel demand throughout a day and the availability of alternative
routes will determine the levels and times of closure most appropriate for a particular
highway project
This abstract accurately represents the content of the candidates dissertation. I
recommend its publication.
Signe
Bruce N. Janson
IV


ACKNOWLEDGMENTS
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| My thanks to each of the members of my doctoral committee for their support during
the completion of this thesis.
j
Very special thanks to my advisor, Bruce N. Janson, for his guidance, his patience,
and his unlimited support. I am very grateful for having the opportunity to learn from
him.
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CONTENTS
CHAPTER Page
1 INTRODUCTION__________________________________________________________ 1
1.1 Research Problem...................................................1
1.2 Previous Attempts to Estimate User Costs Due to Highway Work Zones.2
1.3 Proposed Work......................................................4
1.4 Organization of the Thesis.........................................5
2 PROCEDURE TO EVALUATE TRAVEL TIME COSTS DURING
HIGHWAY RECONSTRUCTION.________________________________________________7
2.1 Introduction.......................................................7
2.2 Evaluation of Traffic Impacts......................................8
2.2.1 Traffic Impacts Analysis Tools and Applications..................8
2.2.2 Experiences with Traffic Management During Reconstruction.......11
2.3 Mitigation of Traffic Impacts.....................................12
2.4 Work Zone Traffic Management Plan.................................13
2.4.1 Evaluation of Travel Time Costs.................................15
2.4.2 Traffic Control Costs...........................................15
2.5 Highway Reconstruction Costs......................................17
2.5.1 Contractors risks..............................................19
2.5.2 Expediting Highway Reconstruction...............................19
2.6 Safety............................................................21
2.7 Proposed Procedure................................................25
3 TRAFFIC FLOW___________________________________________________________29
3.1 Traffic Flow......................................................29
3.1.1 Traffic Flow Theories...........................................30
3.1.2 Fluid Approximation of Traffic Flow.............................31
3.2 Overview of Traffic Assignment Techniques.........................36
3.2.1 Traffic Assignment.............................................3 7
3.2.2 Equilibrium on Transportation Networks..........................38
3.2.3 Route Choice....................................................39
3.2.4 Solution Procedures for the UE Problem..........................40
3 3 Travel Time Functions.............................................42
3.4 Dynamic Traffic Assignment Models................................ 46


CONTENTS (Continued)
CHAPTER Page
4 DESCRIPTION OF MODEL USED IN THIS RESEARCH_________________________50
4.1 Description of DYMOD..........................................50
4.1.1 DYMODs Formulation.........................................51
4.1.2 Optimality Conditions of DUE................................57
4.1.3 Convergent DTA Algorithm....................................59
4.2 Model Implementation..........................................60
4.2.1 Study Network...............................................61
4.2.2 Origin-Destination Matrices.................................63
4.2.3 Trip Tables Used............................................64
5 QUEUING AT HIGHWAYS________________________________________________67
5.1 Queuing In General............................................67
5.2 Queuing At Highways...........................................68
5.2.1 Deterministic Queuing Method................................69
5.2.2 ShockWaves in Highways......................................71
5.3 Proposed Approach To Account For Queuing Effects Within DYMOD.76
5.3.1 Approximation of Shock Wave Effects in DYMOD................79
5.3.2 Capacity Adjustment at Upstream Links.......................82
5.4 Speed-Flow Relationship Used in DYMOD.........................84
6 WORK ZONE SCENARIOS________________________________________________87
6.1 Lane closure strategies.......................................87
6.2 Scenarios Modeled.............................................89
7 ANALYSIS AND INTERPRETATION OF RESULTS_____________________________93
7.1 Presentation of Results.......................................93
7.1.1 Case 1: Closing One of Three Freeway Lanes..................96
7.1.2 Case 2: Closing Two of Three Lanes.........................105
7.2 Comments....................................................109
8 CONCLUSIONS_______________________________________________________111
APPENDIX________________________________________________________________114
REFERENCES,
116


FIGURES
Figure Page
2.1 Flow Chart of Proposed Procedure...................................28
3.1 Conservation of Vehicles in a Stretch of Road......................33
3.2 Speed-Flow Relationship............................................42
3.3 Basic Form of Speed-Flow-Density Relationships.....................43
3.4 BPR Travel Time Function...........................................45
4.1 Study Network......................................................62
4.2 Intersection Representation........................................63
4.3 Average-Hour Trip Matrix...........................................65
4.4 24-Hour Profile of Freeway Volumes.................................66
5.1 Freeway On-Ramp Merge..............................................70
5.2 Graphical Evaluation of Queues at Freeways.........................70
5.3 Shock Wave Due to Change in Densities..............................72
5.4 Shock Wave Description.............................................73
5.5 Time-Distance Diagram of Platoon Formation.........................74
5.6 Freeway Section with Off-Ramp......................................83
5.7 Speed-Density Relationship used in DYMOD...........................85
5.8 Speed-Flow Relationship used in DYMOD..............................85
5.9 Travel Time-Flow Relationship used in DYMOD........................86
5.10 Flow-Density Relationship used in DYMOD............................86
6.1 Work Zone Locations on Study Network...............................90
6.2 Areas in a Traffic Control Zone....................................91
7.1 Excess Travel Times for Scenarios with One Lane Closed.............97
7.2 24-Hour Flow Through the Work Zone with One Lane Closed...........100
7.3 24-Hour Flow Through the Work Zone with One Lane Closed...........100
7.4 Travel Time on all Links not on the Northbound Freeway............102
7.5 Travel Time on all Links not on the Northbound Freeway............102
7.6 24-Hour Flow on the Four Northbound Links with One Lane Closed....103
7.7 24-Hour Flow on the Four Northbound Links with One Lane Closed....104
7.8 Links on Alternatives Routes Going Northbound.....................104
7.9 Excess Travel Times for Scenarios with Two Lanes Closed...........105
7.10 24-Hour Flow Through the Work Zone with Two Lanes Closed..........107
7.11 Travel Time on Links not on the Northbound Freeway w/2 Lanes Closed... 108
7.12 24-Hour Flow on the Four Northbound Links with Two Lanes Closed...108


CHAPTER 1
1 INTRODUCTION
1.1 Research Problem
Planning agencies need to estimate the impacts on travelers due to different lane-
closure strategies in order to determine the best traffic control plan. Extensive
maintenance or reconstruction activities at most urban area highway are taking place
due to many reasons: resurfacing, safety and/or capacity improvements, and repair or
replacement of deteriorated pavements and structures. Maintenance and
reconstruction activities at highway generally imply closure of lanes and reduction of
existing capacity. Since high levels of traffic flow are prevalent at most urban roads,
closing highway lanes for rehabilitation and reconstruction purposes causes a lot of
inconvenience to road users and to the community where these works take place.
Accidents, uncertainty in travel conditions, disruptions to businesses, and delay are
some problems due to freeway work zones. Of these traffic impacts, work zone
delay is generally the most important to highway users. The delay attributed to a
work zone is the difference between the normal travel time on the roadway and the
estimated travel time through the work zone. Accurate estimation of travel time
delays due to highway work zones are needed to develop work-zone traffic
management plans as well as to evaluate how reconstruction/rehabilitation projects
can be scheduled to minimize traffic disruptions and construction costs.
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1.2 Previous Attempts to Estimate User Costs Due to Highway Work Zones
Although a lot of work has been done to estimate adverse impacts due to highway
work zones, most of the proposed methods present limitations or can only be applied
to particular problems. Most planning agencies have relied on judgment and
Highway Capacity Manual (HCM) guidelines, which are considered by many to be
too simplistic (Cohen, 1987), and some have experimented with a variety of
approaches using rules of thumb, simulation, modeling, and statistical analysis to
estimate these impacts. Although some of these approaches have helped planners to
different degrees under different settings, most of them either lack consistency in their
formulation, make assumptions that are to limiting, or can only be applied to very
small problems.
Modeling approaches that have been tried to determine work zone impacts vary from
quick estimation to complete simulation. Applied models include:
Sketch planning estimates of route diversion based on capacity evaluations of
alternative routes (JHK, 1981).
Input-output approaches for estimating delays and queue lengths based upon
the difference between cumulative arrivals and cumulative departures at the
work zone (Abrams and Wang, 1981; Morales, 1986).
Microscopic freeway simulation models to estimate highway queuing
characteristics. (Nemeth and Ruphail, 1982; Rathi and Nemeth, 1986; and
Dixon et al., 1995) implemented microscopic approaches to simulate the
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movement of vehicles through freeway work zones and evaluate the effects on
traffic performance of different traffic control schemes.
Macroscopic freeway simulation approaches to estimate additional road user
cost based on speeds and queuing characteristics through the work zone
(Memmot and Dudek, 1984; Dudek et al., 1985; Cohen and Clarck, 1987;
Krammes et al., 1993).
Network equilibrium models that estimate volumes on alternative routes based
on link capacities and travel time through the work zone (Janson et al., 1986).
The data requirements of the above mentioned models vary greatly, however, the
usefulness of a particular model do not necessarily increase with greater data
requirements and theoretical complexity.
Among the models listed above, QUEWZ (Queue and User Cost Evaluation of Work
Zones) is probably one of the most complete (Memmot and Dudek, 1984). It is based
on average hourly volumes and a geometric description of the freeway subsection. It
computes speeds through the work zone using a relationship between speed and
volume-to-capacity (V/C) ratio similar to that presented in the 1985 HCM. The
minimum speed is predicted with a linear regression model that uses the average
speed and V/C ratio through the work zone. The average speed through the queue is
estimated using a kinematic wave model based on Greenshields speed-flow-density
relationship (Greenshield, 1935). Vehicle hours of delay and the length of the queue
are computed using an algorithm similar to the traditional input-output approach.
Although road-user costs predicted by QUEWZ and similar approaches are acceptable
for rural settings where ramps are spaced apart, these methodologies do not work well
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on high-volume urban freeways where diversion of traffic to alternative routes is
possible. These models generally overestimate the delays and queues that actually
develop upstream of work zones. The main reason why QUEWZ and similar
approaches poorly predict the traffic impacts of reconstruction projects in urban areas
is that they do not properly account for the diversion of traffic to alternative routes.
Diversion to alternative routes reduces the amount of freeway traffic trying to pass
through a work zone. The exact pattern and levels of traffic diversion onto arterials
streets are difficult to predict in advance. These will depend on characteristics of the
alternative routes such as existing levels of service, number of signalized and
unsignalized intersections, and configuration of the routes relative to the affected
highway (Janson et al., 1989).
Appropriate modeling of impacts due to highway reconstruction or severe lane-
closing events should include the effects of diverted traffic on alternative routes. This
will require a valid modeling approach capable of analyzing travel demand for
alternative routes through a corridor surrounding the work zone. A dynamic network
equilibrium modeling approach, such as DYMOD (Janson and Robles, 1995), can
improve the prediction of these impacts.
1.3 Proposed Work
This dissertation project formalizes a procedure to evaluate the impact of traffic
management plans on traveler costs during major highway work. The proposed
procedure is based upon the use of DYMOD, a dynamic traffic assignment program.
DYMOD is used to calculate differences in travel times for alternative traffic
4


management scenarios, and the implications of these travel times on the construction
costs and traffic safety are discussed.
Another objective of this research work is to explain the approximation of queuing
effects currently used by DYMOD and propose a better queuing approximation
technique. A description of how queues develop in traffic networks will be given
along with a mathematical derivation of the queuing phenomena taking place in a
small example section of highway.
In this dissertation, all traveler costs are assumed to be a function of estimated travel
times only. That is, the difference in travel time between two scenarios will indicate
the increase in traveler costs (i.e. stops, queuing delay, travel delay, fuel consumption,
and others) between those two scenarios.
1.4 Organization of the Thesis
This dissertation consist of eight chapters. Chapter 1 presents the problem and the
motivation for this work as well as an overview of previous efforts related to the
solution of this problem and the scope and limitations of this dissertation.
In Chapter 2, a review of the main aspects involved in evaluating impacts due to
highway reconstruction is presented before explaining the proposed procedure. The
review consists of a discussion concerning (1) evaluation and mitigation of traffic
impacts due to highway reconstruction activities, (2) traffic control, construction, and
user costs, and (3) safety considerations. A flow chart of the proposed procedure is
included along with a description of each step.
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Chapter 3 includes a general review of traffic flow models. Differences between
microscopic and macroscopic approaches are highlighted and a description of the
fluid approximation of traffic phenomena is given. The second part of Chapter 3
reviews aspects of traffic assignment techniques including network equilibrium, travel
time functions, and dynamic traffic flow considerations.
In Chapter 4, a full description of the dynamic traffic assignment model used in this
dissertation and the necessary steps for its implementation are presented. The
description includes assumptions and formulation of the model as well as optimality
conditions and the solution algorithm used. Also, data requirements concerning
supply and demand characteristics of the study network are given.
Chapter 5 begins with a general description of queuing phenomena and common
methods to estimate traffic queues at highways. An example of how shock waves
form in a traffic stream is given, and the proposed approach to account for queuing
effects in a network (in the context of DYMOD) is explained in the last part of the
chapter. The explanation of the proposed queuing procedure includes a detailed
description of the queuing subroutine implemented in DYMOD which, given the
limitations of the travel time function used, approximates shock wave effects by
adjusting capacities of links upstream of bottlenecks or very congested links.
In Chapter 6, general comments about work zone scenarios are presented and the
work zone cases evaluated in this dissertation are explained. An analysis of the
scenarios evaluated follows in Chapter 7, where results are tabulated and figures are
used to describe the main findings. The analysis includes a discussion of the
influence of some network and work zone characteristics on total travel time. Finally,
conclusions are given in Chapter 8.
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CHAPTER 2
2 PROCEDURE TO EVALUATE TRAVEL TIME
COSTS DURING HIGHWAY RECONSTRUCTION
2.1 Introduction
Adequate planning is required to mitigate corridor-wide impacts of major
urban highway reconstruction projects. Activities spanning a longer time
frame than the actual duration of reconstruction should be carefully
coordinated so as to minimize disruption of traffic patterns and other impacts
to society. Janson et al. (1989), based on a study of 25 highway
reconstruction projects, recommended a process for managing transportation
during highway reconstruction. The process is used to develop and implement
a transportation management plan and consist of the five sequential phases
enumerated next.
(1) Identifying the extent and needs of the plan
(2) Develop and approve the plan
(3) Preparation to implement the plan
(4) Implementing and monitoring the plan
(5) Post-construction activities
Each phase comprehend a series of related tasks (see TRB Special Report 212,
1987 and Janson et al., 1989). Tasks contained in phase 2 include (1)
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evaluation of work zone traffic control schemes, (2) determination of the
adequacy of alternative routes to accommodate additional traffic, and (3)
identification of alternative transportation system management strategies that
can be implemented, among others. The procedure proposed in this
dissertation deals mainly with activities related to phase 2 of the above
process (development of a traffic management plan).
2.2 Evaluation of Traffic Impacts
Most highway reconstruction projects have a unique set of characteristics that
require site-specific analyses and customized solutions. However, many of
the factors that should be considered in evaluating the travel impacts of
highway reconstruction are common to most projects. The travel impacts of
work zone management options may include changes in any of the following:
(1) travel patterns (i.e., destinations, routes, modes, and/or departure times of
trips); (2) traffic conditions (i.e., travel times or average speeds), and (3)
accident rates. In some cases, the travel impacts are confined to the highway
being reconstructed, but at many major projects, the impacts extend to
alternative routes and modes in the corridor (see TRB Special Report 212,
1987). The magnitude of the impacts must be estimated in order to determine
the proper allocation of roadway space to traffic at different times of day.
2.2.1 Traffic Impacts Analysis Tools and Applications
Available analysis tools for the evaluation of highway travel impacts are
grouped next according to their specific application (see Krammes, 1989):
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(1) Network-based planning models perform travel demand estimation
functions (i.e., trip generation, trip distribution, mode split, and traffic
assignment) using a link-node representation of the highway and
transit networks in an urban area. Network-based planning models are
particularly useful if a corridor-wide evaluation is required. The
primary role of planning models would be in the traffic assignment and
mode split analyses required to estimate changes in corridor travel
patterns.
(2) Quick-response estimation techniques perform some or all of the same
travel demand estimation functions as network-based planning models
using simplified analyses that are less time, labor, and data intensive.
Quick-response estimation techniques could be used as an alternative
to network-based planning models if travel impacts are more localized
to the work zone area.
(3) Highway capacity analysis procedures are used to estimate the
capacity of the reconstruction zone and of alternative routes and
modes. Capacity analysis procedures translate roadway, traffic, and
operational control conditions into estimates of capacity, level of
service, average speed, and other measures of effectiveness (MOEs).
(4) Traffic simulation models evaluate the time-varying nature of traffic
flows and the complex interactions among highway geometric
elements in estimating MOEs as a function of roadway, traffic, and
operational control conditions. Traffic simulation models may be used
to simulate existing traffic conditions in the corridor and to estimate
operational MOEs for alternative traffic management plans. Traffic
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simulation models with traffic assignment capabilities may also be
useful in evaluating changes in travel patterns resulting from a
reconstruction project.
(5) Traffic signal optimization models are used to develop optimal signal
phasing and timing plans for isolated signalized intersections, arterial
streets, or signalized networks. The principal role of these models in
the travel impact evaluation process would be to develop
improvements on alternative routes, such as signal re-timing or
parking restrictions, to be included in a traffic management plan.
Current knowledge is limited about how motorists adjust their travel patterns
in response to a major highway reconstruction project. Furthermore, there
have been few reconstruction-related applications of most of the tools
identified above. As a result, it is difficult to make definitive statements about
how accurately these tools would forecast travel demand changes and impacts
due to highway reconstruction. Analysts should clearly understand the
reliability of the tools used before interpreting their results.
Among the major decisions that must be made in selecting the appropriate
analysis tool, the most critical one involves the scope of the evaluation, i.e.,
whether a corridor-wide evaluation is required or whether the evaluation may
be restricted to the highway being reconstructed. Two types of analysis tools
should be considered for an evaluation restricted to the highway being
reconstructed: highway capacity analysis procedures and traffic simulation
models. For major highway reconstruction projects, the impact evaluation
should be corridor-wide. A major issue in a corridor-wide evaluation is how
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travelers may change their routes through the corridor. Therefore, a corridor-
wide evaluation requires an analysis tool with traffic assignment capabilities.
2.2.2 Experiences with Traffic Management Daring Reconstruction
Various efforts to mitigate the impacts of highway work zones have been
proposed and implemented (Janson et al., 1987). These include; innovative
scheduling of construction activities, new materials and placement techniques,
contract incentives, transportation system management (TSM) strategies, and
traffic management plans among others. TSM consists of implementing
measures to maintain a reasonable level of service through and around the
reconstruction work site. TSM actions for handling traffic in reconstruction
corridors include: additional bus service, park-and-ride lots, HOV actions,
vanpooling, expansion of alternative routes, and public information.
Experiences from five major projects: 1-376 in Pittsburgh (1982), 1-93 in
Boston (1985), 1-76 in Philadelphia (1984), US-10 in Detroit (1986), and I-
394 in Minneapolis (1986), are summarized next (see TRB Special Report
212, 1987; Krammes, 1989). The approaches taken to evaluate the potential
impacts included the use of network-base planning models, quick-response
estimation techniques, and highway capacity analysis procedures. In
Pittsburgh, quick response estimation techniques were used in the early stages
of the planning process to evaluate corridor-wide impacts. In Boston, the
results from a recent origin-destination survey, along with capacity analysis
and manual traffic assignment, were used. In Philadelphia, an origin-
destination study was conducted in the corridor; the results from the study,
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coupled with manual traffic assignments, formed the basis for the travel
impact evaluation. In Detroit, the regional network-based planning model was
used to perform traffic assignment analyses for a preliminary assessment of
the travel impacts of the traffic-handling options being considered. In
Minneapolis, planning for the reconstruction period was performed as part of
the broader effort to develop a long-range transportation systems management
plan for the 1-394 corridor.
The traffic management strategies that resulted horn these evaluations were
also different. The experiences from these five projects demonstrate that
major urban freeway reconstruction can be conducted without intolerable
disruptions in corridor traffic flow. The travel impact evaluations for each
project were successful in that they helped to design more effective corridor
traffic management plans.
2.3 Mitigation of Traffic Impacts
An effective traffic management plan should be implemented during highway
construction to minimize traffic delay and to provide the required level of
safety at the construction site. This may include optimizing of work zone
traffic control design and practices, including optimal control device design,
optimal lane-closure configuration, and optimal work-zone length. Other
traffic handling strategies include complete closure or partial closure of the
facility, off-peak hours construction, provision of effective detours, and
implementation of TSM plans mentioned above.
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In planning a major highway reconstruction project, an acceptable balance
must be reached between (1) maximizing the safety and efficiency of the
reconstruction activity, and (2) minimizing the adverse impacts on motorists
and affected communities. The impacts of most highway reconstruction
projects on traffic patterns and economic activity generally translate into direct
costs to the community affected. Normally, users' time or delay costs
resulting from recurrent congestion constitute by far the largest share of users
costs. Consequently, only users time costs are frequently cited as a severity
index or a measure of users losses resulting from reconstruction operations.
2.4 Work Zone Traffic Management Plan
A Work Zone Traffic Management Plan (WZTMP) can be defined as the basic
scheme of moving traffic through or around a construction area. Its elements
include; (1) the type and length of work zone, (2) time of work, (3) number of
lanes, (4) width of lanes, (5) speed control method, (6) and right-of-way
control method. Of these elements, the type of work zone is usually the most
significant.
Two steps are important to develop an effective WZTMP for a particular
reconstruction project.
(1) Determine the nature of construction work and identify alternative
work zone types: (a) consider all elements of the proposed WZTMP,
(b) consider the potential effects on traffic flows due to other works on
the vicinity of the project.
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(2) Perform a traffic control analysis and evaluation of costs and impacts
to select the best traffic control strategy. For work zones where high
flows or severe capacity reduction are present, it may be necessary to
perform a thorough analysis of the entire highway corridor to
determine what alternate highway facilities can accommodate the
excess traffic demand during these periods
For example, a procedure outlined by Abrams and Wang (1981) executes a
series of steps by which characteristics of the work zone are considered to
define feasible TMPs, after which a quantification of impacts is performed
and a benefit-cost analysis is used to select the preferred alternative. In this
study, delays and queue lengths resulting from restricted capacities at work
zones are calculated using the traditional input-output approach shown in
Chapter 6 of the Highway Capacity Manual (TRB Special Report 209, 1985).
Safety, traffic delay, and project cost are the three main factors used to select
work zone strategies, but in some cases, other measures of effectiveness may
be important to quantifying the impacts of alternative TMPs. A list of
impacts is as follows:
Traffic Impacts: Time delay, stops, fuel consumption, operating costs, and
accidents;
Project Cost Impacts: Cost of traffic control, cost of construction;
Environmental Impacts: Air pollution and business losses.
14


2.4.1 Evaluation of Travel Time Costs
In normal highway operations, the three major types of user costs are
generally: (1) vehicle operating costs, (2) accident costs, and (3) user travel-
time costs.
Besides accidents, motorist perceive excess travel time as the main
inconvenience of work zone traffic disruptions. Excess travel time, or delay,
usually represents the biggest increase in users cost for travelers going through
or avoiding a work zone area. The delay attributed to a work zone is the
difference between the normal travel time on the roadway and the estimated
travel time through or around the work zone.
Road user costs are mainly affected by travel demand and roadway capacity.
Highway reconstruction or maintenance activities generally imply disruptions
to normal traffic flow. These disturbances cause vehicle speed fluctuations,
stops and starts, and travel time losses. In general, traffic delay costs are
related to traffic volume, road geometry, time-of-day and duration of the
rehabilitation or reconstruction work, characteristics of the work zone, and the
traffic management plan implemented. This travel time losses are generally
converted into monetary value for use in cost effectiveness or cost benefit
analyses.
2.4.2 Traffic Control Costs
There are many variables associated with construction work zones that limit
the development of guidelines for selecting the most cost effective traffic
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management plan. Some of the factors that influence construction and traffic
management costs are:
(1) type of construction (bridge reconstruction, pavement reconstruction);
(2) location of construction (urban or rural)
(3) length of bridges or roadway under construction;
(4) type of traffic management plan;
(5) traffic control plan layout and devices;
(6) construction phasing details;
(7) total construction cost;
(8) bid item quantities for traffic control measures;
(9) construction procedures and equipment;
(10) construction quality issues;
(11) other factors (available work space for construction, factors affecting
capacity).
Traffic management costs include not only the cost of the traffic control
devices, but also the cost of their installation, maintenance, and removal.
Traffic management programs for some reconstruction projects may account
for 10 to 20 percent of the total project costs.
Its important to recognize that no two construction projects that are similar in
their type of construction are necessarily comparable in their traffic
management needs. In a study to determine construction and road user costs,
and safety impacts associated with traffic control through work zones (on rural
four-lane highways), Bums et al. (1989) found a wide variation in costs. For
example, one-way median crossover costs ranged from $7,111 to $70,605, and
16


two-way median crossover costs ranged from $12,989 to $188,555. This
variation is due to many cost variables, including geometric design, pavement
design, median width, bidding practices, etc. The study used data from 51
construction projects in 11 states and seven types of construction projects
were analyzed for two traffic control alternatives. Bums et al. (1989) found
that the type of construction was the most important aspect affecting the
selection of the most cost-effective traffic control strategy.
Because of many work items and the wide variation in the basis of payment
for traffic control, it is very difficult to find sufficient uniformity among the
states to develop unit price ranges for all items of work that could be applied
on a general basis. Comparisons among projects is difficult since significant
differences are likely to exist Each highway construction project will have a
unique set of conditions and constraints that require individualized analysis
and customized solutions.
2.5 Highway Reconstruction Costs
Typical costs elements of most highway reconstruction projects can generally
be grouped into construction and traffic control activities. Common work
categories are the following (see TRB Special Report 214, 1987).
Site preparation and earthwork, clearing, excavation, placement of
new embankment, and grading;
Drainage: construction of ditches, drains, culverts, and other minor
structures required for drainage;
17


Pavement: all pavement construction -sublease, base, and top course-
on lanes and shoulders;
Structures: rehabilitation or replacement of bridges and larger culverts;
Traffic and safety: placement of permanent traffic control devices,
lighting, signing, fencing, striping, markers, and similar driver aids;
Traffic control: temporary traffic control measures during
construction;
Miscellaneous: other items such as utility pole relocation or curb and
sidewalk construction;
Right-of-way: purchase of land or use permits; and
Engineering, mobilization, and other: engineering design and
oversight, allowances for contractor front-end costs to assemble
necessary equipment and personnel on-site mobilization.
Of these, the pavement category generally accounts for the greater share of
project costs (40 to 80 percent).
The total cost of a project should be carefully evaluated during the planning
and design stage so that the most cost-effective construction methods and
traffic-handling strategies are identified. Construction costs should be
estimated for alternative strategies and combined with user costs to determine
the total project cost.
18


2.5.1 Contractors risks
Contracting for highway construction always includes an element of risk.
Risks are those events and conditions a contractor may face that are difficult
to predict (particularly their timing, cause, and effect). The most common
risks are associated with weather, traffic operations, conflicting utility
services, convenience of property access, and third-party involvement. When
bidding on a project, a contractor is expected to predict various unplanned or
unscheduled events that would interfere with construction activities.
Evaluating the economics of a construction project requires an analysis of all
the potential alternatives that are capable of providing the minimum required
performance. There are several economic analysis methods that compare
alternatives, of which the discounted cash-flow methods (e.g., present worth,
annualized cost, and rate of return) are the most frequent means of
comparison. In the discounted cash-flow scheme, the factors that will
influence the analysis results include inflation, the discount rate, and the
analysis period. Other economic analyses such as benefit-cost ratios, break-
even analyses, pay-back periods, and capitalized cost methods are used less
frequently.
2.5.2 Expediting Highway Reconstruction
Impacts due to work zones can be lessened if the required reconstruction or
maintenance work is done in the shortest possible time. Project time can be
reduced by implementing efficient traffic management strategies that are
necessary to provide a safe working space at the work zone. Besides good
19


traffic management, the required reconstruction or maintenance work can be
accelerated through: (1) highway construction design, (2) construction
techniques and equipment, (3) innovative materials, and (4) project
management (see, Ward and McCullough, 1993).
Expediting through highway construction design'. Expediting highway
pavement construction should be considered at all stages, from the preliminary
planning and design of the project to completion. Properly designed plans
help to reduce construction time and costs, and minimize the effects associated
with the operation. Innovative design methods that accelerate pavement
construction include the use of full-depth material, minimum number of
layers, and thicker pavements.
Expediting through construction techniques and equipment: Some techniques
identified in the literature include the use of prefabricated elements, vacuum
treatment of Portland cement concrete, in place recycling of pavement
materials as polymer and fly ash. Advanced construction equipment can
considerably reduce the duration of highway construction and, therefore,
minimize the delay associated with the operation. Fast-track equipment
include slip-form pavers, automatic dowel bar inserters, improved pulverizers
and strippers, and diamond wire saws.
Expediting through innovative materials: Fast-tracking materials may be
consider to reduce delays caused by construction operations. For example,
high-early strength materials are effective in reducing the time required to
reopen a facility to traffic. And, although substantial costs may be associated
20


with the use of these materials, the cost-effectiveness, in many cases, is
justifiable.
Expediting through project management: Efficient project management,
including multiple contract letting, incentives/disincentives, and computer
applications are proven methods that help in reducing the time to complete a
project. Accelerating projects is a cost-effective approach when the benefits
of the approach (reduced road user costs) exceed the costs of implementation.
In general, project acceleration measures are recommended when (1) high
traffic volumes cannot be easily diverted, (2) significant road user costs can be
saved, and (3) funds are available to cover added contractor costs (see TRB
Special Report 212).
2.6 Safety
Accident rates within construction or maintenance work zones have been
found to be higher than for similar periods before work zones were established
(Nemeth and Migletz, 1978; Hall and Lorenz, 1989; Pigman and Agent,
1990). Some of the factors explaining the increase in accident rates at work
zones are; poor traffic management, inadequate layout of the overall work
zone, inappropriate use of traffic control devices, and a general
misunderstanding of the unique problems associated with construction or
maintenance work zones.
The purpose of traffic control devices is to help ensure the safety and efficient
movement of all traffic throughout a road transport system. During work zone
activities, traffic is dependent on the design, placement, and uniformity of
21


traffic control devices for guidance and warning through and around the work
zones. The primary traffic control devices used most often in work zones are
signs, channelizing devices, and pavement markings. The installation and
maintenance of the devices may be the responsibility of the road authority,
contractor or police. Traffic control devices should be simple to understand
and readily seen and followed by the motorists, and should be located
sufficiently in advance to provide motorists time to react to the situation.
Traffic safety in the work zone should be an integral and high priority element
of every project from the planning stage through the design and operation
phases. Common impacts of most highway work such as; increased traffic
congestion, danger to drivers facing unfamiliar traffic and road situations, and
the potential risk for workers, can adversely affect traffic flow and safety.
Management of roadworks should address these impacts in a way that is
economically acceptable and satisfactory to road users and construction
workers. Implementation of the safety management scheme chosen involves
temporary traffic guidance devices and signs, as well as safety equipment and
driver information.
Safety management principally attempts to reduce the ocurrence of accidents.
Important aspects are to identify, investigate, set priorities, and correct
hazardous locations and features. Managing safety implies that it can be
observed, measured, and optimized. However, safety comparisons of
competing alternatives is difficult because of the lack of a clearly defined
safety scale. Safety is usually accounted for by satisfying the appropriate
State standards. A design is considered safe enough if it meets or exceeds all
22


individual safety criteria. Because relationships between design elements and
safety are not well understood, safety aspects are sometimes overlooked.
When determining the form and timing of highway work zones, full account
must be taken of their effects on the safety of road users and of the need to
minimize possible disruptions of traffic. The aim of traffic management at
major road works is to obtain a balance between providing adequate space for
the contractor to carry out the work safely, economically and speedily, while
also allowing traffic to flow safely and without excessive delay.
The assessment of traffic accident data concerning roadworks remains one of
the major difficulties when attempting to determine the scale of the safety
problem. There are two major types of statistical procedures for accident data
analysis: (1) the Bayesian approach, and (2) the non-Bayesian or frequency
approach which can be broken down into: before-and-after study with no
comparison group, before-and-after study with a comparison group, and
before-and-after study with a comparison group and a check for comparability
(TRB Circular 416,1993).
The frequentist or accident data based models use historical data from
reported accidents to develop multiple regression models for predicting
roadside accident frequencies as a function of roadway characteristics. The
appeal of this approach is that accident frequencies are predicted directly as
opposed to the indirect approach used with the encroachment probability
model.
Accidents are discrete random events that are caused by the combined
interaction of driver, vehicle, and highway characteristics. Some of the
23


techniques proposed to estimate the number or severity of accidents employs
Bayesian conditional probabilities. For example Glennon (1974) developed a
hazard index that includes (1) the expected number of vehicle encroachments,
(2) the probability of a collision given that an encroachment has occurred, and
(3) the probability of an injury given a collision has occurred.
A probabilistic approach to quantifying safety could identify the sites where
the occurrence of severe accidents are likely. Safety implications of design
changes can be compared to determine the safest alternative. Unlike accident
data based models, encroachment probability models can be used to predict
accident frequencies for a variety of roadway geometries and traffic
conditions. It is the only method for predicting accident frequencies for newly
constructed or reconstructed roadways.
The drawback to this procedure is the need to obtain and manipulate a great
deal of data about the geometric, operational, and safety characteristics of the
highway. Obtaining encroachment characteristics such as distributions of
encroachment speeds and angles, lateral vehicle movements, sizes of
encroaching vehicles, and others, is difficult.
Most of the existing cost-effectiveness procedures are based on the concept of
benefit-cost analysis. Benefits derived from a safety improvement are
measured in terms of reduced accident frequency and/or severity. Costs
associated with a safety improvement include increases in the cost for initial
installation and maintenance. The cost of an accident varies with its severity
in terms of both the number of vehicles and people involved. The average
24


casualty costs in U.S. in 1986 dollars is as follows: fatality ($ 1500,000),
injury ($ 390,000), and PDO ($ 12,000) (Road Transport Report, 1989).
2.7 Proposed Procedure
The main steps to developing a work zone traffic impact evaluation plan
usually consist of (1) inventory of the transportation corridor system, (2)
identification of goals and constraints, (3) evaluation of impacts and
identification of possible mitigation measures, (4) quantification of
contributions and estimated costs of mitigation measures, and (5) selection of
final plan. The proposed procedure developed in this dissertation to evaluate
traveler and contractor costs at work zones is based on insights from previous
works by Abrams and Wang (1981), Janson et al. (1989), and Krammes
(1989). A flow chart of the procedure is shown in Figure 2.1.
The procedure begins with the identification of the required improvement
action (maintenance or reconstruction) and an inventory of the affected
corridor. The inventory may include: (1) the transportation facilities, (2)
current usage, (3) the boundaries, and (4) the operational MOEs of existing
conditions in the corridor.
At Step 2, all feasible WZTMPs and reconstruction techniques are
considered. TMPs include different work zone types and lane closure
strategies (i.e., closing one lane, closing two lanes, crossovers, and different
work zone lengths). Reconstruction and/or maintenance techniques include
(1) use of different kinds of pavements, (2) use of prefabricated elements, (3)
25


different project schedules, and (4) innovative reconstruction and traffic
control methods.
The capacity of the reconstruction zone is a major determinant of travel
impacts. Traffic performance measures such as delays, stops, fuel
consumption, and operating costs are directly related to the capacity and
operating speed on the roadway segment. If the level of service through a
given work zone type is acceptable, then traffic impacts will probably be
restricted to the highway being reconstructed. If the level of service is not
acceptable and alternative routes exist, some traffic will probably be diverted
from the highway and the entire corridor should be evaluated. If a corridor-
wide evaluation is not needed, capacity analysis and simulation procedures
can be used to estimate the MOEs.
If the entire corridor needs to be evaluated, then a dynamic traffic modeling
procedure called D YMOD is used in this dissertation to determine whether the
available capacities of alternative routes can compensate for the capacity
reductions on the highway being reconstructed. The execution of D YMOD
requires a network representation of the affected corridor as well as flow
patterns and other travel characteristics. DYMODs output for a given
network configuration is used to calculate traveler costs and, if the overall
impact on the network is not acceptable, capacity improvements on alternative
routes and/or other TSM strategies are implemented. DYMOD is run for the
new network conditions and the new costs are computed. If the impacts are
still greater than allowed, the process is repeated until the impacts on the
network are considered acceptable.
26


Once MOEs and traveler costs for a particular WZTMP and construction
method have been computed, they are added to construction and traffic control
cost for the method selected. From all feasible combinations of construction
methods and WZTMP alternatives, the one with the lowest total cost is chosen
and the final WZTMP is implemented.
27


"Problem Identification
(Reconstruction/Maintenance) and
Inventory of Area
s,i i
Identify all feasible construction
methods (including time length and
construction costs for each) and all
WZTMS
YES
NO
Analyze only highway
being reconstructed using
simulation or capacity
analysis procedures
Evaluate entire
corridor using
DYMOD
PREPARATION
- Dcveiop/Modify
Network
- On g in- Desinali on
Patterns
- Capacities. Travel
Time Functions
Improve capacity on
alternative routes and/or
other TSM strategies
Calculate Construction Costs and Travel
Time Costs for alternative (ij)
Select Lowest Cost Alternative and Finalize Plan
Figure 2.1. Flow Chart of Proposed Procedure
28


CHAPTER 3
3 TRAFFIC FLOW
3.1 Traffic Flow
Traffic congestion is a common problem to most urban areas. Drivers have to
balance the conflicting objectives of reducing travel time and avoiding accidents. It is
not well understood what policies drivers actually choose, but they develop habits that
keep the traffic moving while making accidents uncommon. Engineers are given the
task of designing and managing the road system to achieve efficient and safe
conditions for all drivers.
Since human behavior is variable, there is no exact relationship describing the driving
behavior of all individuals. Therefore, a traffic model can either describe what will
happen in an average sense or attach probabilities to various scenarios. A model that
considers averages makes sense when heavy traffic conditions exist, since there is less
variability.
Up to now, there exist mainly two different approaches (with different goals) for the
description of the traffic flow on freeways. On the one hand, there are many
microscopic and macroscopic models that describe the dynamics of traffic flow on a
road (e.g. Gazis et al, 1961; Payne, 1979; Kuhne, 1984; Cremer and May (1985);
Kuhne and Beckschulte, 1993). Most of these dynamic traffic flow models lack route
choice capabilities and describe traffic flows on just one route without taking into
account the interdependencies between different roads of a highway network. On the
29


other hand, there are steady-state and dynamic models to calculate traffic flows on
transportation networks. Although the theory of steady-state traffic assignment is
well developed (see Sheffi, 1985), the dynamic evolution of flows is still neglected
and most dynamic representations of the traffic assignment problem lack general
acceptance.
3.1.1 Traffic Flow Theories
Traffic flow theories seek to describe the precise mathematical way in which vehicles
interact on streets and highways. Traffic flow can be studied as:
(1) Microscopic analysis of traffic flow focuses on the movement of individual
vehicles in traffic, each vehicle being treated as a single mass particle. The
general objective of this approach is to know the position, velocity, and
acceleration of individual vehicles for all time that each vehicle is on the
section of highway being studied. The simplest of these models is concerned
with unidirectional traffic in a single lane with no passing. Vehicles move
along the lane at constant speed u0. At time /=0 +, the lead vehicle begins to
deviate from this constant speed motion. This deviation generates a
disturbance that propagates upstream as each successive vehicle tries to adjust
for the disturbance created by the lead vehicle. For this vehicle-following
model, two types of stability questions are of interest: (a) under what
circumstances does a maneuver by a lead vehicle cause an overcompensation
by a following vehicle (local stability)? and (b) under what circumstances
does a small disturbance by a lead vehicle get amplified as it travels
(propagates) through a line of vehicles {asymptotic stability)?
30


(2) Macroscopic analysis of traffic is analogous to theories of fluid dynamics or
continuum mechanics. Macroscopic analysis characterize phenomena that are
observed in traffic streams, that is, the traffic characteristics as seen by an
observer on the side of the road. It is based on a set of three fundamental laws
of the traffic process: the conservation equation, the volume-density
characteristic, and the mean speed dynamics. A macroscopic model of traffic
flows does not keep track of each individual vehicle movement. Instead, it
pays attention to each section of the road network and describes average
traffic conditions along each of these sections as time goes by.
3.1.2 Fluid Approximation of Traffic Flow
Assuming that freeway traffic behavior at a given point in time and space is only
affected by the state of the system in the neighborhood of that point, partial
differential equations (PDE) have been used to represent traffic phenomena. The first
and most important contribution to the macroscopic theory of traffic flow was made
by Lighthill and Withman in 19SS, and by Richards in 1956, who drew an analogy
between the flow of traffic and the behavior of flood movements in rivers. The
fundamental hypotheses of the LWR theory is that at any point of the road, the flow
of vehicles (veh/hour) is a function of the concentration or density (veh/mile) and that
slight changes in flow are propagated back through the stream of vehicles along
kinematic waves. This problem was first understood in other areas such as gas
dynamics, water waves, and hydraulics. The basic physics of all these problems is
described by hyperbolic (wave propagation) equations, the simplest of which is the
first order PDE. The LWR theory showed that if a traffic stream can be treated
mathematically as a compressible fluid, it should be feasible to start with some initial
31


conditions and functional relationships and predict subsequent values of flow,
density, and speed in time and space.
For a long and crowded road, the conservation equation can be characterized by two
quantities; the flow rate (number of vehicles passing through a freeway location per
unit time), and the vehicle density (number of vehicles per unit length). A second
equation is needed for the simultaneous determination of these two quantities. This
additional equation is obtained by postulating an equation of state relating flow and
density, leading to the traffic equation; a first order nonlinear PDE which (after
attaching appropriate subsidiary conditions) determines the traffic state under most
circumstances. This traffic equation indicates that the traffic flow information will
propagate in the traffic stream with the local wave speed (slope of the density-flow
curve). Exact solutions for this equation are obtained using conventional methods for
hyperbolic partial differential equations. The basic equations are explained next:
The flow rate q(x,t) at a point x and at a time t is the number of vehicles passing
through x per unit time at instant t with
jq(x,t)dt = number of vehicles passing through x over time interval (tx,U)
\
The vehicle density at point x and instant t is the number of vehicles per unit road
length at time t and at point x on the road with
'2
jk(x,t)dx = number of vehicles in a stretch of road (xvx2) at instant t
'i
32


For a stretch of road (xx ,x2;, as shown in Figure 3.1, without an entry or exit, the
following postulate states that the number of vehicles must be conserved
Figure 3.1. Conservation of Vehicles in a Stretch of Road
Number of cars entering Number of cars leaving No. of cars No. of cars
the stretch of road at x, > * the stretch of road at x2 S < in(xy,x2) * in(xltx2J
over the period(tx,t2) over the period (tx ,t2) at time t2 at time tx
Under suitable differentiability conditions on k and q, we can write
k,(x,t) + qM(x.t) = 0
where the subscripts t and x indicate partial differentiation with respect to the
subscripted variable.
Starting with an initial configuration at a reference time t = 0, we are interested in the
quantities k and q at all points of the road for later time. A relationship between k
or,
jq(xx,t)dt- jq(x2,t)dt = jk(x,t2)dx- jk(x,tx)dx
i
33


and q (equation of state), which can be found by controlled experiments, is used to
get a single equation for k(x,t) alone. An adequate first approximation of the exact
equation of state may be q = Q(k).
The resulting first order partial differential equation is complemented by some initial
or boundary conditions for a complete determination of k(x, t). For example, the
vehicle concentration at some initial time may be given and the complete initial-value
problem will be
k,(x,t) + ^kx= 0 and
ak
k(x,0) = kQ(x)
Given an equation of state and a well-posed set of initial or boundary conditions, it is
possible to determine the density of the traffic stream in space and time k(x, t). This
traffic equation, however, produces multiple solutions in circumstances in which
vehicles move into zones of greater congestion. Under such circumstances,
discontinuous shock solutions result.
A conventional method of solving the LWR model uses the method of characteristics.
A characteristic is a curve (with space as a Auction of time) with the property that the
traffic density along the curve does not change. Characteristics indicate which points
in time and space influence which others. For uncongested conditions, the LWR
theory will produce a unique solution, but under heavy traffic, multiple solutions can
occur.
According to the LWR theory, when a stream of traffic encounters more congested
conditions, the characteristics lines will intersect and a steep density profile (shock
34


formation) will occur. More than one value of density will be predicted for the
overlapped region, which, in physical terms, doesnt make sense. This difficulty is
not due to the solution scheme for solving the traffic equation, but instead, to the
assumptions made and to the equation of state employed. By postulating q = Q(k), it
is assumed that vehicle speed at any position along the road depends only on the
vehicle density at the same point in space and time. In real life, drivers look out for
traffic conditions ahead and slow down if the traffic ahead is heavy and slow. Such
driving habits eliminate the need for abrupt changes in speed at the last second.
Besides a coarse representation of the shocks, other deficiencies with the L WR theory
include its failure to describe platoon diffusion properly and its inability to explain the
instability of heavy traffic, which exhibits dramatic oscillatory behavior. Some
attempts to investigate the structure of shocks and stability phenomena have led to
higher-order relations, analogous to the conservation of momentum in fluids. Payne
(1979) discretized the LWR model and added another equation representing the
dynamics of mean speed. Further improvements were made by Papageorgiou (1983)
and Cremer and May (1985) among others. Gas dynamic analogies considering
relaxation time and viscosity effects have also been attempted to improve the
limitations of the LWR model (Kuhne and Beckschulte, 1993). But, as explained by
Daganzo (1995), besides the added complexity of these higher order models, they fail
to improve the predictions of the more simpler LWR model. They include ambiguous
assumptions, require engineering fixes, and lead to unrealistic results in many
instances.
Current efforts to model freeway traffic networks numerically using finite difference
approximations of fluid models are faced with discretization and aggregation
problems. To be practical, these models must discretize freeway links into sections
35


that are large compared with the width of a shock. Thus, even if a numerical model
could be constructed that was consistent with the LWR theory and also captured the
shock wave structure, the improved accuracy would be lost in the subsequent
aggregation of the data within each freeway section.
3.2 Overview of Traffic Assignment Techniques
The urban transportation planning process (UTPP), which has resulted after years of
experimentation and development, is commonly used to assist decision makers to
select a course of action for improving transportation services. The process consists
mainly of four phases; data collection and inventory, model analysis, travel forecasts,
and network evaluation. The model analysis phase of the UTPP consist of the
following four stages:
1. The decision to travel for a given purpose (trip generation)
2. The choice of destination (trip distribution)
3. The choice of travel mode (modal choice)
4. The choice of path or route (trip assignment).
Urban travel demand forecasting is a complex process because demand for travel is
influenced by the location and intensity of land use, the socioeconomic characteristics
of the population, and the extent, cost, and quality of transportation services. The
purpose of the travel forecasting phase of the UTPP is to perform a conditional
prediction of the travel demand in order to estimate the likely transportation
consequences of several transportation alternatives that are being considered for
implementation.
36


3.2.1 Traffic Assignment
The final step in the model analysis process is to determine the actual street and
highway routes that will be used and the number of vehicles that can be expected on
each highway segment. The procedure used to determine the expected traffic
volumes is known as traffic assignment. This problem can be defined as follows:
Given: (1) a link-node representation of the urban transportation network; (2)
the associated link performance Junctions; and, (3) an origin-destination
matrix; find the flow and travel times on each ofthe network links that satisfy
a given traffic objective.
This step may be viewed as the equilibration between the demand for travel and the
supply of transportation in terms of physical facilities, transit services and operational
controls. The main objectives of traffic assignment are to:
(1) obtain aggregate network measures, (e.g. total highway flows,
total revenue by bus service),
(2) estimate zone-to-zone travel costs for a given level of demand, and
(3) identify congested links on the network.
The requirements to perform traffic assignment are the following:
(1) knowledge of how many trips will be made from one zone to another
(this is determined in the trip distribution phase),
(2) knowledge of the available routes between zones and how long it will
take to travel on each route, and
37


(3) a decision rule (or algorithm) that states the criteria by which motorists
will select their routes.
Three basic traffic assignment techniques include: (1) all-or-nothing assignment; (2)
deterministic equilibrium assignment (capacity restraint and mathematical
optimization); and (3) stochastic equilibrium assignment (multinomial logit and
multinomial probit). The most popular traffic assignment algorithms generally
consist of variations of the traditional iterative capacity restraint approach. These
algorithms consist of a series of all-or-nothing assignments combined with
computations to improve estimates of link impedances and link volumes. In capacity
restrained assignment, traffic is assigned to the network in iterations. After each
iteration, the link travel times are adjusted using link performance functions and link
volumes of successive iterations are combined using some type of linear
combinations method. This technique is most applicable for peak-period assignments
(Ortuzar and Willumsen, 1990).
3.2.2 Equilibrium on Transportation Networks
The equilibrium considered for transportation networks is between performance and
demand or between level-of-service versus flow. Potential travelers represent the
demand, whereas the supply is represented by the network itself, with prices given by
travel costs. The interdependencies between the components of the network
necessitates a system-based approach in which the equilibrium flows and travel times
throughout a network have to be determined simultaneously (Sheffi, 1985).
This notion of equilibrium stems from the dependence of link travel times on link
flows. Assume that a given origin and destination pair is connected by many paths,
38


and that the number of travelers between these two points is known. Motorist would
take what may initially be the shortest path in terms of travel time. As a result, the
travel time on this path might increase to the point where it is no longer the minimum
travel-time path. Some of these motorists would then use an alternative path. The
alternative path, however, might also be congested, and so on.
The determination of the flows on each of these paths involves a solution of a
demand/performance equilibrium problem. The flow on each link is the sum of the
flows on many paths between many origin-destination pairs. A performance function
is defined independently for each link, relating its travel time to its flow.
3.2.3 Route Choice
The basic premise of traffic assignment is the assumption of a rational traveler, i.e.
one choosing the route which offers the least perceived individual cost. A number of
factors are thought to influence the choice of route when driving between two points;
these include travel time, distance, monetary cost, congestion and queues, type of
road, scenery, sign-posting, road works, reliability of travel time and habit. The
production of a generalized cost expression incorporating all these elements is a
difficult task. For urban vehicle traffic, the most common approximation is to
consider only time as a surrogate measure for all costs combined.
Wardrop (1952) proposed a model of route choice behavior that forms the basis of
many network performance models used today. He established two principles of
route choice. According to his first principle, users choose the route that minimizes
their own travel time. This is known as user equilibrium (UE), where the travel time
on each used paths is equal and less than the travel time on any unused path.
39


The user-equilibrium definition implies that motorists have full information and that
they consistently make the correct decisions regarding route choice. It also assumes
that all individuals are identical in their behavior. Another kind of equilibrium model
called stochastic equilibrium recognizes that users have only limited information
about the network and that they will choose paths based on their perceptions of travel
costs.
A Wardrop equilibrium assignment can be stated as a mathematical program in which
an objective function is minimized subject to constraints representing properties of
flows. The objective function corresponds to the sum of the areas under the cost-flow
curves for all links in the network. A general requirement for convergence of
Wardrops equilibrium to a unique solution is that the objective function be convex.
This means that the cost-flow curves must be non-negative, continuous, and
monotonically non-decreasing. This mathematical problem is nonlinear and the most
common method used to solve it is due to Frank and Wolfe (1956). The Frank-Wolfe
algorithm uses a steepest descent approach to minimize the objective function.
3.2.4 Solution Procedures for the UE Problem
The solution of the mathematical programming formulation of the UE problem
involves minimization of a nonlinear objective function subject to linear constraints.
Solution of this non-linear program (NLP) involves calculating a direction in which to
search and a step size to make, both of which must remain within the problem
constraints. Methods for doing this are called feasible-direction methods. The
Frank-Wolfe algorithm may be described as a linear approximation method. The
steps are as follows:
40


StepO Find a feasible starting point or initial feasible solution to the NLP.
Step 1.1 Evaluate the partial derivatives of the objective function at the initial feasible solution.
Step 1.2 The evaluated partial derivatives will be the coefficients of a linear equation. Solve for the minimum of this equation subject to the same constraints as in the original NLP. The line connecting the initial feasible solution to this newly calculated minimum gives the direction of search.
Step 2 Find the minimum of the original NLP objective function along the line determined in Step 1.2. The location of this minimum gives the length of the step size in the descent direction. This minimum is the new solution to the NLP.
Step 3 Calculate the convergence measure. If the desired convergence is met, the solution is completed; if not, the new solution is substituted for the previous solution (or the initial feasible solution) and the procedure repeated from Step 1.1.
The F-W algorithm is called a method of convex combinations because of the
calculation performed in Step 2. Here, a minimum value of the NLP objective
function is determined along the line between the previous feasible solution (prior
minimum A) an the minimum of the LP problem of Step 1.2 (prior minimum B).
This minimum, being found along a straight line projected onto the NLP objective
function is, in fact, a weighted average of the two minima (that is, prior minimum A
and prior minimum B). As the algorithm nears the NLP minimum, the relative
weighting of the two minima should tend toward smaller contributions by the new LP
41


minimum. In the particular case of the F-W algorithm as applied to the UE problem,
the change in the flow on a specific link should decrease as the number of F-W
iterations increases. This may be measured by calculating a mean percentage flow
deviation for all links at each iteration.
33 Travel Time Functions
A critical relationship in traffic engineering is the function relating flow and speed on
a link. Generally, speed decreases with increasing flow with the rate of reduction in
speed being higher when flow approaches capacity. Maximum flow is attained at
capacity and, as seen in Figure 3.2, an unstable region with low speeds and low flow
is reached when attempts are made to force traffic volumes beyond this capacity.
BASE FREEWAY SEGMENT
* capacity
**vfc ratio based on 2000 pcphpivaBd only for 60-and 70-MPH design waads
Figure 3.2. Speed-Flow Relationship, (from TRB Special Report 209,1985)
42


Several mathematical formulations describing the relationships among flow, speed,
and density have been studied by traffic flow theorists. Greenshield (1935), for
example, assumed the speed of the traffic stream to be a linear function of the
density, which leads to a parabola for the speed-flow and density-flow curves as
shown in Figure 3.3. Other models are more complex. Greenberg (1959) proposed
the first model for the flow-density curve derived entirely from hydrodynamic theory.
He assumed the equation of motion of a one-dimensional fluid and the continuity
equation to derive a logarithmic model for the speed-density relationship. Edie
(1961) proposed a two-regime relationship, using logarithmic and exponential forms.
Figure 3.3. Basic Form of Speed-Flow-Density Relationships, (from TRB Special
Report 165, 1965)
43


The exact calibration of the flow-speed-density relationships describing an
uninterrupted flow may vary from location to location and even over time at the same
location. The form of these relationships may be determined by trying several
different forms and applying standard regression techniques. Most analysts have
developed calibrations of speed-density relationships because speed-density curves
are monotonically decreasing and involve simpler mathematical forms than flow-
speed or flow-density curves.
In traffic assignment, this relationship is handled in terms of flow versus travel time
per unit distance. The many names given to this function are; cost-flow, link
performance, link impedance, link capacity, volume delay, travel time, or congestion
function. When congestion effects are taken into account in traffic assignment
methods, a set of functions are needed to relate link attributes (free flow speed,
capacity) and flow on the network to the resulting speed and time cost. The travel
time function approximates the way travel time increases as the traffic flow increases
on a link. Without a proper travel time function, it is impossible to accurately
represent the drivers' route choice behavior, since such choices are largely based on
travelers perception of travel time.
Two main approaches are generally used to define the proper travel time function
(Branston, 1976). The mathematical function approach usually provides a relatively
simple relationship between time and flow. However, changes in link characteristics
are not easy to implement. The semi-theoretical approach, in the other hand, allows
one to incorporate changes in network characteristics, but it often results in a more
complex form and the derived coefficient may only be valid for a particular network.
44


Of the several link capacity functions that have been proposed, the one developed by
the Bureau of Public Roads (BPR)in 1964 is shown in Figure 3.4 and is expressed as:
T=T0*(\+y*(V/C)p )
where,
T = link travel time
T0 = link ravel time under free-flow conditions
V = link volume
C = link capacity
Figure 3.4. BPR Travel Time Function
The BPR function is almost totally an empirical relation. First, there is the question
of the design or practical capacity for each link. Second, there is the question of
45


both the meaning and the derivation of the function parameters a and p. The
definition of capacity is ambiguous due to the practical inability to determine the
vehicle capacity of a length of roadway. The difficulties with the function parameters
result both from the virtual impossibility of obtaining, by any statistically reliable
method, link-specific values, and from the general inability to represent different
classes of roadway facilities. Thus, rather general approximations of the values are
usually accepted.
This BPR travel time function, with slight modifications, has been extensively used
over the last thirty years. It performs acceptably for most steady-state assignment
procedures. However, its use in dynamic traffic assignment models has been
questioned by many researchers (see Daganzo,1994), especially when high levels of
congestion are present on the network. The BPR travel time function also creates
problems when queuing delays are present on the network. Since a queue generated
on one link may back up onto other links, the assumption that the cost depends only
on the link's flow is not always satisfactory. In addition, high travel times and very
low flows due to oversaturation cannot be represented with this type of travel time
function alone.
3.4 Dynamic Traffic Assignment Models
Steady state models assume that demand is constant over time. These models are
obviously unable to represent time varying effects, which are needed to properly
evaluate the performance of a traffic system under heavy demands. One of the most
important dynamic effects on routing is the fact that the attractiveness of competing
routes may vary during the analysis period. Independently of the criterion used to
46


compute an optimal route during a particular time interval, this route may not be
optimal for trips departing in subsequent time intervals.
Another shortcoming with steady-state equilibrium models is the assumption that
demand is instantaneously propagated across the network. Motion in space and time
through the network is not represented. Effects such as time-dependent queuing
cannot be represented in steady-state models.
The need to represent queuing effects and time-varying demand has motivated
researchers to develop a dynamic generalization of the steady-state user-equilibrium
and system-optimal models. Previous studies of the dynamic traffic assignment
(DTA) problem can be grouped into analytical models and simulation models.
Although simulation models represent some traffic dynamics more accurately, they
do not have the attractive analytical properties of the analytical models. In particular,
their solution quality is highly variable because they are not intended to converge on a
solution with given properties. Thus, the convergence characteristics and the solution
quality cannot be determined.
Analytical models can be classified into three categories: optimal control approaches;
variational inequality approaches; and mathematical programming approaches. Most
optimal control problems are associated with time-dependent processes which are
optimized through extensions of the calculus of variations methods. The objective is
to determine control strategies that cause a process to satisfy physical constraints and
at the same time minimize or maximize some performance criterion (i.e., minimize
total travel times in a highway system). Optimal control approaches to DTA usually
present continuous-time formulations that are solved by discrete-time algorithms (Ran
47


I
et al., 1993; Wie et al., 1995). However, due to the mathematical complexity of these
models, practical solution procedures have not been yet developed.
Some alternative dynamic models describing the behavior of traffic network users are
expressed as variational inequalities (VI), where, as in the optimal control approach,
the solution space is infinite-dimensional because of the continuous-time variable t.
Variational inequality theory is capable of formulating and analyzing more general
problems than the constrained optimization approach. For example, if the Jacobian of
the vector of link functions is not symmetric (i.e., the link travel time also depend
upon flows on other links), the problem can be formulated as a variational inequality,
but an equivalent convex programming problem may not exist. Although
behaviorally more realistic than alternative dynamic equilibrium models, they require
the solution of complex dynamics expressible as a system of simultaneous integral
equations that are difficult to approximate. It is possible however, to discretize these
continuous-time models and apply the theory and algorithms of finite-dimensional VI
(Friesz et al., 1993; Wie et al., 1995).
Although the aforementioned dynamic traffic modeling approaches (simulation,
optimal control, and variational inequality models) have Improved our understanding
of how dynamic travel demands affect daily traffic flows and impedances on
alternative routes, they are limited to restricted network configurations. None of these
research efforts has produced a model that is compatible with the large transportation
planning applications of U.S. metropolitan planning organizations. Optimization
approaches have been developed, however, that meet this requirement.
Many works in the dynamic traffic assignment literature use optimization algorithms
of one form or another to estimate travel patterns over transportation networks. These
48


approaches essentially attempt to generalize the static assignment problem, with
suitable modifications, to the dynamic case. In these works, feasible (time dependent)
network flows are defined by means of mathematical relations, and an equilibrium
condition that extends the user equilibrium principle (Wardrop, 1952) to the
dynamical case is formulated. To reduce the equilibrium condition to a tractable
form, researchers have assumed that the amount of time that a vehicle entering a link
at time t spends on that link can be expressed as a function of the state of the link at
time t. This assumption allows the formulation of a convex programming problem
whose solution exists and is unique in terms of link flows. The first mathematical
approach to dynamic traffic assignment was formulated by Merchand and Nemhauser
(1978). Their model was formulated as a discrete time, nonlinear, and nonconvex
programming program. Carey (1987) presented an extended formulation of the
Merchand and Nemhauser model that is convex and nonlinear. Janson (1991a,b)
proposed a bi-level programming formulation for the DTA. In his approach, an upper
problem computes the spatial variables (assigned link flows) while the temporal
variables are held fixed. The lower problem then, determines when the nodes of the
network are reached by the most recent assigned flows. The solution algorithm
iterates until the spatial and temporal variables attain the desired convergence. Other
refinements, including flow propagation and FIFO constraints, were added by Janson
and Robles (1993, 1995).
49


CHAPTER 4
4 DESCRIPTION OF MODEL USED IN THIS RESEARCH
4.1 Description of DYMOD
DYMOD was originally formulated by Janson (1991a, b) and Janson and Robles
(1993,1995). DYMOD uses a bilevel mathematical programming framework
consisting of two optimization problems where the constraint region of the first is
determined implicitly by the solution of the second. In the upper-level problem (UP),
a traffic assignment problem is solved for a demand that varies for all time intervals
considered. The UP solves for dynamic user-equilibrium traffic assignment subject to
constraints to constraints that ensure non-negativity and conservation of flow. The
lower problem (LP) minimizes the travel time along paths and ensures correct
temporal propagation of flows across the network.
The dynamic user-equilibrium version of DYMOD is defined as follows:
Given a network with link impedance junctions, and given a set of zone-to-zone trip
tables indicating the number of vehicle trips departing from each origin zone and
headed toward destination zones in successive time intervals, determine the volume of
vehicles on each link in each time interval such that, for each 0~D pair of zones, no
path has a lower travel time than any used path for trips departing within a given
time interval.
50


D YMOD uses zero-one variables called node time intervals to track trips across the
network in both time and space. Trips depart from any node or zone as a uniform trip
rate within a given time interval. All trips flows are tracked through the network in
quasi-continuous rather than discrete time, since link volumes are fractionally split
between discrete time intervals. This improves (1) spillback queuing effects and
dynamic link capacity adjustments, (2) FIFO trip ordering between all O-D pairs, and
(3) link volume and speed transitions between time intervals.
Using the Frank-Wolfe algorithm, the upper problem is solved for dynamic user-
equilibrium flows subject to nonnegativity and flow conservation constraints, with the
node time intervals held fixed. By adjusting the node time intervals, the LP
minimizes the travel time along routes used by flows departing zone r in time interval
d and arriving at node n in time interval t. The link flows calculated in the UP are
held fixed when solving the LP. Link capacity adjustments are done between
executions of LP and UP.
4.1.1 DYMODs Formulation
DYMODs assumptions are the following:
In most time-dependent traffic assignment models, times at which vehicles leave
each zone and where they want to go are assumed to be given.
Link travel times are calculated with the same B PR-type function despite of the
type of link (intersection, arterial, or highway), although other monotonically non-
decreasing functions can be substituted for different link types.
51


The program assumes that travelers know of future conditions and events taking
place in the network, and also that travelers know and will always follow the least
expensive route to their destination.
The formulation shown here assumes monotonically nondecreasing impedance
functions dependent on each links volume in each time interval.
{UP) Min X 2 f* f'jW*1"
ijeK leT
subject to
4=2 VijeK.teT
reZ dsi

rzd
y va a* Vv1'/
miTrt mjrm
meK
njeK
Vn eN,reZ.d eT
v* C >0 VreZ.ijeK.deT.teT
-bd -bd~l VreZ.ieN.deT,and6 =b'-St
and, Vr eZ,i eN.d eT.t eT inequations (6.1-6.3)
idt-k
in
idt-k
idt-k
*ri
min {l. (bri---------Jf
at for = 0
A//Abd]at Vk> 0 for which bd~l -(t-l-k)&t<0
(to(t-k)-bd-1)}
max<0,


for min k for which bd~x (t -1 k)At > 0
where all jar* j and j^j are optimal for:
(6.1)
(6.2)
(6.3)
(1)
(2)
(3)
(4)
(5)
52


(7)
OP) MaxY.lL 5>*
reZ (CAT rfer
subject to
a* =(0,\)
2>2=i
VreZ.ieN.deT.teT
Vr eZ.i eN,d eT.teT
t*T
^*-fAf]a|*£0 VreZ.ieN.deT.teT
[#-fr-UA/Ja* !>0 VreZ.ieN.deT.teT
bd-d^t VreZ.deT
bd max[ed, bd~l + M/] VreZ.ieN.deT, and b =bxh A/
f^-f'-UA//
*2 =
A/
a:
Vr eZ,/eJV,*/ eT.r eT
4
Vr eZ.iy eAT, - /wax {*', (7 UAf + A/yp }] a* < g%a*
VreZ,ijeK,deT,teT.p = t-\.Sf*=fi?(x?j)-g%
(8.1)
(8.2)
(9.1)
(92)
(10)
(11)
(12)
(13)
(14)
where,
N = set of all nodes
Z = set of all zones
K = set of all links
At duration of each time interval (same for all r)
T = set of all time intervals in the full analysis period
x'- number of vehicle trips between all zone pairs assigned to link ij in time
interval t (variable)
v% = number of vehicle trips departing zone r in time interval d assigned to
link ij at some time (variable)
53


f>j (x' qdm = number of vehicle trips from zone r to node n departing in time
interval d via any path; zero for any node n eZ (variable)
ed = time (including dAi) at which last trip departing zone r in time interval
d crosses node / less FIFO delay time at node / (variable)
bd = time (including dSt) at which last trip departing zone r in time interval
d crosses node i via its shortest path (variable)
a* = zero-one variable indicating whether last trip departing zone r in time
interval d crosses node i in time interval/ (henceforth called a node
time interval) (0= no; 1= yes) (variable)
* = fraction of all trips departing zone r in time interval d to cross node i
in time interval / (henceforth called a trip flow fraction) (variable)
0* fraction of a time interval At into time interval / that last trip departing
zone r in time interval d crosses node / (variable)
g% average travel time on link (i,j) of last trip departing zone r in time
interval d adjusted for the time into interval / versus / -1 that this trip
enters the link (variable)
h = minimum fraction of time interval that trips departing zone r in time
interval d must follow trips departing in time interval d-1.
Equations (1-6.3) define the upper-level problem. Equation (2) defines total flow on
link ij in time interval t to be the sum of flows departing any zone r in any time
interval d £ / using link ij in time interval / in order to formulate the objective
function as given by equation (1). Conservation of flow equation (3) constraints
inflow minus outflow at each node and zone in each time interval to sum to the proper
54


trip departure totals in each time interval between each 0-D pair, and equation (4)
requires all link volumes to be nonnegative. DUE requires nonlinear mixed integer
constraints with node time intervals and trip flow fractions indicating the time
intervals in which trips from each origin cross each node so as to insure temporally
continuous trip paths and to spread the trips more continuously in time over these
intervals.
Node time intervals indicate the time interval in which trips departing zone r in time
interval d last cross node i. Each node time interval acts as an if-then operator to
activate certain constraints as needed. A node time interval applies to the last vehicle
of each departure time interval, with the time of the last vehicle departing zone r in
time interval d to cross node i via its shortest path given by bd. The difference
between node / crossing times oflast vehicles departing in successive time
intervals, defined as Abd in equation (5) is used in equations (6.1-6.3) to determine
the temporal spread or dispersion of trips crossing node i from the same origin.
Equations (8-9) compute node time intervals that define temporally continuous trip
paths with the zone-to-node travel times. Equation (8.1) defines the last time interval
t in which arcs incident from node / are used by trips departing zone r in time
interval d. Equation (8.2) allows only one interval t in which trips departing zone r
in interval d can last cross node i.
According to equations (8-9), links are traversed within time intervals that trip paths
cross their tail nodes. If the travel time from zone r to node i is within tAt, then
a* equals 1, since these trips must cross node r in time interval /. Because of
tracking last vehicles, if any path crosses a node at the exact start of a time interval,
55


then the solution algorithm sets a* = 1, but all trips from that origin for that
departure interval will be assigned to the link over previous time intervals.
In equation (10), (equal to the start time of the last vehicle departing zone r in
time interval d) is set to dte to correctly set the clock to the end of each time
interval, and also to prevent the LP maximization from having an infinite solution.
Equations (11-14) impose first-in first-out trip ordering between all O-D pairs
according to their travel times in successive time intervals. Vehicles are assumed to
make only one-for-one exchanges of traffic positions along any link, which is
acceptable and expected in aggregate traffic models. Equation (11) is a vehicle
following constraint that prevents later trips from getting too close to trips departing
earlier from the same zone in successive time intervals so as to prevent these trips
from bunching. The value h is the fraction of a time interval that the last trip
departing from zone r in time interval d must follow the last trip departing zone r
in time interval d -1.
Since equation (11) does not insure FIFO trip ordering between all O-D pairs,
equations (12-14) are also required. Equations (12-13) compute an average travel
time on link (i,j) of the last trip departing zone r in time interval d adjusted for
the time into interval t versus t -1 that this trip enters the link.
Equations (12-13) smooth out speed transitions between time intervals in a quasi-
continuous manner so that vehicle speeds do not abruptly change if they enter links
just split seconds before or after a time interval change.
56


Constraint (14) does not entirely replace the need for constraint (11). Constraint (14)
allows trips between different O-D pairs to become concurrent while sharing the same
path, while constraint (11) insures a minimum separation of last vehicles departing
from the same zone in successive time intervals.
4.1.2 Optimality Conditions of DUE
Although DUE is nonconvex over the domain of feasible node time intervals for all
trip departures to all destinations, DUE is convex with a unique global optimum for
any given set of fixed node time intervals. The optimality conditions of DUE can be
derived from UP for a given set of node time interval as given by an optimal solution
to LP. LP is a shortest path linear program for which there exists an optimal solution
for any given solution to UP. Any set of node time intervals resulting from LP
defines a directed network for which UP is a convex nonlinear program with a global
optimum solution. Since node time intervals from LP are uniquely determined by a
given set of link volumes resulting from UP, they can be assumed to be known in the
derivation of optimality conditions for DUE.
Equation (17) is the Lagrangian of UP with fixed integer node time intervals. For a
given set of fixed node time intervals, the bordered Hessian matrix of equation (17) is
positive definite, which means that there is a unique global optimum with no local
optima. This Hessian matrix is positive definite only if each impedance function is a
monotonically nondecreasing function of flow at link ij in time interval t alone.
57


L(X,V,X,fi,T) = Yd £ P f!j(y>)dw
ijcJC teT
-Z ZA' 4-Z Zv^*
jfrfT ieT L reZ rfsr
( m\
+Z Z 3 1 M - Tvrf 6* ^ rmjrm
reZ rtejV +ZII KK^)
reZ jfeAT (17)
The optimality conditions are given by equations (18 20)
*Jijxy ~Xij VifeK,teT
actj
dL d ,.d / 5/ dt\ idt
K -+[Mr, Mri )^ = (h-Trij) ri
dL _V T* Vd AJt = 0. ( 'o'' Al Vr
dtt ^ Trijvrij9ri
(18)
Vr eZ.ij eK.d eT.t eT (19)
eZ.ij eK.d eT.t e.T (20)
where,
t*j =0 if >0, nonnegative otherwise; equals impedance difference
from node / to node j via a used path versus by link ij (used or unused) in
time interval / for trips departing from zone r in time interval The last part of equation (17) insures nonnegative link flows and results in a third
optimality condition given by equation (20), which requires r* to be zero if any trips
departing zone r in time interval d are assigned to link ij in time interval t, and
nonnegative otherwise. According to equation (18), the optimal solution has a unique
58


equilibrium impedance for each link in each time interval. According to equations
(19-20), for any given pair of nodes, all used paths for a given origin and departure
time must have equal impedances, and any unused path between these nodes cannot
have a lower impedance.
4.1.3 Convergent DTA Algorithm
The two subproblems of DUE are solved successively by a convergent dynamic
algorithm (CDA) as follows:
Step 1. Read; network supply data, temporal trip matrices, initial link flows, and
calculate initial node time intervals by solving LP with initial link flows. Set
iteration counter = 0.
Step 2. Increment iteration counter by I.
Step 3. Solve UP: Use F-W algorithm to find dynamic equilibrium assignment ofall
trips s.t. fixed node time intervals from LP.
Step 4. Adjust capacities of links in time intervals effected by spillback queues,
incidents, other events. These capacity adjustments will be explained in
section 5.2.
Step 5. Solve LP: Recompute node time intervals via shortest paths s.t. UE flows and
FIFO constraints.
Step 6. If few node time interval changed since previous LP solution STOP,
else, go to step 2.
59


4.2 Model Implementation
The model reads data from three input files as follows:
1) It reads the values for ID, From Node, To Node, Free-flow Travel Time, and
Capacity from the Link File.
2) It also reads the O-D volumes for all zones from the given Trip Table.
3) From the Control File it reads Number of links, Number of nodes,
Convergence limits, and the percentages of trips departing every zone during
each time interval.
4) It also reads link IDs and time intervals when the capacity of those links is
affected if an Accident File is used.
Figures showing the link and control files used in this study are found in the
Appendix.
Profiles of average observed speeds were used for calibrating the parameters of the
travel time function employed in the model which is similar to the BPR function
described in Chapter 3. Different values for the alpha and beta parameters were tried
until the observed speed profiles were closely approximated. The values of alpha and
beta that gave the best fit for the observed speed data were, 0.7 and 6, respectively
and the travel time function used in this study was:
T = T0 Vl + 0.7*(V/C)6)
This function is of similar shape to speed-flow relationships used in the Highway
Capacity Manual (TRB, 1985) and other transportation planning and traffic
engineering procedures (see Branston, 1986).
60


4.2.1 Study Network
Figure 4.1 shows a 10-zone network used in this study consisting primarily of a main
highway corridor including four interchanges and two main arterials adjacent to each
side of the highway. Travel between zones 1 and 10 at the North and South extremes
of the network represents the majority of the traffic flow using this network. Of the
total flow on the network, 22 percent leaves from zone 1 and 22 percent leaves from
zone 10 to all other zones, whereas only 7 percent of the total, leaves from each of the
other zones.
The network consists of links connected by nodes. The nodes are classified as
intermediate nodes and zone centroids. Zone centroids are nodes that represent urban
areas where traffic is generated or attracted, and which are connected to other nodes
by zone connectors. There are 222 nodes in the network (including 10 zone
centroids) and 350 links (including 20 zone connectors). All links are unidirectional,
that is, each two-way street is represented by two separate links connecting two
different node pairs.
A transportation Geographic Information System software package (TRANSCAD)
was used to code and display the network. The use of this GIS platform facilitated
the tasks of analyzing and editing the network links, as well as for manipulating their
attributes. Every link in the network can be attached a series of attributes besides its
geographic representation. The data attached to every link may include; ID, Length,
Capacity, Free Flow Travel Time, and other data as needed.
61


10
Jr
Figure 4.1. Study Network
At an intersection, every legal turn or through movement is represented by a separate
link (see Figure 4.2). This means that for an intersection of two two-way streets, up
to twelve links are needed to indicate the twelve possible movements, assuming that
U-tums are not allowed, and that instead of having only one node representing this
intersection, as many as eight nodes are needed.
62


Figure 4.2. Intersection Representation
The main advantages of representing the network with this level of detail are that (l)
the delay incurred by vehicles at each turn movement (right, through, and left) at
different intersections can be estimated more accurately, and (2) illegal turn
movements cannot occur in the model.
4.2.2 Origin-Destination Matrices
One of the requirements for applying a traffic assignment model is a good estimate of
the origin-destination (O-D) information, indicating the patterns of flows through the
network. This information is compiled in a trip matrix where the number of trips
going from each zone to all other zones is indicated.
63


Trip tables can be obtained mainly by origin-destination surveys and the use of
growth factors and synthetic techniques. Obtaining O-D data by means of surveying
trip makers presents many problems and is very expensive. Growth factor models are
used to update old O-D matrices, but these methods do not account for changes in
network conditions. Growth-Factor and gravity models are the most widely used trip
distribution models. Gravity models consider the attraction and distance between two
zones to estimate the number of trips that travel between them. Alternative O-D
estimation approaches include intervening-opportunities models and other synthetic
techniques that rely on traffic counts on some links of the network. Common
methods to estimate an unknown O-D trip matrix using traffic counts as the prior
information include maximum likelihood, entropy maximizing, generalized least
squares, and Bayesian inference estimation techniques. See, for example, Cascetta
and Nguyen (1988) for a comprehensive review. Most of these modeling approaches
attempt to produce an unbiased trip matrix when reproducing observed values on a
subset of links. A matrix of route choice proportions (e.g., estimated from an old O-D
trip matrix) permits the inclusion of information in addition to traffic counts into the
estimation process.
4.23 Trip Tables Used
A trip table was not estimated for this study using the above mentioned techniques.
Instead, an average 24-hour volume obtained from 5-min observed 1-25 data is used
to represent the level of flow leaving the two zones at the end of the highway
corridor. For the other eight zones, the number of trips is specified so as to have
some flows in most links of the network. The base 24-hour trip table used is shown
in Figure 4.3. A series of trip tables are needed to represent the departure rate for
64


each time interval when implementing DYMOD. These trip tables are obtained by
separating the base 24-hour trip matrix into successive matrices for each interval
according to an average profile of daily traffic volumes.
To represent the variability of flows departing from any origin zone in the study
network, a 24-hour profile of traffic volumes based on observed highway links flows
was used. This average 24-hour profile of volumes was developed from morning and
afternoon peak observed 5-min volumes from I-2S links. Data provided by the
Colorado Department of Transportation were used to determine the typical weekday
distribution 5-minute traffic volumes for ramps and 1-25 through-lane locations.
These 1-25 locations, southeast of the Denver central business district, have induction
loop detectors at the on-ramps and through lanes. These data consist of volume,
occupancy, and speed for five-minute intervals, for the morning and afternoon peak
periods (5:00 am to 10:00 and 2:00 to 7:00 pm). The rest of the 24-hour profile (7:00
65


pm to 6:00 am and 10:00 am to 3:00 pm) was approximated. The average 24-hour
profile is shown if Figure 4.4.
Time of Day
Figure 4.4. 24-Hour Profile of Freeway Volumes


CHAPTERS
5 QUEUING AT HIGHWAYS
S.l Queuing In General
Queuing theory deals, in general, with mathematical procedures to analyze the flow of
objects through some network. The network may have one or more locations at
which some restrictions in time or rate of flow are present. At these locations, the
facility will process the objects as rapidly as the restriction will permit and the objects
that cannot immediately pass this restriction are stored in a real or fictitious reservoir
until they can.
Queuing theory formulations attempt to predict how a proposed system will behave.
Generally, costs are associated with any delay and the provision of a higher service
rate at any restriction. When analyzing systems with different service components or
strategies, the usual problem is to compare operating costs and delays.
In most conventional queuing systems, the following occurs. The customers (objects)
arrive at some service point at some specified times. The service facility (the
restriction) requires some time to serve each customer. If customers arrive faster than
the facility can serve them, they must wait in a queue (the reservoir).
In a typical queuing problem, one specifies the arrival rate of customers, and the
service times. One wishes to evaluate (among other things) the average queue length
67


and/or the average delay per customer. The basic characteristics of a queuing system
are as follows:
a. The input or arrival pattern of customers. The interarrival time may be
deterministic so that it is the same between any two consecutive arrivals, or it
may be assumed to follow some specified stochastic behavior.
b. The pattern of service. The time may be constant (deterministic) or it may be
stochastic.
c. The number of servers of service channels.
d. The capacity of the system. A system may have infinite capacity, that is, the
queue in front of the server(s) may grow to any length; against this there may
be limitation of space.
e. The queue discipline. This indicates the manner in which the units are taken
for service. The usual queue discipline is first come, first served.
5.2 Queuing At Highways
Highway bottlenecks are generally defined as portions of a highway with lower
capacity than the incoming section. Insufficient capacity conditions at a highway
section can be due to a variety of factors such as lane-blocking events or the merge of
incoming streams. Bottleneck congestion results in queues on the highway or
network links leading to the highway. Analyzing delays and other queuing impacts
on a highway requires an understanding of the processes that lead to the occurrence of
traffic queues.
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Traffic through a highway is a complex phenomenon for which a number of nonlinear
models exist, however, approximations to the bottleneck problem are rather simple.
Two approaches dealing with bottleneck problems at highways are common. The
first one uses simple deterministic models and treats traffic passing through a
bottleneck as a standard queuing problem. The other approach models changes in
traffic densities and speeds through the bottleneck using wave representations of the
traffic stream, by considering traffic analogous to fluid flow. Simple continuum
models that assume linear shock-wave analysis are generally used, despite the
essential nonlinearity of wave propagation.
5.2.1 Deterministic Queuing Method
In the conventional deterministic queuing analysis of highway bottlenecks, curves
representing the cumulative number of vehicles which would arrive at x by time t if
there were no queue A(x,t), and departure curves D(x,t), representing the
cumulative number of vehicles which actually pass x by time t are used to
geometrically represent trip time and queues (Newell, 1971).
In the next example the on-ramp merge, point b in Figure 5.1, becomes a bottleneck
during the rush hour causing a queue to form on both the freeway and the ramp.
The departure curve is given by the capacity of the bottleneck section, and the vertical
distance Q(x,t)~ A(x,t)-D(x,i) shown in Figure 5.2 is called the queue. This
queue indicates the difference between the number of vehicles which would like to
pass x by time t and the number which actually pass, it is not the number of vehicles
in the physical queue. The total delay to all vehicles that pass the bottleneck is given
by the area between A(x,t) and D(x,t) in Figure 5.2. The priority rule at the merge
69


Cumulative vehicle count
does not affect the total delay to the traffic passing, it does however, affect how the
queue is partitioned between the freeway and the ramp.
i
o
T
Figure 5.1. Freeway On-Ramp Merge
Figure 5.2. Graphical Evaluation of Queues at Freeways
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Deterministic queuing analysis of traffic flow does not rely on a speed-flow
relationship that is necessary in shock wave analysis. As a result, queuing analysis
cannot model how vehicles approach the bottleneck section. Vehicles passing
through the section are simply 'stacked' in a queue waiting to be served at a service
rate based on the bottleneck section's capacity. The vehicle queue is a standing one,
and the movement of vehicles within the queue is not considered.
In contrast, given a speed-flow function, shock-wave analysis can model the
movement of vehicles approaching the bottleneck. The different representations of
the queue will affect the derivation of the physical length of queues and the
interpretation of the results. Thus the physical length of a moving queue cannot be
easily derived from queuing analysis. The distinction between the standing queue in
queuing analysis and the moving queue in shock-wave analysis will also affect the
computations of the delay (see Chin, 1995).
5.2.2 Shock Waves in Highways
Using a hydrodynamic analogy, a shock wave is said to exist whenever traffic streams
of different conditions meet (Figure 5.3).
In shock wave theory the speed at which discontinuities in traffic flow states (i.e., the
growth of a queue upstream of a bottleneck) travel through the stream depends on the
flows and vehicle densities existing on each side of the wave. A common description
of shock waves in traffic as presented in most traffic engineering textbooks is given
next. Figures 5.4 and 5.5 as well as part of the material in this description are
extracted from Papacostas and Prevedourous, (1993).
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Figure 5.3. Shock Wave Due to Change in Densities
Suppose that a traffic stream is moving on a roadway at a given flow, speed, and
density as shown on the flow-density diagram of Figure 5.4. Point 1 corresponds to a
flow of 1000 veh/hr, a density of 25 veh/mile, and a mean speed of 40 mi/hr. Now,
assume that a truck in the stream decides to slow down to 10 mi/hr. If no passing is
permitted, the following vehicles will also have to travel at the speed of the truck. At
any instant, the last vehicle to join the platoon will be traveling at 10 mi/hr, but
farther upstream vehicles would continue to approach the platoon at the original
conditions. The stream conditions within the platoon are represented by point 2 on
Figure 5.4, where the slope of chord 0-2 is the platoon speed and the flow and density
are shown to be 1200 veh/hr and 120 veh/mile, respectively.
72


I
!
Figure 5.4. Shock Wave Description
A change in stream conditions can also be caused by a sudden reduction of capacity
on a highway (bottleneck condition) due to a reduction in the number of lanes,
accidents, or work zones. When such a condition exists and the flow and density on
the highway are relatively large, the speeds of the vehicles will have to be reduced
while passing the bottleneck. The point at which the speed reduction takes place can
be noted by the brake lights of the vehicles coming on. An observer will see that this
point moves upstream as traffic continues to approach the vicinity of the bottleneck
indicating an upstream movement of the point at which changes in flow and density
occur. This phenomenon is usually referred to as a shock wave in the traffic stream.
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In the above example, there are two shock waves. One is seen between the platoon
and the free flow conditions in front of the platoon (line AA, Figure 5.5). The other is
seen between the approach and the platoon conditions (line BB, Figure 5.5). The
shock wave at the front of the platoon is defined by the truck, whereas the shock wave
at the rear of the platoon is defined by the last vehicle to join the platoon. Figure 5.5
shows that the shock waves AA and BB displace with time in relation to the roadway.
The rate at which the platoon grows is related to the speeds of the two shock waves
AA and BB. If, after a time, the truck driver decides to either accelerate or to exit the
highway, the vehicles stuck behind it will be free to increase their speeds, and another
shock wave will begin between the release conditions and the platoon conditions.
Figure 5.5. Time-Distance Diagram of Platoon Formation
74


The speed of a traffic stream shock wave is given by the slope of the chord
connecting the two stream conditions that define the shock wave (points 1 and 2 in
figure 5.4). Labeling the two conditions as a and q in the direction of traffic
movement, the magnitude and direction of the speed of the shock wave between the
two conditions are given by
where,
W speed of the shock wave,
qq = freeway flow rate in the queue just downstream of the shock wave,
qa = freeway flow rate approaching the shock wave and the queue,
kq = vehicle density in the queue just downstream of the shock wave, and
ku vehicle density approaching the shock wave and the queue.
The equation above indicates that larger differences in flow rates or smaller
differences in vehicle densities on each side of the wave correspond to higher shock-
wave speeds. If the sign of the shock wave is positive, the wave is traveling in the
direction of the stream flow; if it is negative, the shock wave moves in the upstream
direction. The example shown in figure 5.5 shows two shock waves, AA and BB,
both traveling in the direction of the stream of vehicles. The speed of the shock wave
at the rear of the platoon is
W(BB)=
1200-100
120-25
= 2.1 mi / h
The rate of growth of the platoon is given by the relative speed between the front and
rear of the platoon, or 10.0 2.1 = 7.9 mi/h. At any point on the flow-density
75


diagram, the speed of the wave is theoretically less than the space mean speed of the
traffic stream.
When the flow on a highway is greater than the capacity of a bottleneck, the speed of
the wave is not only less than the space mean speed of the vehicle stream, but it
moves backward relative to the road. As vehicles enter the restricted area, a complex
queuing condition arises, resulting in an immediate increase in the density in the
upstream section of the road and a considerable decrease in speed. Shock waves are
created by backward moving waves from a bottleneck meeting forward moving waves
indicating the end of a physical queue.
S3 Proposed Approach To Account For Queuing Effects Within DYMOD
The way in which queuing effects were modeled in DYMOD prior to this dissertation
work was to subtract the flow in excess of capacity at the bottleneck from the capacity
of each upstream link in proportion to the inflow of feed link. The speed at which the
queue lengthened was not estimated, only the maximum extent of the queue (i.e., the
queue developed to its full length in one time interval). Subtracting the excess flow
from the capacities of upstream links was not proportional to the capacity of each
link, and did not achieve the same V/C ratio on each link.
A fundamental property of fluids is that the pressure (characterized here by the V/C
ratio) is the same throughout a pressurized region. Queues are now estimated in
DYMOD to satisfy this principle, and DYMOD estimates the speed at which the
queue lengthens.
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The main indications of traffic queuing consist of low flow and high travel time.
These effects can be represented if a parabolic-shaped curve of density-flow similar to
the one first proposed by Greenshield is used (Greenshield, 1935). Since the
impedance function used on DYMOD is restricted to be monotonically non-
decreasing, lower flows and higher travel times cannot be directly represented and
have to be induced indirectly. The proposed approach to account for queuing effects
in the context of DYMOD is executed outside of the upper or lower problems. The
effects of queuing are accounted for by adjustments in the links capacities. Links
flows are assigned in the upper problem based on lowest travel times given by the
impedance function used, but without considering any bounds on the volume-to-
capacity ratio. Therefore, for any given assignment of flows to the network, the
volume-to-capacity ratio may exceed unity at some links.
To represent queuing effects during congested conditions, subroutine QUECAP
checks all the networks nodes and affects the capacity of inflow links preceding links
whose V/C ratio is greater than unity. The revised capacities are fed back to the upper
program where a new set of flows are assigned to the links based on the revised travel
times. This process is repeated until the specified convergence parameters are met.
Flows assigned to each link during the last execution of the upper problem are passed
to QUECAP. This subroutine is executed between the upper and lower problems of
DYMOD to recognize exogenous link capacity changes, and to compute endogenous
link capacity variations. Three essential steps of QUECAP are: (1) to track the
locations of multiple spillback queues in a network, (2) to weight the effects of
multiple spillback queues when jointly affecting inflows to any node, and (3) adjust
the capacities of inflow links to each node in proportion to the fractions of flows
affected by each queue. Endogenous link capacity changes stabilize with successive
77


solutions to the two subproblems of DYMOD, and are only made between
subproblem solutions so that the upper problem remains convex for each given set of
supply specifications.
The queue propagation and capacity adjustment procedure is explained next. First
note that all nodes are configured such that every node has only one outflow link (a
merge node) or one inflow link (a diverge node). Capacity adjustment steps starting
from original unadjusted capacities are:
1) Reset capacities of links to original capacities, or to capacities exogenously
reduced for specified time intervals;
2) Increment iteration counter by 1 and initialize queue on arcs k to 0;
3) Increment the time interval from 1 to T.
4) Increment the node number from l to N.
5) For time interval 1 to T, and node 1 to N, process all the inflow and outflow
links as follows:
5.a) For all outlow arcs (arcs leaving the node being examined), compute:
5 .a. 1) The queue that has accumulated on the arc by that time interval.
S.a.2) The oversaturation flow that link k can absorb based on flow
speeds and densities to track queues end point in each interval.
S.a.3) The time interval amount that flow on link k is affected by
queues in interval t
78


5.a.4) Weighted V/C ratio and percentage of volume affected by
queuing.
S.b) For all inflow links (arcs going into the node being examined), if
weighted v/c ratio of outflow links exceeds 0.9S, compute:
5.b.l) Adjusted v/c ratio,
5.b.2) Revised capacity (k,t)
6) Return to step 4 until all nodes are processed, then return to step 3 until all time
interval are processed. If, for any link and time interval, the difference between
the capacity just calculated, and the previous capacity is less than a specified
percentage, then exit, else go to step 2 for another pass.
In order to capture the effects of multiple queues spilling back from several places in
the network, and queues spilling back farther than one link in any time interval,
multiple passes of the of the above steps are performed until all capacities in all time
intervals do not change significantly.
53.1 Approximation of Shock Wave Effects in DYMOD
As explained in Section 5.2, a bottleneck condition exists when a capacity reduction
on a link forces a change in the density of a stream of traffic. An upstream moving
shock wave is formed at the point where the speed reduction takes place. In the
context of DYMOD, a bottleneck may also occur when the flow assigned to a link is
higher than the available capacity, even if no capacity reduction has taken place.
79


The speed of this upstream moving shock wave will indicate how the queue grows
depending on the existing flow. As before, the wave speed would be given by the
difference of the flows at and before the bottleneck, divided by the difference of the
densities at the two sections.
where,
uw = wave speed with negative values for the upstream direction
qx =flow upstream, before the bottleneck
q2 = flow downstream, at the bottleneck
kx = density of traffic at the upstream section
k2 density of traffic at the bottleneck
In D YMOD, only one level of flow is available at each time interval and therefore, a
direct calculation of the wave speed, using the above formula is not possible. Instead,
the fraction of the link that is occupied by a queue is estimated and the extent of the
queue is tracked along upstream links.
To determine whether a link is totally or partially affected by a queue, the computed
queue is compared with the amount of flow the link can absorb. The absorbed
volume represent the amount of vehicles that would be stored in the link as the stream
of traffic compresses to a higher density. Thus, the absorb value depends on the
length of the link and the difference in densities computed using the original capacity
and the revised capacity of the link in question as follows:
absorb (kr k0)*Link length
80


where the density values are obtained by dividing the flow over the speeds of the
traffic streams.
k0 = density of traffic stream considering the original capacity,
kr = density of traffic stream at the section with revised capacity
Speeds are computed dividing the free-flow speed over the travel time function
used. Two different speeds are obtained by using two different capacities in the travel
time function, the original and the revised capacity.
speeda = link length / travel time for flow with original capacity
speedb = link length / travel time for flow with adjusted capacity
An equivalent calculation implemented in DYMOD uses the difference in travel time
taken by the two traffic streams to cross the link.
absorb=(timer time0)* vol( k,t)f t_int
where the times are obtained as the product of the free-flow travel time and the
travel time function used.
time0 = FF7T*|l + y{vol( k,t)/ cap0 J
time, = FbTl *£ 1 + y{yol(k,t)/cap^f j
where,
vol(k,t) the flow at link k at time t
t_int = the length of the time interval considered
cap0 and capr are the original and revised capacities, respectively, and
y and p are the travel time function parameters as previously defined.
81


The fraction of the link affected by a queue is given by the ratio of the computed
queue and the flow absorbed by the link. If this ratio is greater than one, the time
intervalfraction (TIF) that the flow on link k is affected by queuing in time interval
t is computed. The affected TIF equals (the time that queue reaches links tail minus
the interval start time) divided by the time interval duration. If TIF equals one, the
whole link is affected and the queue will spill over the next upstream link.
53.2 Capacity Adjustment at Upstream Links
Exogenous changes (e.g., accidents, weather effects, signal timing changes, or time-
of-day road restrictions for special events or construction) can be input to the program
for specific links and times of day. Endogenous link capacity changes occur when
spillback queues reduce the capacities of upstream links. When accidents or recurrent
congestion create oversaturated conditions, upstream link capacities are reduced by
spillback queues using wave propagation speeds. These capacity losses create further
upstream effects to the extent and duration of the oversaturated condition.
The propagation of the queuing effects are directly dependent on the time interval
fraction that the flows on a link are affected by queuing as explained in the next
example.
At each time interval, a weighted V/C ratio is calculated for the two outflow links, a
and b, shown in Figure 5.6. This average V/C ratio, weighted by the TIF variable is
computed as follows:
WVC
_yf vol(k.t)1
* vcapr(k,t)
TIFk
/PCV
82


where,
WVC=the weighted volume-capacity ratio of all outflow links k
PCV = ^vlCk,t)*TIFk
k
k consists of links a and b in this example
Figure 5.6. Freeway Section with Off-Ramp
Therefore,
vol(a.t)1 TIFa + vol(b.t)2 TIF
If the weighted V/C ratio is greater than both, 0.95 and VC0, an adjusted V/C ratio for
the inflow link (link c in Figure 5.6) is computed as follows;
A VC = VC0 + (WVC -VC0)*PCV
where,
AVC=adjusted V/C ratio of inflow link
VCQ = inflows V/C ratio using original capacity
wvc=
vol(a.t)2 7TFa vol(b.t)2 TIF
capr(a,t)
capr(b,t)
83


And the revised capacity for the inflow link is the minimum of the following
arguments;
capr(k.t) = Min[capa, Max(capmm,
where,
caPmm = 311 specified lower bound for the capacity of link k.
cap0 = the original capacity of link k as explained before.
5.4 Speed-Flow Relationship used in DYMOD
In order for the DYMOD formulation to converge to a unique solution the solution
space must be convex. To ensure this, the travel time impedance function used to
calculate the link travel times must be monotonically non-decreasing. A BPR-like
function is usually chosen for highway links because of its widespread use and
simplicity. This convexity condition precludes the direct use of a flow-density
relationship which allows for two levels of flow for each density value.
This limitation, however, can be partially overcome by adjusting the capacity of links
upstream of a bottleneck. By treating the capacity of links as a variable, low flows
and high travel times can be approximated in most instances within the model. When
the capacity of upstream links is reduced to reflect the higher density at a bottleneck,
the assigned flows are lowered in successive executions of the assignment program,
and the computed travel times for those links increases due to the lower capacity.
Figures 5.7 to 5.10 plot speed-density, speed flow, travel time-flow, and flow-density
relationships for an example link as executed in DYMOD.
84


Figure 5.7. Speed-Density Relationship used in DYMOD
85


UnK Travel Time (mint)
Figure 5.9. Travel Time-Flow Relationship used in DYMOD
Figure 5.10. Flow-Density Relationship used in DYMOD
86


CHAPTER 6
6 WORK ZONE SCENARIOS
6.1 Lane closure strategies
On highway reconstruction projects, such questions as project length, level of
capacity reduction, staging, sequencing and designation of period when work may be
conducted must be considered for their impact on safety and smooth operations.
Much research have been conducted concerning a variety of issues related to work
zones and their impacts such as: (1) work zone length (Paulsen et al., 1977; Martinelli
and Xu, 1996), (2) different work schedules (Rouphail et al., 1987; Shepard and
Cottrell, 1983), (3) capacity through a work zone (Dudek and Richards, 1982; Dixon
et al., 1996), and many others. However, few studies have been done to develop a
systematic method for selecting appropriate lane closure strategies. One example is
the work of Dudek et al. (1985), which evaluated lane closures strategies for work
zones on four-lane divided highways. Although, the selection criteria for an
appropriate lane closure strategy reached by the study are not conclusive.
There are mainly two lane closure strategies for work zones on highways; crossover
and partial lane closure. Crossovers are constructed to bring both directions of traffic
to one side of a highway. The other side is completely closed for rehabilitation
activities. In a partial lane closure one or more lanes are closed in one or both
directions. In such an arrangement, more attention is given to providing safe ingress
87


and egress for employees vehicles and construction equipment along the work zone,
particularly on high speed, high volume roadways, since the proximity of passing
vehicles endangers the work zone crew. Productivity and the quality of the work are
more likely to suffer in partial lane closures as compared to completely closing one
side of a highway.
Handling traffic through work zones is affected by many factors that must be
considered when designing a traffic control scheme. Some of these are:
(1) number of lanes and widths to be maintained,
(2) whether to use shoulders for traffic or not,
(3) method of separating opposing traffic flows,
(4) method of separating traffic from construction,
(5) contractor access to travel lanes, and
(6) maintenance, closure, or restriction of ramps.
The Manual on Uniform Traffic Control Devices for Streets and Highways (MUTCD,
1978) states the basic principles that govern the design and usage of traffic control
devices, and includes a separate chapter detailing Traffic Controls for Construction
and Management Operations.
The major elements of a Traffic Control Plan for large construction projects are a set
of drawings that depict the work to be done along with the traffic control devices that
are to be used and a set of notes that give details on the overall operation. The notes
for these plans may include: (l) special provisions, (2) sequence of work, and (3)
quantity of materials table. The sequence of work identifies the stages of operation so
that traffic can be maintained during the life of the project
88


Traffic control costs are usually higher when a crossover is used. Traffic control costs
at work zones include installation of signs, markings, lighting, barricades, and
channelizing devices, and other personnel and material costs. In addition, when a
crossover is used there is the cost of constructing the crossover facility.
Work zone capacities will vary depending upon the nature of the work being done,
the number and width of lanes available to traffic, the size of equipment at the site,
and the location of equipment and crews with respect to open lanes or traffic.
Most state DOTs use the capacity values in the HCM (TRB Special Report 209,
1985) as a guide for analyzing work zone lane closures. These values were primarily
collected by the Texas Transportation Institute (TO) for urban freeways in Texas.
TTI updated these capacity values in the early 1990s to 1575 vehicles per hour per
lane for a two-lane to one-lane urban closure and 1460 vehicles per hour per lane for a
three-lane to two-lane urban closure.
6.2 Scenarios Modeled
Only partial lane closures are considered in this study. Because of the limitations of
the network used, a complete closure of the highway in one direction cannot be a
adequately modeled. The place on the network where the work zones are located is
shown in Figure 6.1.
The scenarios modeled are the following:
a. One of three lanes closed for 24 hours;
b. Two of three lanes closed for 24 hours;
c. One of three lanes closed at night (8:00 p.m.-6:00 a.m.);
89


d. Two of three lanes closed at night (8:00 p.m. 6:00 a.m.);
e. One of three lanes closed during day off-peak (9:00 a.m. 3:00 p.m.);
f. Two of three lanes closed during day off-peak (9:00 a.m. 3:00 p.m.).
Figure 6.1. Work Zone Locations on Study Network
This work zone location on Figure 6.1 area comprehend a 0.7S mile stretch of
highway. Four runs are made for each of the scenarios described above as follows:
(1) a 0.25 mile work zone beginning at the end of the on-ramp (segment A), (2) a 0.25
90


work zone area in the middle of the 0.75 stretch of highway(segment B), (3) a 0.25
work zone immediately before the off-ramp (segment C), and a work zone covering
the totality of the 0.75 section of highway (segments A, B, and C). The length of
each work zone is assumed to represent the buffer and construction area only as
shown in Figure 6.2.
Figure 6.2. Areas in a Traffic Control Zone
Also, for the 24-hour closures, other cases are analyzed: (1) totally closing the two
on-ramps immediately upstream of the work zone, and (2) closing only the on-ramp
91


just before the work zone. For the night and off-peak cases, the capacity of the on-
ramps was not affected.
For the cases where two out of three lanes are closed, the capacities of the two off-
ramps prior to the work zone were increased form 17S0 to 2600 vph to represent the
conversion from one lane to two lanes necessary to accommodate traffic diverting
from the highway to avoid the work zone. Capacities of the intersection movements
at the ends of the two off-ramps were increased by 50 percent to reflect signal timing
changes made at the intersection to give preference to traffic diverting from the
highway. These capacity adjustments represent traffic management measures likely
to be implemented for a work zone closure of this magnitude.
Overall, 40 different work zone scenarios were analyzed, as well as the case were no
work zones are present. The analysis of the results is presented in the next chapter.
92


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