A multiple-path routing strategy for vehicle route guidance systems

Transpn. Res..C. Vol. 2, No. 3. pp. 185-195. 1994 Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0968-090X/94 $6.00 + .OO

Pergamon

0968-090X(94)00007-7

A MULTIPLE-PATH ROUTING STRATEGY FOR VEHICLE ROUTE GUIDANCE SYSTEMS CHI-KANG

Department

of Transportation

LEE

and Communication Management, National Cheng Kung University, University Road, Tainan, Taiwan 70101

1,

(Received 19 October 1992; in revised form 5 May 1994) Abstract-This paper analyzes the transportation effectiveness of a multiple-path routing strategy using traffic simulation from the perspective of planning and designing a vehicle route guidance system. The test results indicate that the multiple-path routing strategy performs better than the commonly used shortest-path routing strategy.

INTRODUCTION

Research

background

route guidance systems (VRGS) have been given prominence in recent years as a result of various technological and practical developments. Several demonstration projects with diverse system architectures and technologies have been tested in Europe, the United States, and Japan. However, route planning in these systems is primarily based on the shortest-path strategy. Several researchers have noted the importance of dynamic, real-time, and multiple-path routing strategies, so as to improve the effectiveness of vehicle route guidance systems (e.g. Boyce, 1988; Ben-Akiva, de Palma and Kaysi, 1991; Bell, Busch, Rossner and Heymann, 1992; Jayakrishnan, 1992). In this context, this paper analyzes the effectiveness of a multiplepath routing strategy. Vehicle

A multiple-path

routing strategy

A traffic centre-based vehicle route guidance system, such as the EURO-SCOUT system (Hoffmann, 1991), is assumed. Three kinds of computational tasks are defined for the traffic centre. One collects, organizes, and updates traffic information during every short time interval (e.g. every 30 seconds). Another computes K shortest paths (e.g. K = 5, for each O/D pair based on recent traffic conditions during an operational time interval T, e.g. every 15 minutes). The third sorts the K shortest paths based on the latest traffic information during short time intervals (e.g. 1 minute). In general, the time period for updating traffic information is smaller than or equal to the time period for sorting the K shortest paths, and both of these are usually shorter than the operational time period for solving the K shortest-paths problem. Such a system may be appropriate for micro and/or workstation computer-based traffic control centres. The example shown in Fig. 1 indicates the difference between the commonly used shortestpath routing strategy and the proposed multiple-path routing strategy. It is assumed that the shortest-path problem or the K shortest-paths problem is solved every 10 minutes, T = 10; and the time period for updating traffic information or sorting the K shortest paths is 2 minutes. Hence, the route suggestions generated by the shortest-path routing strategy, for equipped vehicles on the network during 12:00 to 22:00, are based on the traffic conditions during 0O:OO to 02:OO. Moreover, all vehicles for the same O/D pairs during 12:00 to 22:00 will receive the same suggestion. However, with the multiple-path routing strategy, K shortest paths for each O/D are solved every 10 minutes. If the value of K is sufficiently large, the real shortest path at any time slice from 12:00 to 22:00 is likely one of the K shortest paths from 0O:OOto 02:OO. Therefore, the multiple-path routing strategy sorts the K shortest paths every 2 minutes to find the near real-time shortest path. For example, the route suggestion to an equipped vehicle at 17:00 is one of the K shortest paths found on the basis of traffic conditions from 0O:OOto 02:00, and it is the shortest path on the basis of traffic conditions from 12:00 to 14:O0. Furthermore, the multiple-path routing strategy generates route suggestions every 2 minutes; thus, the 185

186

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(Minute:Second)

00 : 00 L Route : Suggestions : Update

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& Route Suggestions : Update

Suggestions:

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[The

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r--it--------J Shortest

Path

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[The Multiple-Path

Routing]

Fig.1.Shortest-path andmultiple-path routing strategies

equipped

vehicles

operational

Research

time

passing period

environments, routing

roadside

unit with the same destination

allocated

to different

during

a lo-minute

paths.

objectives

In order to evaluate Then,

a specific are possibly

the

simulation

strategy,

multiple-path

the effectiveness

an experimental results

a comparison

routing

strategies,

design are

of this hypothetical of a simulation

described,

between

including

the

the commonly

and the effects

routing

strategy

study is first proposed

of system

effectiveness

used

of

shortest-path

environment

for various

traffic

in the next section. the

hypothetical

and the proposed

factors

on the perfor-

A multiple-path

routing strategy

187

mance of a vehicle route guidance system. In brief, this paper analyzes the concept and effectiveness of a multiple-path routing strategy using traffic simulation from a perspective of planning and designing a vehicle route guidance system.

A SIMULATION

STUDY

TrafJic simulation model A mesoscopic and stochastic traffic simulation model developed by Lee, Ho, Fu and Fang (1992) is used to estimate the performance of a vehicle route guidance system. As shown in Fig. 2, it consists of six basic components. Vehicle traffic generation assigns vehicles from each centroid. Similar to the MPSM model developed by Chang, Mahmassani and Herman (1985), the highway traffic simulator records each movement of individual vehicles and moves a packet of vehicles with one speed along a link; a deterministic speed-density or delay function is used to compute the speed of a vehicle on a street segment or at an intersection. A traffic information interface simulates the process of collecting and updating traffic data in the traffic centre. Route suggestion generation computes route plans based on a routing strategy of interest. Drivers’ route choice behaviour simulates drivers’ adaptive route choice behaviour. Traffic signal control computes signal timings based on a signal control strategy of interest. Because of the limit of existing knowledge, several assumptions are made to simplify the modelling and experimental work. They are: 1. The signal timing for every intersection is fixed, and there is no interaction between traffic signal control and generation of route plans. 2. All drivers of equipped vehicles follow route plans, and the drivers of non-equipped vehicles have map information and use the shortest distance routes. 3. There is no error in the traffic information interface, and the traffic control centre can obtain accurate network traffic information. 4. There is no consideration of day-to-day dynamics, which means the time-dependent departure pattern and total number of vehicle trips from each centroid are fixed; in other words, only the spatial spreading effect is considered.

Initial

El

t

Conditions

t

Signal Control

Performance

Output

Fig. 2. Structure of the traffic simulation model.

188

C.-K. LEE

(1) t

(13) t

-

(L*) t (:9R)

(7)

-

(:8*) t

-

(8)

-

(:3*) t

-

(9)

-

(:8*) t

-

(:3)

-

Number of intersections

t

(9) t

(2) t

-

-

(:6*) t (L2) t (:6R)t (:lt) t

(is*) t (iOf) t (f5) t (iOf) t (:5*) t

(L#) t t (:a*) t 4, 6, (i9/)

Percentage of roadside unit

(:6*) t

Location roadside

(17) t

-

&2*, t

-

(10)

(:7*) t

-

(11)

(:2*) t

-

(12)

(:7)

of units

12

48%

intersections

marked

*

20

80%

intersections

marked

* and

25

100%

all

R

intersections

Fig. 3. Test network and location of roadside units.

System per$ormance measure Various measuresof a vehicle route guidancesystemare importantfor evaluatingits performance(e.g. drivers’ acceptance, safety implications to the drivers as well as to the whole network, etc.). In this study, mean travel time is selected to measure the benefits of such a system to the whole network on average. For each simulation run, the model starts with an empty network at 7:00 a.m., and the travel time for each vehicle departing from 7:30 to 8:30 a.m. is recorded; after that, the mean travel time is computed as the total travel time divided by total number of vehicle trips. The reduction in travel time is used to measure the difference between the network with and without a vehicle route guidance system. It is defined as Average travel time without VRGS -

Average travel time with VRGS 9

Average travel time without VRGS

where the average travel time is the average of mean travel times for several computer

runs.

A test example As shown in Fig. 3, a small grid networkis used in this study. The length of each street segment is randomly generated within the range from 500 to 700 meters. The size of the network is 5 nodes by 5 nodes with 25 signalized intersections. There are 12 traffic zone centroids Table 1. Levels of system design and system environment System design factors

System environment

factors

The K value The T value The D value Equipped vehicles Network congestion

factors

1, 3, 5 5, 10, 15 (minutes) 48%. 80%. 100% 25%, 50%. 75%, 100% 75%, loo%, 125%

A multiple-path

routing strategy

189

located at the peripheral nodes of the grid. All links in the network are two lanes each, and all intersections allow all feasible turning movements. An O/D trip matrix of 12 zones by 12 zones for a morning peak on a weekday with its associated departure time distribution is defined as the second input. If the O/D trip data are used to solve a static user equilibrium flow pattern, about 35% of links on the network have a volume/capacity ratio (V/C) z 0.80. Therefore, the network is partly congested. Moreover, in order to reduce the possible effects of poor signal control, the user equilibrium flow pattern and the SOAP package are used to determine the best signal timing plan for each intersection (FHWA, 1982). Experimental design Two kinds of experimental factors may affect the performance of the multiple-path routing strategy. First, from the system designers’ point of view, three variables are chosen: the number of shortest paths to be computed (the K value); the length of the operational time interval to solve the K shortest-paths problem (the T value); and the density of roadside units (the D value). These variables can be decided by the designers before system implementation. Secondly, two environment variables are considered: the percentage of equipped vehicles and the level of network congestion. These environment variables generally cannot be controlled by the designers. Based on preliminary testing with various values of the above mentioned experimental factors, and the geometric consideration of the network, levels for each test factor are defined and summarized in Table 1 and Figure 3, where the index of network congestion level is a multiplier for the base O/D trip matrix. Accordingly, there are 12 specific system environments and 27 system alternatives. Therefore, 324 experiments are required to test the performance of different system alternatives under various system environments; and 12 experiments are used to create the comparative base of no vehicle route guidance system under various system environments. Furthermore, because the simulation is stochastic, the model must be solved several times in order to obtain a reliable performance estimate for one experiment. After preliminary runs, with a significance level of 5% and a confidence interval of half estimated variance, 16 samples are required for every experiment described in this study (Kleijnen, 1987). Moreover, to get a good measure of reduction in travel time, the method of common random numbers is used to perform the sampling for different experiments (Kleijnen, 1987).

EVALUATION

OF THE ROUTING

STRATEGY

Experimental results The results of the percentage reduction in travel time for the 336 experiments described previously are shown in Table 2, where the three numbers in a cell are, respectively, for D = 48%, D = 80%, and D = 100%. When the percentage of equipped vehicles is not high (25% or 50%), different system alternatives have similar performance, even under various system environments. However, if the penetration of the system is high, the K value and the T value have significant effects, but not the D value. As expected, an increase of the K value and a decrease of the T value usually improves the performance of the system. Furthermore, similar to several previous studies (Reekie, Case and Tsai, 1989), the test results indicate that the shortest-path routing strategy (K = 1) is not suitable if the penetration of a vehicle route guidance system is high. Such results may arise when many vehicles are assigned to similar paths during one operational period (see Boyce, 1988); or it may be caused by the stability of the routing strategy (see Bertsekas and Gallager, 1987). In order to present the overall performance of the vehicle route guidance system over different system environments, the results in Table 2 are replaced by descriptive symbols to form Table 3. It is clear that if the multiple-path routing strategy is considered (K > l), the performance of the system is relatively good for the system environments in the lower right comer of Table 3. That is, the multiple-path routing strategy effectively spreads vehicles on the network spatially, even if the congestion of the network and the penetration of the system are high.

125%

15

10

10

T=

T=5

15

T=

T=5

T = 10

T=

T=5

T=

T = 15

16.83 17.23 17.09 17.04 17.13 16.98 17.13 17.45 17.08

K=3

K=l 16.66 16.87 16.81 17.09 17.07 17.00 17.18 17.59 17.59

10.12 9.84 10.00 9.94

10.07 10.09 9.78 10.05 9.90

K=3

2.90 2.87 2.87 2.88 2.81 2.87 2.88 2.89 2.86

K=3

10.00 10.02 10.26 10.25

9.86 9.90 10.34 10.12 10.12

K=l

2.17 2.75 2.15 2.80 2.19 2.19 2.82 2.81 2.81

K=l

17.36 17.29 17.17 17.50 17.02 16.97 17.46 17.13 17.03

K=5

10.24 9.96 9.89 9.96

10.25 10.05 9.96 10.32 9.86

K=5

2.84 2.81 2.84 2.85 2.86 2.84 2.85 2.85 2.83

K=5

20.65 20.86 22.43 22.67 21.94 23.07 23.96 23.60 23.65

12.31 12.82 12.80 12.90 K=l

12.10 12.29 11.74 12.54 12.45

K=l

3.11 3.10 3.10 3.21 3.11 3.06 3.45 3.26 3.30

K=l

23.75 23.82 23.99 23.87 23.67 23.71 23.88 23.26 23.96

13.20 13.54 13.28 13.37 K=3

13.33 13.37 13.33 13.37 13.18

K=3

3.16 3.15 3.19 3.76 3.12 3.68 3.69 3.71 3.76

K=3

50%

24.45 23.92 24.34 23.99 23.62 24.34 24.40 23.83 24.15

13.43 13.55 13.33 13.32 K=5

13.46 13.37 13.43 13.44 13.17

K=5

3.71 3.70 3.61 3.68 3.64 3.63 3.64 3.68 3.68

K=5

for D = 48%. 80%, and 100%.

T = 5

T = 10

T = 15

T = 5

T = 10

T = 15

T = 5

T = 10

T = 15

Percentage reduction in mean travel time, respectively,

3

g % 3

‘JE B 2 8 *$

100%

15%

25%

T = 5

T = 10

T = 15

T = 5

T = 10

T = 15

T = 5

T = 10

T = 15

13.92 13.15 12.44 17.86 17.52 18.09 21.82 22.12 22.09

K=l

10.67 6.48 10.38 10.30

3.43 3.15 3.10 7.01 7.48

K=l

24.31 24.39 24.65 24.10 24.74 24.60 24.03 24.85 24.19

K=3

13.14 13.86 13.03 12.98

12.73 12.74 13.01 12.92 13.00

K=3

3.38 3.26 3.31 3.44 3.42 3.38 3.26 3.30 3.19

K=3

15%

0.01 0.30 0.04 0.11 0.14 -0.5 2.00 1.44 1.54

K=l

of equipped vehicles

reduction in mean travel time

Percentage

Table 2. Percentage

24.70 25.13 25.06 24.56 25.08 25.36 24.50 25.14 25.17

K=5

13.18 12.92 13.12 13.09

12.91 13.09 13.12 13.09 13.19

K=5

3.27 3.24 3.24 3.31 3.29 3.10 3.13 3.11 3.12

K=5

T = 5

T = 10

T = 15

T = 5

T = 10

T = 15

T = 5

T = 10

T = 15

-2.6 -4.4 -8.8 7.71 5.45 3.10 16.18 16.00 16.18

K=l

-7.4 0.28 -0.1 4.08

-36.3 - 50.7 - 14.3 -18.5 -25.5

K=l

-8.7 - 12.4 -9.8 -7.1 -1.2 -9.6 -2.4 -3.3 -3.6

K=l

21.07 21.55 21.33 20.85 21.74 21.73 21.15 22.47 22.18

K=3

10.78 0.10 10.32 10.45

9.51 10.04 9.92 9.03 9.95

K=3

1.33 1.47 1.59 1.46 1.51 1.51 1.09 1.20 1.24

K=3

100%

22.01 23.09 23.01 21.96 22.93 23.28 22.08 23.17 23.18

K=5

10.92 10.49 10.95 11.06

10.21 11.00 11.04 10.46 16.49

K=5

1.35 1.19 1.44 1.51 1.45 1.42 1.21 1.23 1.25

K=5

9

125%

100%

Descriptive

0

F

.s c

75%

K=5

K=l

K=3

K=5

T= 15

10

T=

K=3

symbols for various percentage

T=5

15

T=

K=l

10

T=

K=l K=3

in mean travel time:

T=5

15

T=

reductions

K=5

15

T = 10

T=

K=5 15

T=5

T = 10

T=

T=5

K=3

K=5

T=5

15

K=3

T=5

T=

K=l

T = 10

K=l

K=5

K=

K=l

K=l

Percentage of equipped vehicles

T = 10

15

K=3

50%

table

T = 10

T=

K=l

25%

Table 3. Descriptive

I

75%

K=3

K=3

K=3

K=5

K=5

K=5

10

15

10

T=

T=5

15

T=

T = 10

T = 15

T=5

T=

T=

7d7;iJ

K=

00000

00000

K=l

I

‘2 2 3 u 2 7,320 3J7X9

-----

q EIOOO

00000 00000 00000

K=l

100%

K=3

K=3

K=3

K=5

K=5

K=5

C.-K. LEE

-*

K=5

_+K=l

I

25

I

I

50

75

Percentage

of

Equipped

Vehicles

I

100 (%)

Fig. 4. Effect of traffic incidents. Traffic incidents In order to compare the capabilities of different system alternatives with nonrecurrent traffic incidents, the network input data is varied. Let 10% of the links on the network randomly selected have nonrecurrent traffic incidents. One traffic lane of such a selected link is closed at a randomly selected starting time for a randomly selected duration of 10 to 30 minutes. For two 3 +-k=5

,k=l

2.5

2

1.5

1

0.5 0 -0.5 SO Percentage

75 of

Equipped

Vehicles

(%)

Travel Time Difference = (Average Travel Time with Traffic Incidents - Average Travel Time without Traffic Incidents) Travel Time without Traffic Incidents)

Fig. 5. The trip predictability.

/ (Average

A multiple-path

193

routing strategy

selected system alternatives (K = 1, T = 10, D = lOO%, and K = 5, T = 10, D = 100%) and the base travel demand pattern, a generated pattern of traffic incidents is tested with various percentages of equipped vehicles. Each line in Fig. 4 shows the reduction in travel time of one system alternative under the system environment with traffic incidents. Obviously, the multiple-path routing strategy (K = 5, T = 10, D = 100%) is better than the shortest-path routing strategy (K = 1, T = 10, D = lOO%), especially when the penetration of the system is high. Moreover, to understand the average trip predictability of one system alternative, the results of the new experiments with traffic incidents are compared to the corresponding results of the experiments without traffic incidents. As shown in Fig. 5, system alternative (K = 1, T = 10, D = 100%) is sensitive to the environment change, especially when the penetration of the system is high; however, system alternative (K = 5, T = 10, D = 100%) always gives stable mean travel times with and without nonrecurrent traffic incidents. That is, the multiple-path routing strategy appears to have better trip predicability than the shortest-path routing strategy, on average. The variability of travel demand pattern In order to study the effect of travel demand pattern on the performance of a vehicle route guidance system, the O/D trip matrix input data is changed. The base O/D trip matrix is varied randomly to check the effect of travel demand pattern. For a fixed system alternative (K = 5, T = 10, D = 100%) and a fixed percentage of equipped vehicles (loo%), Tiinew= TVbase(1 + r), where Tune, and Tijbaseare corresponding elements in the new and base O/D trip matrices, and r is a random number ranging from -0.25 to +0.25 for the case of 25% variation. For each case of trip matrix variation, 10 experiments are tested. As tire results show in Table 4, it is clear that the performance of the system alternative varies as the travel demand pattern changes. However, the results in Table 4 appear to suggest that the reduction in travel time is correlated with the total number of trips.* Therefore, each experiment with a newly generated trip matrix is further compared to the experiment with the base trip matrix. As the comparison results show in Fig. 6, there is no clear relationship between the total number of trips and the percentage reduction in travel time. In summary, the results discussed above suggest that the O/D trip matrix determines the spatial traffic pattern, which affects the spatial spreading performance of a vehicle route guidance system. System design Besides the issues on the demand side, such as the identification of important environment factors, their characteristics, and their possible effects on the performance of a vehicle route guidance system, it is also necessary to examine the issues on the supply side, such as possible system alternatives and their characteristics, as well as how to find a good design. In the simple Table 4. Effect of travel demand pattern 25% Variation Total Vehicle Trips 15048 14692 14414 15406 15000 15797 14527 14416 14682 14528

50% Variation

75% Variation

Decreased Travel Time (o/o)

Total Vehicle Trips

Decreased Travel Time (%)

Total Vehicle Trips

Decreased Travel Time (8)

22.16 17.59 17.34 15.15 14.78 8.31 7.16 6.99 5.54 4.61

15253 15262 15799 10469 14997 14469 14585 14146 14586 14148

17.62 17.36 13.88 10.75 8.31 5.03 4.76 3.91 3.83 2.21

16281 16272 16005 16000 14315 14049 14047 14313 14047 13219

18.85 18.81 10.84 9.02 5.44 5.38 3.17 1.98 1.83 1.02

* The percentage reductions shown in Fig. 6 should not be interpreted as benefits from VRGS, as the comparisons made for different O-D matrix cases to the single base case.

are

C.-K. LEE

194

5o

I

+ +

40

$

*

30

*

X

+ +

m

20

10

X X

u

4

* 25% -40

x 50%

+ 75%

-30

-20

Difference Difference New O/D Matrix)

%A A *s

of

-10

Total

Number

of

0 Trips

10

(%I

of Total Number of Trips = (Total Number of Tripe in the Trip Matrix - Total Number of Trips in the Base O/D Trip / (Total Number of Trips in the Base O/D Trip Matrix)

Travel Time Difference = (Average Travel Time with a New O/D Trip Matrix - Average Travel Time with the Baee O/D Trip Matrix) / (Average Travel Time with the Base O/D Trip Matrix)

Fig. 6. Travel time difference

and total number of trips.

example discussed above, it is easy to use Table 2 to find a good system alternative for a specified system environment; more generally, some quantitative methods are needed to identify important design factors, to understand the effects of the factors, and to find a good design. The results of the analysis of variance shown in Table 5 give a clear indication that the D value may not be an important design factor for a specified system environment; but the interaction between design factors T and K does seem important. Furthermore, the regression analysis shown in Table 6 provides a performance function of the design factors for one specified system environment; this function indicates some nonlinear relationships between the design factors and the system performance. Moreover, optimization techniques can be used to find a good design using the function described above by the response surface method (Kleijnen, 1987). In summary, many experimental results can be obtained from such a simulation study, and various nonlinear and interaction relationships among the design factors and the system performance may exist. Methods for identifying these relationships are required for planning and designing a vehicle route guidance system.

Table 5. Analysis Factor K T D KT KD TD KTD

Error Total

df 2 2 2 4 4 4 8 405 431

of variance:

equipped vehicles

ss 121001 828330 I132 219004 2216 9384 3291 68585 1252914

= lOO%, network congestion

= 125%

MS

F Value

Pr > F

60501 414150 566 54750 554 2346 411 169

357.26 2445.58 3.34 323.31 3.27 13.85 2.43

0.0001 0.0001

0.0363 0.0001 0.0117 0.0001 0.0142

A multiple-path Table 6. A performance

function: equipped vehicles congestion = 100%

Dependent

= 0.9883;

195 = 1001,

network

Variable: Mean Travel Time

Estimated Coefficient

Independent Variables Constant K T D KK TT DD KT KD TD KTD R-squared

routing strategy

Standard Error

61.847 -3.126 79.450 110.14 0.002 - 4.201 -7.419 - 1.389 - 1.410 - 6.242 - 1.397 no. of observations

CONCLUDING

18.481 0.655 7.385 13.897 0.001 0.548 1.106 0.601 0.781 2.626 0.779 = 324.

t Statistic 3.346 -4.770 10.758 7.925 3.595 -7.661 -6.707 -2.314 - 1.805 -2.377 - 1.793

REMARKS

The paper reports a planning process using traffic simulation to evaluate a multiple-path routing strategy for vehicle route guidance systems. Based on the reported test results, the multiple-path routing strategy performs better than the commonly used shortest-path routing strategy. Moreover, the test results indicate that the performance of a vehicle route guidance system is dependent on some important system design and system environment factors. Therefore, further studies on these factors from both aspects of system environment and system design will be required for practical applications. Acknowledgements-The improve the paper.

author

is grateful

to the two anonymous

reviewers

for their helpful

suggestions

to

REFERENCES Bell M. G. H., Busch F., Rossner F. and Heymann G. (1992) The synthesis of dynamic route guidance and urban traffic control. Paper presented at the 25th ISATA Conference, Italy. Ben-Akiva M., de Palma A. and Kaysi I. (1991) Dynamic network models and driver information systems. Transportation Research, 25A(l), 25 l-266. Bertsekas D. and Gallager R. (1987) Dam Networks. Prentice-Hall International (Englewood Cliffs, NJ). Boyce D. E. (1988) Contribution of transportation network modelling to the development of a real-time route guidance system. Paper presented at the First International Conference on Transport for the Future, Sweden. Chang G.-L., Mahmassani H. S. and Herman R. (1985) Macroparticle traffic simulation model to investigate peak period commuter decision dynamics. Transportation Research Record, 1005, 107-121, Hoffmann G. (1991) Up-to-the-minute information as we drive: How it can help road users and traffic management. Transport Reviews, 11(l), 41-61. Federal Highway Administration (1982) Handbook of Computer Models for Truffic Operations Analysis. FHWA-TS82-213. U.S. DOT (Washington, D.C.). Jayakrishnan R. (1992) In-vehicle information systems for network traffic control: A simulation framework to study alternative guidance strategies. Ph.D. Thesis. University of Texas at Austin. Kleijnen T. P. C. (1987) Statistical Tools for Simulation Practitioners. Marcel Dekker (New York). Lee C.-K., Ho C.-H., Fu C.-T. and Fang Z.-F. (1992) A traffic simulation model for testing the effectiveness of vehicle route guidance systems. Transportation Planning Journal, (2). 163-188 (In Chinese). Reekie D. H. M., Case E. R. and Tsai J. (Eds.) (1989) Proceedings of 1st Vehicle Navigation and Information Systems Conference, Ontario, Canada.