A dynamic route assignment model for guided and unguided vehicles with a massively parallel computing architecture

M&l.

Pergamon

Cornput. Modelling Vol. 22, No. 4-7, pp. 377-395, 1995 Copyright@1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0895-7177/95-$9.50 +- 0.00

0895-7177(95)00145-x

A Dynamic Route Assignment Model for Guided and Unguided Vehicles with a Massively Parallel Computing Architecture G.-L.

CHANG AND T. JUNCHAYA Department of Civil Engineering University of Maryland College Park, MD 20742, U.S.A. gangQeng.umd.edu

Albstract-This article presents a dynamic assignment model which has been developed primarily for ATMS/ATIS real-time applications. In order to satisfy the real-time computational requirements, the proposed model has been developed on the Connection Machine, CM-2, a massively parallel computer. Its implementation on the Connection Machine is aimed to exploit the parallel nature of the problem and to take advantage of the underlying computing architecture. The model follows an integrated assignment-simulation framework which assigns both guided and unguided vehicles to the network dynamically in both spatial and temporal dimensions. It uses a learning process in which the new route assignment uses information gained from previous iteration in computation of new paths. During each iteration, guided vehicles follow routes according to time-dependent shortest paths while unguided vehicles follow static shortest paths. Several numerical examples have been carried out to illustrate the potential applications in analyzing the network performance under various ATh,lS scenarios. Keywords-Dynamic

assignment, Parallel processing,

Route guidance.

1. INTRODUCTION With Advanced Traveler Information Systems (ATIS) and Advanced Traffic Management Systems (ATMS), the traffic control center has the ability to provide real-time traffic and offer routt: guidance instructions to ATIS-equipped vehicles. The anticipated benefits of these route guidances depend on the complex interactions arnong various factors, including the market penetration rate of ATIS-equipped vehicles, driver behavior and compliance rate, level of congestion, dynamic nafure of traffic patterns, and signal control strategies. Any dynamic assignment model destined for successful applications in ATMS/ATIS must be able to address all those critical issues. Furthermore, since such a dynamic assignment model will be used m real-time environment involving thousands of streets, intersections and vehicles, its computational speed poses another critical research issue. Although advanced parallel computing architectures can offer the required computational power with an economical price/performance factor, it will require a fundamental change in the modeling methodologies so as to achieve the full benefit of parallel processing. This article describes our efforts in exploring parallel computing architectures for the de\~elopmpnt of a real-time

dynamic

route assignment

model.

This article is organized as follows: the next section briefly reviews the existing dynamic assignment models along both mathematical formulations and assignment-simulation directions. Section 3 describes our modeling approach to the proposed dynamic route assignment model. Typeset

by A&T-QX

378

G.-L. CHANG

AND T. JUNCHAYA

Section 4 reports the parallel description of the parallel variables and their relationships. Section 5 explores the potential model applications with several experiments and their preliminary results. The conclusion and future research are outlined in the last section.

2. LITERATURE

REVIEW

Even without considering factors such as the market penetration

rate of ATIS-equipped

vehi-

cles, or driver compliance rate; the presence of time-varying travel demand and time-dependent link flow have already made the dynamic assignment model much more difficult to solve than the static one. There are relatively few optimization models developed to date which examine dynamic traffic assignment on networks and most of them emerged in the recent year. Each of such dynamic models is based on a different set of decision variables and behavioral basis; it offers different capabilities in formulating dynamic traffic systems or determining control actions. The first mathematical

programming approach to the dynamic assignment problem is due to

Merchant and Nemhauser [l]. Their model was formulated as a discrete-time,

nonconvex, and

nonlinear program, in which congestion is represented explicitly in the constraints. An efficient solution procedure for a stepwise linear version of this model was given by Ho [2]. Carey [3] reformulated the Merchant-Nemhauser problem as a convex nonlinear program, which offers the analytical and computational advantages over the original non-convex formulation. However, in extension to multiple destinations or traffic types scenarios, the need to satisfy the first-in, first-out (FIFO) traffic relation has resulted in a nonconvex constraint set [4]. Apart from the mathematical programming formulation of the discrete time problem, Friesz et al. [5] proposed dynamic extensions for both static system and static user optimization models, using the continuous time optimal control formulation which assume both the O-D demands and link flows to be continuous time functions. Their dynamic system optimization model can be considered as a continuous time version of the Merchant and Nemhauser model. Furthermore, their dynamic user optimization model is the first dynamic generalization of Beckmann’s equivalent optimization problem for static user equilibrium. Wie [6] extended the above models to include elastic time-varying travel demand which implicitly considers departure time choices. Ran and Shimazaki [7] also developed a dynamic system optimal assignment model for a simplified network with multiple origins and multiple destinations using optimal control theory. Along the same line, Ran and Shimazaki [8] and Ran, Boyce and LeBlanc [g-11] proposed several extensions of their dynamic user optimal traffic assignment models, including choices for both departure and route preference. Another research direction, introduced by Papageorgiou et al. [12], was derived from classical automatic control methods. They used a macroscopic modeling framework for time-varying traffic demands with a feedback regulation to establish dynamic traffic assignment conditions. This is carried out by using independent splitting rates of each traffic subhow with a given destination. In terms of the ATMS applications, the optimal control approach remains to be improved. For instance, it lacks an explicit first-in, first-out (FIFO) requirement, and thus models traffic congestion unrealistically. Most importantly, it does not have an efficient solution procedure for a network of realistic size. The feedback regulation approach also does not meet FIFO requirement. Even though the feedback regulation concept is attractive from the control standpoint, the formulation does not establish the underlying mathematical basis in order to estimate some key model parameters for system operations. A detailed review of these models can be found in [13]. Since it is often difficult to take all flow interactions into account with theoretical formulations, assignment-simulation framework emerged as one of the main tools to overcome those limitations. An example assignment-simulation model in the ATIS context was proposed recently by Mahmassani and Jayakrishnan [14]. Their model explicitly represents path selection decisions of individual motorists (both guided and unguided) along their path in response to supplied information. At every node where an alternate path is available, the user will switch routes if

A Dynamic Route Assignment Model the benefit modeled

from switching

with

approach

a traffic

proposed

assignment-simulation by Kaufman

t,raffic simulation computed

following

The

[17] to create

with the in-vehicle

these paths,

Otherwise

the iteration

condit,ions

until a termination

creating

condition

particularly

bilities

and route choice behavior,

mine the traffic control

estimating

trip time.

fully integrates

The next

assignment

above models

link trip time

access to an external camlot

assignment

offers several vehicles

be projected, assignment

section

an external

DRAM queues,

(Dynamic and paths

unguided varyhlg

Assignment

traffic

performance

criterion

performance

until the criterion

mathematical In DRAM,

route guidance

the above

model to clct,er-

model is down or not availthle,

to have a rudimentary a dynamic

route

in an integrated

simulation

assignment

model to determine

model

that

the anticipated standpoint,

then time-varying

It is thus desirable

for estimating

for

model.

Prom the controller

strategy.

capability.

mechanism

ii the

link trip t,ime for the dynnmic,

trip time.

Model)

is an assignment route

guidance

a road network.

[18] which considers

performance

that

predicts

control

It is an extension DRAM

reiterates

predicts

assignment

the predictions

is met.

Note that the iteration

solution

due to the difficulty

Given

paths

and .work t,ime-

for vehicles

mechanism. of paths

process is employed in obtaining

flows,

center)

of the earlier

only one type of vehicles.

using event-driven

the model

model

from traffic

IF the

and traffic

only to search

an elegant

and efficient

solution. unguided

and static

shortest

both the signal queuing

In each iteration, the initial

vehicles

are assigned

link travel time obtained

bot,h time-dependent

for actual

However,

traffic simulation

guided vehicles are assigned to follow time-dependent

the instantaneous

While

efficiency.

APPROACH

can receive

is not satisfied,

link t.ravel time, whereas

mcluding

and the new set of traffic pro-

for both guided and unguided vehicles,

network’s

for the optirnal

(e.g.,

and Zhuang

O-D demands

and estimates

are

capa-

decisions.

mechanism

as they travel through

Junchaya

paths

Concept

of guided

vehicles

by Chang,

Route

vehicles

shortest

and thus lacking the “look-ahead!’

of route guidance

Modeling

it simulates

over the mathematical

mechanisms traffic

model to have a rudimentary

Fundamental

advantages

model

3. MODELING 3.1.

based on the shortest

then the loop is terminated.

guidance

model is down or not available,

for evaluation

for

with various classes of route guidance

will introduce

from the assignment

simulation

be projected

policy

occurs.

and simulation

require

resulting

paths,

was recently

routing

The time-dependent

if the access to a simulation

for the dynamic

are

(MPSM)

INTEGRATION

The

Using INTEGRATION,

models still depend on an external

then all future link trip times cannot It is t,hus desirable

data.

and also in terms of computing

perspective,

procedure

system is then determined

the new route

in terms of representing

two assignment-simulation

system

Model

by running

set of forecast

trip time.

framework

grams,

begins

agree with the previous

continues,

assignment-simulation

process

new link trip time.

If the new paths

of the traffic

Simulation

based on an iterative

iterative

information

and produces

The dynamics

logic.

an initial

under time-dependent

then recomputed.

The

threshold.

from the Macroparticle

model framework

et al. [16].

model

vehicles equipped

The

a certain adapted

[15], and follows a fixed time-step

Another

paths

exceeds

simulator

379

shortest

to follow static

at the time of their departure.

path takes

into account

delay and the oversaturated

paths using predicted

shortest

the vehicle

spillback

paths

based on

The computation intersection

deployment,

and later departed used to progressively

may yield paths

it provides

traffic

an insight

along different

reduce

and network

uncertainties

performance

into the complex

paths.

that

interaction

The new time-varying

in a subsequent

prediction

delay

delay.

guided and unguided vehicles are assigned on both spatial and temporal iterations

of

basis.

are not satisfactory between

the current

link information of paths

for both

is then guided

380

G.-L.

CHANG AND T. JUNCHAYA

and unguided vehicles. Ultimately, near user-optimal paths for guided vehicles are produced at the end of the iterative process. More specifically, the proposed model differs from other assignment- simulation in the following respects: 1. DRAM

has a route loading mechanism to estimate link trip time rather than depending

on an external simulation model. 2. It uses speed-concentration

function to estimate links trip time and realistically consider

the link capacity constraints. 3. It incorporates a learning process, where information from previous iterations are used to obtain better path selections and assignment. 4. Intersection control and queue buildup are explicitly represented in the model. Thus, the signal delay and queuing delay can be realistically estimated. 5. The model assigns ATE-equipped vehicles to routes with the “look ahead” feature. Nonequipped vehicles are assigned to other predetermined routes.

PROPOSED METHODOLOGY

I

4

PHASE 1 PATH PREDICTION

+

PHASE 2

+

INTERNAL ASSIGNMENT

Figure 1. Outline of DRAM

3.2.

overall structure

Model Structure

The overall structure of DRAM is outlined in Figure 1. The inputs are time-varying traffic demand for both guided and unguided vehicles. Predefined route plans for unguided vehicles can also be used instead of assuming static shortest paths. DRAM consists of four phases: path prediction, internal assignment, evaluation and learning, and implementation. These phases are described as follows: PHASE

1:

PATH

PREDICTION

The current version of DRAM consists of two classes of vehicles: guided and unguided vehicles, depending on whether drivers have direct access to the real time information. In this study, guided vehicles are assigned paths that are fastest in a time-dependent network using predicted information. Unguided vehicles are assigned static shortest path computed from instantaneous link information at their departure time. For example, the travel time of time-dependent shortest path through link 1, 2, and 3 at time t is computed as follows:

*DTTl-3(t)

=

c1[tl + cz [t + cl(t)]

+ c3 [t +

cl(t) + c2 (t + q(t))]

.

A Dynamic

Route

.4ssignment

delay nodeexit time by At

Model

381

Assign trafiic to next link on path

3 compute signal b

updatenode exit the

queue delay

ib Figure

2. Internal

Assignment

Logic

Flowchart.

Similarly, the travel time of static shortest path through link 1, 2, and 3 using instantaneous travel time information at time t is represented as follows: STTl_3(t)

=

c1[t] + ca[t] + cg[t],

where TDTTr_a(t)

= Time-dependent travel time from 1 to 3 at time t,

SSTl-3(t)

=

G(t)

= Travel time on link i at time t.

Static travel time from 1 to 3 at time t,

From this phase, an initial route plan will be est,ablished for each O-D pair at each time period. Each O-D pair will also have a pointer that keeps track of its current position on its route plan that will be updated during the course of internal loading. 2: INTERNAL ASSIGNMENT(AN EVENT-DRIVENsIh4uLmIoN MECHANISM) The internal assignment phase assigns time-varying travel demands along the predicted paths for both classes of vehicles. The assignment procedure is similar to event-based simulation where the assignment takes place along both spatial and temporal dimensions. After each loading and assignment step, DRAM predicts time-varying flows, queues, and travel time in the network according to the time-varying distributions and evolution of vehicles. These results will be used internally for evaluation in the next phase. A logic flowchart of the internal loading process is shown in Figure 2. The process starts by finding all those vehicles with their earliest link exit time. Each vehicle has a pointer to its PHASE

382

G.-L. CHANG

AND T. JUNCHAYA

vehicle enterlink

2 Figure 3. A graphical illustration of link travel time and intersection components in overall link trip time.

delay time

current link along the path and the exit time from this link. When its exit time is the earliest, one, this vehicle is loaded to the next link in the route plan, provided that there is enough available capacity left on that link. The process continues by loading all vehicles with the same link exit time. If some of the vehicles cannot be loaded because of link concentration constraint,, their exit, times from the current link is delayed by one time interval. When all vehicles with the same link exit time have been loaded, we would like to estimate the time these vehicles will stay in this link and their new link exit time. Assume that subsequent vehicles have no effect on the speed of current group of vehicles in the link, we model link trip time as the sum of two components: a travel time along the link and the time delayed at the downstream intersection (Figure 3). The travel time of vehicles along a link depends on the average speed given by the macroscopic speed-concentration relation [15]. The function used in DRAM is given by

t

?I; = (Vf -v,)

2

P.

( >+ 1-

w,,

(1)

0

where Vf

=

the mean speed in Section i during the tth time step;

vf, v, = the mean free-flow speed and minimum speed on the facility, respectively; k,

= the maximum or jam concentration;

k;

= the concentration at the entry time; and

Pi

= a location dependent parameter.

The intersection delay is further separated into two components: signal delay and queue delay. The delays are estimated at the time when vehicles arrive at the end of the link. The total signal

A Dynamic Route Assignment

Model

383

delay varies with their arrival time at the intersections. For instance, if vehicles arrive during the red phase, they will be subjected to signal delay and queuing delay, depending on the number of vehicle waiting in queue at that time. Finally, DRAM can update the new departure time of these vehicles from their current links after computing link travel time, signal delay and queuing delay. These vehicles are subjected to another assignment when their departure time become the earliest one among all time-varying demands. DRAM will repeat the internal loading process for another group of vehicles with the same link exit time. If the link exit time is still within the projected time horizon, these vehicles are assigned to links. Otherwise, the internal assignment stops and we continue to the next phase. The procedures for internal loading phase is summarized in Table 1. Table 1. Internal Assignment Step

1

Select an event with the earliest link entry time from the event list. If the link entry time is greater than the projected time period, then go to Step 7, else go to Step 2.

Step

2

Check the link’s capacity for the selected event (i.e., O-D travel demand) to enter. If there is enough capacity, assign this event to the downstream link, otherwise increment the link entry time by one time interval.

Step

3

Check the event list for another event with the same link entry time. If there is one, repeat Step 2, otherwise go to Step 4.

Step

4

Compute the link speed and travel time subjected on each link.

Step

5

Compute

Step

6

Compute the intersection delays (i.e., signal, queue, and queue spillback delays), depending on the projected intersection arrival time computed in Step 5.

Step

7

This step updates various information down into the following processes:

.

PHASE

Procedure.

to the assigned number of vehicles

the arrival time of vehicles at the downstream

intersections.

for the next assignment of events. It is broken

Update the new link entry time of events selected in Step time and intersection delays;

.

Reinsert the events selected in Step

.

Update the number of vehicles in the corresponding ing to the duration that those vehicles will stay.

2 based on the link travel

2 into the event list; and links at each time period accord-

Step

8

Go to Step

Step

9

Finish the assignment of all travel demands within the projected time period, compute network system performance data during one iteration of the iterative route assignment model.

3:

EXALUATION

1.

AND LEARNING

At the beginning of the previous phase, the paths for guided vehicles are based on projected time-dependent link travel times. However, after the completion of the internal loading phase, all the projected time-varying O-D demands have been loaded to the network and the time-dependent travel times are now based on “actual” traffic loads. The anticipated travel time can then be compared with the “actual” travel time. If the difference is above the predefined threshold, then the assigned. routes for guided and unguided vehicles have not reached stable traffic patterns, and the iterative process continues in repeating the path prediction module. However, if the difference is below the threshold, the predicted routes for guided vehicles have stabilized and can be broadcast to guided vehicles as the real route guidance instructions. The motivation of an iterative assignment process is due to the significance of information gamed from each path confirmation, namely the “actual” time-dependent travel times. By assigning the time-varying O-D demands to the network along anticipated paths, some traffic patterns will be developing and will not change in the next iteration. Thus, this information can be used to improve the prediction of paths for guided and unguided vehicles in the next iteration. Thus, the uncertainties in the prediction of time-dependent travel time are progressively reducing as more tral% patterns are developed.

384

G.-L.

CHANG AND T.

JUNCHAYA

Currently, traffic signal operation in DRAM is modeled as an isolated intersection with simple two-phase operation with the same cycle length for all intersections. The phase duration is adjusted at the end of each iteration based on the corresponding traffic flow volume. Using the following algorithm that relates the duration of a phase split to the projected volume of incoming flow, we can adjust the phase splits after each iteration. At each intersection,

there are four approaches: north-south, south-north,

east-west, and west-

east. The total flows in each approach are tabulated and the maximum flow between the total flows of north-south and south-north direction, (fmax~_s), and between east-west and westeast direction,

(f max~_W), are computed. The green time for the north-south direction can be

computed as follows:

= f

QN-S

m=N-S

f m=N-S

+f

m=E-W

rN_s = Cycle length - gN_s.

1

* Cycle length.

The green and red time in the east-west direction are simply the reverse for the north-south direction. Note that, the above algorithm is rather simple and will be used only to illustrate the capability of DRAM in adjusting signal splits based on predicted flow. A more elaborated algorithm that make better use of the information from preceding iterations to adjust signal splits and/or cycle length can certainly be incorporated

as required.

PHASE 4: IMPLEMENTATION When the differences between anticipated and actual travel times are minimized through successive learning and iteration, the traffic control center will be able to predict the traffic patterns and network performance based on the current predicted time-dependent shortest paths. These final time-dependent shortest paths can then be recommended to ATIS-equipped vehicles.

4. DEVELOPMENT OF A DATA PARALLEL ASSIGNMENT MODEL One of the most critical components in designing a parallel model for SIMD (Single Instruction/Multiple Data) computers is the specification of parallel variables. These parallel variables will determine the effectiveness of the parallel model since different selections of parallel variable will lead to different communication patterns, and/or algorithms. 4.1.

Parallel

Variables

There are four main parallel variables used in the proposed model: network link performance, intersection, dynamic demand, and time-varying path parallel variables. Their data structure and relationships are discussed below. 4.1.1.

Network

Link Performance

Parallel

Variable

(NLP-PV)

The NLP-PV is shaped as a two-dimensional parallel variable as illustrated in Figure 4. The first dimension is equal to the number of maximum links in the network, and the second dimension is equal to the number of predetermined projected time periods. The NLP-PV is used to hold time-varying link information during the projected time period. The information for each link includes length, capacity, maximum and minimum speed, beginning node and ending node of each link. 4.1.2.

Intersection

Parallel

Variable

(I-PV)

The intersection parallel variable is also shaped as a two-dimensional parallel variable. The first dimension is equal to the number of maximum nodes in the network, and the second dimension is

A Dynamic Route Assignment

. . . . . .

385

Model

llnk h

..I...

T+AT

I

T+2At

I

I

I

T+SAT if 3

:

if II

T+ PAT

EACH CELL CONTAINS:

tink ID Be&nw-&~~M~e Maximum & Minimum Speed Link Speed Equation Parameters Current Number of Vehkles Current Speed & Travel Time Queueing & Signal Delay

Figure 4. A graphical illustration of network link performance

NodeNcde2

. . . . . .

parallel variable.

Nodeh

T+AT 3

4

T+2AT T+3AT

ii

:

%

:.

a

T+ pAT

. . . . . .

EACH CELL CONTAINS:

Link Indices for all links feeding this node all links exiting this node Signal Control

Figure 5. A graphical illustration of intersection parallel variable.

equal to the number of projected time periods. Each element in the parallel variable corresponds to an intersection. The information in each element includes signal control information, indices for links enter and exit a node, and the time-varying number of vehicles of incoming links. Figure 5 shows the general layout of I-PV. 4.1.3.

Dynamic Demand Parallel Variable (DD-PV)

The dynamic demand is structured as a three-dimensional parallel variable as shown in Fig;ure 6. The first and second dimensions are equals to the number of maximum nodes in the network. The third dimension is equal to the projected time periods. Each element of DD-PV holds information about origin-destination (O-D) trips in each time period. This information consists of the number of vehicle trips for each O-D, current link in the path, and exit time from current

386

G.-L. CHANG

AND

T. JUNCHAYA

T+PAT

DYNAMIC DEMAND PAFIALLEL VARIABLE ynb&ofv&k&w~m~and CurrentLhk Current and Next Node Spillbach Delay Trip Statistics TIME-VARYING PATH PARALLEL VAFIIABLE Link ID Pati Travel Time Next Node on Shortest Path Figure 6. A graphical illustration of dynamic demand and time-varying path parallel variable.

link. Each type of vehicle (e.g., guided and unguided) will require separate type of vehicle will have different demand and will follow different path. 4.1.4.

Time-Varying

DD-PV,

since each

Path Parallel Variable (TP-PV)

The time-varying path parallel variable is also structured as a three-dimensional parallel variable and share the same dimensions as the DD-PV. Each element holds path information for each vehicle type. The information includes path costs, and next node to follow. Thus, for each vehicle type, there is a corresponding pair of DD-PV and TP-PV. In this study, we have two pairs of DD-PV and TP-PV, one pair for guided vehicles and the other for unguided vehicles. 4.2. Parallel Processing In this section, we will discuss the main operations of Phase 1 and Phase 2 that has been parallelized. The main task in Phase 1 is to compute path for each type of vehicle, while Phase 2 assigns vehicles according to path predicted in Phase 1. 4.2.1.

Path prediction

There are two types of path computation in this study: static shortest path and time-dependent shortest path. Habbal et al. [19] have parallelized the all-pairs static shortest path algorithm the SIMD computing architecture using a two-dimensional data structure and network decomposition with the following logic for the parallel model: fork=

l,... processor

, N do in parallel Pi do the following:

Chang et al. [18] extended Habbal’s algorithm to compute all-pairs time-dependent path with the FIFO requirement using a three-dimensional data structure. The parallel all-pairs, time-dependent shortest path algorithm is shown as follows:

shortest logic for

A Dynamic Route Assignment

for t := T + .NPAt, . . , T (End of projection

Model

387

period ---> Beginning of projection

period)

fork = l,...

, N do in parallel processor Pi do the following: &(t)

= min (c$‘(t),

d:;‘(t)

+ d&l (t + d:;‘(t)))

‘d’ifkfj

where

travel cost from node i to node j through node k dfj (t): N: Number of nodes T: starting time of the projection period NP: number of projection periods At: length of one projection period In the beginning, all-pairs, time-dependent shortest paths for time period T + NPAt computed first, followed by the paths computation for the time period T + (NP - l)At, on until all paths in the first projection time period T have been computed. 4.2.2.

are

and so

Parallel internal loading process

In the sequential process, each demand with the same link exit time is loaded in succession. However, with this parallel implementation, all O-D demands with the same link exit time can be loaded in parallel. Each element of the DD-PV has the information about the number of vehicles in each O-D demand and the next nodes on the time-dependent shortest path. Figure 7 illustrates the parallel loading of demands from DDA-PV to links in NLP-PV using this information as indices in the interprocessor communication.

AU O-D DEMANDS WITH DEPARTURE TIME -T ARE AMGNED NEWORK

SIMULATNEOUSLY

UNK PERFORMANCE IS UPDATED FROM ASSIGNMENT IN PARALLEL

Dynamic Demand Assignment Parallel Variable

Network Link Performance Parallel Variable Figure 7. Parallel operations

between DD-PV

and NLP-PV.

After all demands with the same link exit time have been loaded on the network, the the prevailing ispeed and travel time for all links in NLP-PV can be computed simultaneously for the current time period. Furthermore, signal delay, queuing delay, and the links trip time for these demands can be estimated in parallel. Finally, the number of vehicles loaded in this period can be used to update the links information in the subsequent time periods for the duration of their links trip Cme.

G.-L. CHANG AND T. JUNCHAYA

388

4.2.3.

Parallel signal control and adjustment of phase splits

In this paper, signal control at all intersections are assumed to operate individually. With this assumption, the signal control process is inherently parallel and thus the signals can be structured as a parallel variable. With the use of intersection parallel variable, two main operations involving signals can be parallelized.

The first operation involves the internal assignment process where

each O-D demand has reached the end of the link, and need to check the signal control status to determine whether or not it is possible to leave. This checking process is done in parallel where each O-D demand in DDA-PV communicate with I-PVV to check the condition of all signals. The other operation involves the adjustment of phase splits. This operation is parallelized by letting the I-PV communicate with NLP-PV to obtain link flows information from all approaching links. Each element of the I-PV simultaneously determines the maximum flow and computes the flow ratios among all approaches before adjusting the phase splits for the next iteration.

5. ILLUSTRATIVE

EXAMPLES

The proposed dynamic route assignment model can be used to analyze the traffic network performance under various ATMS traffic scenarios. Each scenario can incorporate factors such as (a) the market penetration rate of guided vehicles, (b) levels of traffic congestion, and (c) traffic arrival patterns. The route guidance or signal control strategies can also be chosen as control variables though they are fixed in the current study. The proposed model can generated various measure of effectiveness (MOE) that can be used for policy analyses. For instance, the overall benefits of ATMS/ATIS under different levels of market penetration rate of guided vehicles can be estimated with the total system travel time. While this indicator gives some basic insight about the potential benefits of ATMS-ATIS, there are still several unanswered questions concerning the relative performance of guided vehicles and unguided vehicles. Hence, the following outputs, aggregated over all O-D pairs and the projected time period can be used as the basis for exploratory analyses. (a) total O-D distance travel from all O-D pairs for equipped and unequipped vehicles; TDTe = TDT” =

c allO-D

&LLt pairs

%ant

c

all O-D pairs

(b) total O-D travel time from all O-D pairs for equipped and unequipped vehicles (including signal and queue delay); TTTe =

en*t

c

all O-D pairs

TTT” =

c allO-D

pairs

ttMnt

c allO-D

pairs

(c) total spillback delay from all O-D pairs; TSDe = TSD” =

sG,t

c all O-D pairs

SGmt

A Dynamic Route Assignment

Model

389

(d) average speed (a/(b + c)); AvgSpe = AvgSp” =

(e) number of trips that reach destinations

TDT” TTTe + TSDe TDT” TTT”

+ TSD”

for equipped and unequipped vehicles

(f) total vehicle travel time TVTTe

=

c

4nnt

. cmt

C

n4knt . %A

all O-D pairs

TVTT”

=

all O-D pairs

where e

ZZ

equipped vehicles; unequipped vehicles; dt&,, dt;,,, = distance traveled by a equipped/unequipped vehicle between origin m and destination n starting at time period t; travel time of a equipped/unequipped vehicle between origin m tth,,,, tt;,,, = and destination n starting at time period t; spillback delay of a equipped/unequipped vehicle between sd&tr ad;,,, = origin m and destination n starting at time period t; number of equipped/unequipped vehicles between origin m and nuKnt, m&t = destination n starting at time period t; In addition, DRAM can produce various MOEs output at an individual O-D level. The output at this level may be necessary for some special needs. =

U

Thus, the following questions can be answered with DRAM: What is the optimal market penetration rate of guided vehicles for a particular network with certain O-D trip rate? (b) Who receives the potential benefits under an ATMS/ATIS environment (e.g., guided or unguided vehicles)? Cc)Is there any tradeoff in the network performance (e.g., shorter travel time but longer distance)? (4 What is the distribution of performance gain among O-D pairs?

(4

The dynamic route assignment model have been implemented on the Connection Machine CM-2 configured with 16,384 processors running at the clock speed of 7MHz with code written in C* (Thinking Machine, 1990), an ANSI C standard with parallel extension. 5.1.

Experiment

Designs

To illustrate the potential applications of DRAM, the following experiments have been carried out with the proposed model, and implemented on the Connection Machine CM-2 with the code written in C* (Thinking Machine, 1990), a data parallel C extension of an ANSI C standard. The CM-2 is configured with 16,384 processors running at the clock speed of 7MHz. Note that the model properties have already been investigated by Chang et al. [18]. The model was shown to have a desirable property which converges to a near optimal loading pattern. Furthermore, the running time of a data parallel model shows a dramatic improvement over the sequential counterpart.

G.-L. CHANG AND T. JIJNCHAYA

390

5.1.1.

Network

The test network (Figure 8) is a hypothetical grid network with 30 nodes (5x6) and 98 links. There is a two-way street connecting each adjacent pair of nodes. This two-way street is represented by two uni-directional links. The free-flow speed on all links is specified as 30mph with a minimum speed of 5 mph. Each link has three lanes, and its length is arbitrary assigned to be between 2000-2500ft long. Each intersection in a network is controlled by a two-phase signals. The cycle length is fixed at 90 seconds, and initial phase duration is evenly distributed between green and red at 45 seconds. ing vehicles

for each 5 minutes

phase duration

by 5 seconds

DRAM will adjust time interval.

signal phasing

The adjustment

with the minimum

according

to number

of approach-

will increment/decrement

previous

green set at 20 seconds.

a a a a

ORIQIN

& DESnNATlON

NODE

DEsmNATlON

Figure 8. Test network and O-D trip patterns boundary nodes.

5.1.2.

Origin-destination

NODE

0

from north boundary

THROUGH

NODE

nodes to south

demand

The O-D trip patterns used in this study are fixed between nodes along the boundary of the network and along four directions (north-south, south-north, east-west, and west-east). Figure 8 shows the O-D trips for these nodes and the direction from north to south. In each time period, there will be trips between 78 O-D pairs. There

are two types

of vehicle

currently

supported

in DRAM

(guided

and unguided).

While

we assume the same ratio of guided/unguided vehicles for all O-D pairs. We vary the ratio of guided/unguided vehicles into six levels of market penetration (0, 20, 40, 60, 80, and 100%). Other factors involve in setting the O-D demand is the demand volume and its distribution over the projected time period. The projected time period is 60 minutes long and is divided into two half hour periods. For evaluation purpose, we assume that all O-D trips will arrive during the first half hour and none will arrive during the second half hour. Figure 9 shows the arrival pattern used in this study. This pattern assumes uniform arrival pattern during the first half hour. There are five levels of demand at 10, 20, 30, 40, and 50% of maximum flow within this pattern (abbreviated as U-10, U-20, U-30, U-40 and U-50, respectively. We assume the maximum flow to be lBOOvph/lane. In a real-world scenario, the demand volume and distribution for each O-D pair may vary substantially.

A Dynamic

IO

Route Assignment

20

Model

391

40

50

60

Time pZod (min) Figure 9. Uniform arrival pattern

5.2.

of vehicles at different demand level

Results

Note that the results and interpretations of the potential ment,

applications

and not intended

Table

2 presents

ket penetration. almost

for U-20.

the total system

Otherwise,

penetration

the network

At medium

vehrcles,

the performance

significantly

pattern

hicles for U40,

it improves

degradation

the total system Tables

demand

p.attern

Tables similar

is divided

performance

vehicles.

While

from 12-14%

pattern

of trips finished.

improves

vehicles

is that

all cases

benefits

for a total following

guided

of IVHS,

among guided and unguided

for six demand

and number

distance,

for

at 20% guided ve-

into the potential

5), the guided vehicles

travel similar

slightly

of 100% guided vehicles.

penetration

speed,

de-

(U-40 and U-50’), there

deteriorates

are distributed

vehicles

in

At 100% guided

for all arrival patterns

At low demand,

(distance.

improves

for U-10 and from 7-10%

performance

consists

and unguided

(Table

environ-

levels of mar-

performance

has no improvement.

will give some insight

into six levels of market

both type of vehicles

speed and number

trend

the guided and unguided

in all categories

at different

While the performance

flow, respectively.

demand

pattern

For high levels of demand

when the demand

the guided

under an ATMS

the system

the system

One common

performance

3 and 4 compare

At 30% uniform

improves

to study how these benefits

3-8 compare

a.nd 20% of maximum

degrades.

for U-50.

system

at 100% guided vehicles where the performance

(U-30),

of improvements.

it may also be of interest vehicles.

except

and U-20),

then at SO-SO%, the performance

is no consistent

show the performance

(U-10,

performance

level of demand

guided vehicles,

a network

policy analysis.

travel time for six demand

At low level of demand,

20-40%

While

given in the following section are mainly for illustration in analyzing

to be an indepth

all llevels of market

teriorates.

of DRAM

patterns

of 36 cases.

uniform

demand

and unguided of finished start

Each

have

trips).

to outperform

guided vehicles

at 10%

vehicles

unguided

have slight

edge in

G.-L. CHANG

392

AND T. JUNCHAYA

Table 2. Total system travel time (hr) and relative performance.

Table 3. Guided vs. Unguided vehicles with uniform arrival pattern at 10% demand level. Market penetration

MOE 0

20

40

of guided vehicles (%) 60

80

100

Table 4. Guided vs. Unguided vehicles with uniform arrival pattern at 20% demand level. Market penetration

MOE

Guided

Unguided

of guided vehicles (%)

0

20

40

60

80

100

Distance

0

6072

5931

5936

5992

5992

Speed

0

26.08

26.33

26.87

26.40

26.26

Trips

0

2340

2340

2340

2340

2340

Distance

6040

6066

5926

5932

5974

0

Speed

26.45

26.30

26.38

26.43

26.28

0

Trips

2340

2340

2340

2340

2340

0

Table 6 presents the results at 40% level of maximum flow. Guided vehicles clearly outperform unguided vehicles, covering equal amount of distance in less time thus yielding better average speed. The guided vehicles also finish more trips than unguided vehicles except at 60% market share where the unguided vehicles have a very slight edge. However, total system travel time does not improve with the introduction of guided vehicles and became worse as the network consists of 100% guided vehicles. At 0% level of guided vehicles, the unguided vehicles achieve very good

A Dynamic Route Assignment Model Table 5. Guided vs. Unguided vehicles with uniform level.

393

arrivalpatternat 30% demand

Table 6. Guided vs. Unguided vehicles with uniform arrival pattern at 40% demand level. Market penetration

MOE

of guided vehicles (%)

0

20

40

60

80

100

Distance

0

5962

5735

5949

5495

5704

Speed

0

24.12

23.13

26.08

22.21

21.63

Trips

0

2179

2035

2256

1933

2005

Distance

5992

5711.

5676

5990

5545

0

Speed

25.68

20.86

20.90

24.50

17.15

0

Rips

2302

1980

1984

2265

1898

0

1

Guided

Unguided

Table 7. Guided vs. Unguided vehicles with uniform arrival pattern at 50% demand level. Market penetration

MOE

Guided

Unguided

of guided vehicles (%)

0

20

40

60

80

100

Distance

0

5428

5728

5124

5026

4571

Speed

0

21.23

22.10

19.09

17.65

13.08

Trips

0

1784

1896

1505

1388

1066

Distance

5338

5060

5625

5052

5092

0

Speed

18.12

16.75

18.60

13.80

11.15

0

Trips

1694

1549

1815

1390

1335

0

performance which deteriorate as market share of guided vehicles increase. This implies that guided vehicles improve their performances at the expense of unguided vehicles. The results for 50% maximum flow in Table 7 illustrates similar patterns as Table 6. Guided vehicles outperform unguided vehicles at all level of market share except for 100% level. The guided vehicles travel faster than unguided vehicles while reaching more destinations. The total system travel time improves by 4% at 20% market share, remain stable at other level except at 100% level where the performance worsens. In addition to the simplified arrival pattern and common ratio of guided/unguided vehicles for each O-D pair, this study focuses mainly on the path assignment with minimal attention being focused on the setting of signal control. Signal and queue delays are significant parts of

G.-L. CHANG AND T. JUNCHAYA

394

travel time, and with proper signal control strategies, the network performance m$y improve tremendously. It is anticipated that route assignment when properly coordinated with adaptive signal control will show improved performance over the current results. However, adaptive signal control

is beyond

the scope of this study

and should

be addressed

in future

ATMS/ATIS

studies.

6. CONCLUSIONS This article puting

presents

architecture.

a dynamic

route assignment

Its implementation

model with a massively

on the Connection

Machine,

parallel

in particular

SIMD comof the design

of parallel variables, fully exploits the parallel nature of the problem and takes advantage of the underlying computing architecture. The proposed model also addresses several critical issues that are needed to determine the potential benefits of ATMS/ATIS under various traffic scenarios. In particular, various

market

the model has the capabilities penetration

rates

to analyze

of ATIS-equipped

the network

vehicles,

performance

different

path

with regard

selection

to

strategies

for multiple classes of vehicles, and various levels of congestion. Furthermore, the output can be analyzed at different levels of detail, ranging from an individual O-D pair, vehicle class, to an aggregated network level. The proposed preliminary model does not provide in-depth study in some areas, in particular, the treatment of signal control. However, the model platform is flexible enough so that more complicated models can be utilized in future work to provide more realistic results. Furthermore, the results indicate that most of the improvement received by guided vehicles is at the expense of unguided vehicles. While definitive conclusion cannot be drawn from these results, they provide some guidelines in carrying out future research. We are currently pursuing two lines of research in order to improve the model. The first research direction addresses signal control and route guidance aspects. Signal and queue delays constitute the major part of travel times in an urban street network. Thus, by coordinating route guidance strategies with the adaptive signal control and freeway control strategies, the network performance could be significantly improved. The results also show that a single route guidance strategy may not be adequate for guiding vehicles from all O-D pairs. Hence, a network’s performance may be improved when guided vehicles received route guidance instructions that depend on the distance to destination, levels of congestion, etc. The second line of research explores the development of dynamic route assignment model with MIMD parallel computing architecture. The main computation issues for the MIMD programming model are synchronization and load balancing among processors. An exploration of using the MIMD machines is currently conducted in parallel to the development of a SIMD model. These extensions and further optimization of the parallel models are required before the actual implementation of dynamic route assignment or real-time traffic simulation models can be successfully implemented in an ATMS/ATIS environment.

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