JOURNAL OF TRANSPORTATION SYSTEMS ENGINEERING AND INFORMATION TECHNOLOGY Volume 13, Issue 1, February 2013 Online English edition of the Chinese language journal

Cite this article as: J Transpn Sys Eng & IT, 2013, 13(1), 118−123, 178.

Received date: Sep 18, 2012; Revised date: Nov 8, 2012; Accepted date: Nov 20, 2012 *Corresponding author. E-mail: hxliu@home.swjtu.edu.cn Copyright © 2013, China Association for Science and Technology. Electronic version published by Elsevier Limited. All rights reserved. DOI: 10.1016/S1570-6672(13)60095-1

RESEARCH PAPER

Perception Updating Based Stochastic Dynamic Assignment Model ZHANG Xi, GUO Hongyang, LIU Haixu*, LI Yanlai, PU Yun School of Transportation and Logistics, Southwest Jiaotong University, Chengdu 610031, China

Abstract: Traffic network systems endogenously display both stochasticities and dynamics with vehicle flows formed from aggregated travelers in response to their previous experiences and the information provided by Advanced Traveler Information Systems (ATIS). In consideration of the day-to-day trip-making decisions of travelers, a perception-updating-based stochastic dynamic assignment model is proposed to describe the evolution of the traffic network flow pattern. In this model, the route flows are treated explicitly as random variables and the distribution of them is proved to asymptotically converge to a stationary probability distribution. A simulation algorithm is developed for implementing the model. Numerical results under two scenarios, with and without ATIS, are also provided for comparing the stochastic dynamics of route flows. Key Words: system engineering; stochastic dynamic assignment; perception updating; road network; ATIS

1 Introduction

In a traffic network, the decisions of travelers such as choices of departure time and routes are always random because of the uncertainty of traffic demand and supply. In addition, advanced traveler information systems (ATIS) provide travelers with various ways of accessing traffic information, such as radios, cellular phones, websites, and variable message signs. Travelers usually change their trip-making decisions according to traffic information and their past experiences. This process also involves dynamics from the view of the travel times. Traffic flow usually represents aggregated trip-making decisions of numerous travelers. Therefore a network system with traffic flow defining its state variable is essentially characterized as a stochastic dynamic system. Through recognizing the stochasticities and dynamics of the system, the evolution of the system state can be comprehensively analyzed, providing a reliable basis for traffic planning and management purposes.

Using information technologies, ATIS provide travelers with real-time and reliable traffic information, which greatly increases the convenience for travelers. Such information results in travelers making different decisions, thus the traffic flow and network conditions vary dynamically. The dynamics

of trip-making decisions by travelers has attracted significant research interest from scholars all around the world. Mahmassani and Liu[1] designed several laboratory experiments to simulate travelers’ daily commuting behaviors through an interactive multiple user decision simulator. They found that the provision of information would increase travelers’ sensitivity to the change of travel time and travelers therefore, would switch routes more frequently. Shi[2] established a dynamic choice model based on the forecasted travel time updating process through experiences and real-time information, and found that the expected perceived travel time would stably converge after repeated trips of travelers. Liu et al.[3] established a day-to-day route choice evolving framework which integrated a risk aversion analysis and a perception updating process. They found that the systematic error of ATIS would affect the time needed for the convergence of route flow. He et al.[4] simulated the route choice behavior with mixed guidance modes and generalized the influences of different guidance modes on the route choice behavior of travelers. Xiong et al.[5] classified travelers by their compliances to ATIS and formulated a corresponding perceived route travel time function. Kuang et al.[6] confirmed that whether ATIS exists or not, travelers would unlikely acquire the exact traffic information. They established a

ZHANG Xi et al. / J Transpn Sys Eng & IT, 2013, 13(1), 118−123, 178

stochastic assignment model based on generalized costs and found that the appropriate market penetration of ATIS and quality of information would result in a reasonable flow pattern. Nonetheless, in spite of the recognition of stochasticities and dynamics of trip-making behavior of travelers, the studies mentioned above, usually targeted one characteristic while neglecting another for simplification. Thus, the conclusions obtained are relatively restrictive. Therefore, it is necessary to consider both stochasticities and dynamics simultaneously, and use the stochastic dynamic system approach to study traffic networks and travelers’ trip decision making process. Treating route choices as random variables, Cascetta[7], proposed a kind of Markovian stochastic dynamic assignment model in which the temporal route choice probability was assumed to follow Markovian characteristics. Namely, the present route choice probability could be determined by the previous finite route choice probabilities. The Markovian assignment model implies that the day-to-day trip-making decision can be explained as an accumulating process of travelers’ experiences. Subsequently, Davis and Nihan[8], and Cantarella and Cascetta[9] systematically discussed the relationships between dynamics and equilibrium of traffic systems. They studied the day-to-day dynamics of traffic systems, and reached general conclusions in both theory and by using numerical cases. They also proposed a dynamic simulation algorithm different from the classical mathematical programming algorithm for the implementation of the assignment model. From the perspective of modeling, Watling and Hazelton[10–12] also studied the relationship between equilibrium and dynamic assignment models. They referred that the external information system might have influences on the stochastic dynamic model. To the best of our knowledge, however, there is very limited literature that discussed this issue in detail. Hence, in this paper, considering the stochasticities of the route flow and the perception updating process, a stochastic dynamic assignment model is proposed. In the following sections, the model is first described. It is tested using numerical examples, and finally the comparison of route flows with and without ATIS is carried out.

2 Notations and assumptions

Consider a road network G(N, A) where N is the set of nodes and A is the set of links. Let W be the set of OD pairs and Kw be the set of feasible routes for any OD pair w W∈ . Assume that the demand of the OD pair w W∈ is a given constant and notated as dw. The link-route incidence is expressed as a 0-1 variable δak, which equals 1 if and only if the link a A∈ is a part of the route wk K∈ , otherwise it equals 0, and Δw is the corresponding matrix. Let ac be the designed capacity of link a, and at be the free flow travel time of that link. Let the discrete time domain L={0, 1, ···, M}

be the observation time domain, where M is set to be large enough to ensure that the changes of the network state can be fully observed. Any time point n L∈ is assumed to represent a day.

For any OD pair w in a road network system, let Sw={0, 1, 2, ····, dw} be the sample space of the route flow. Let the random variable ( )n

kF% be established on Sw, representing the flow on route k on day n, and ( )n

w%F be the corresponding vector. Thus,

the stochastic process ( ) ( ) ( ){ }0 1, , , Mw w w w=% % % %LF F F F describes the

stochastic dynamics of the route flow on the OD pair w. Let ( )nax be the flow on link a on day n, and x(n) be the

corresponding vector. ( ) ( )n nw w= %x Δ F holds according to the

link-route incidence. Namely, the link flow vector is a function of a random route flow vector. Assume that the link cost is the function of link flow and capacity, and is notated as

( ) ( )( ),n na a a at x cτ=

Let ( )nkT be the route cost, it is obtained that

( ) ( )n nk ak aa A

T tδ∈

=∑ according to the link-route incidence relationship. Let ( )n

kV be the measurable disutility of route k and ( )n

wV be the corresponding vector. Let the route choice probability function be expressed as ( ) ( )( )n n

k k wP P= V and ( )nwP be the

corresponding vector.

3 Utility updating and route choice

The state of a road network is usually affected by many random factors such as traffic accidents, weather conditions, road reconstruction, and maintenance. Thus it is unlikely that travelers will obtain the exact information about the network performance before their trips. But after repeated trips, travelers may accumulate abundant experiences and become familiar with the network state. Thus, travelers might form certain perceptions of the present network state based on their previous experiences and use them as their trip-making decision basis. 3.1 Utility updating

Route choice is an important part of trip-making decisions. Presently, the discrete choice model based on random utility maximization is capable of appropriately describing the route choice. In order to describe the influence of travelers’ past experiences on their present perceptions, a learning mechanism with the property of exponentially decreasing linear weights needs to be introduced[12]. The measurable disutility of route k on day n can be stated as:

( )( )

( ) ( ) ( )

11

11

11 2

1 1 11 1 1

1

m n iikn i

k m ii

mn n n m

k k km m mi i ii i i

TV

T T T

γ

γ

γ γγ γ γ

−−=

−=

−− − −

− − −= = =

=

= + + +

∑∑

∑ ∑ ∑L

wk K∈ (1)

where m is the effective experience length or memory length. It means travelers would form present perceptions through the past m days’ experiences. )1,0(∈γ is a parameter with

ZHANG Xi et al. / J Transpn Sys Eng & IT, 2013, 13(1), 118−123, 178

1 11

mm ii

γ γ− −=∑

as the learning weights, representing the influence of the past experiences on the present perception. It can be stated that the experienced travel time on a past day has a larger influence as the day is closer to the present.

Travelers are often not unanimous about route cost perceptions due to their individual social economic attributes. Such differences can be described as random utility errors across perceptions, and thus the random disutility ( )n

kV% of route k can be expressed as:

( ) ( ) ( )n n nk k kV V ξ= +% wk K∈ (2)

where ( )nkξ is the perception error among travelers, and is

assumed to follow an independent Gumbel distribution with a mean of 0 and variance of σ(n). It is clear the variance of the distribution is assumed to be dependent on time in this paper. 3.2 Perception error updating

With the advances of information technologies, ATIS is capable of providing comprehensive traffic information to travelers. But the systematic errors of ATIS are unavoidable due to measurements, equipment, and other factors. Thus, the information provided by ATIS may not exactly describe the actual network state. In other words, the released information contains certain noises or errors. In this paper, the information error is assumed to follow an independent normal distribution with a mean of 0 and variance of φ.

With traffic information and accumulated experiences, travelers are gradually forming similar perceptions of the network state. This is equivalent to the process that the variance of the perception error is decreasing over time. According to Refs. [3] and [13], this process can be stated as the following recursive equation:

( )( )

( )

1

1

nn

n

σ ϕσσ ϕ

−

−=

+ (3)

where φ is the variance of the provided information error and is assumed to be fixed over time; σ(n) is the variance of the travelers’ perception error and is assumed to vary over time. 3.3 Route choice

Assume that the perception errors follow independent and identical Gumbel distributions, the route choice probability can be expressed as a Logit probability function, which is:

( 1) ( 1)( )

( 1) ( 1)

exp( )exp( )

w

n nn k

k n nll K

VpV

θθ

− −

− −∈

−=

−∑

wk K∈ (4)

where θ(n–1) is the parameter of the Gumbel distribution, and its relationship with the variance is ( ) ( )1 16n nθ π σ− −= .

As Eq. (4) shows, the route choice probability will be affected by the feasible route set Kw of the OD pair w. Different feasible route sets stand for different route choice behaviors, thus different flow patterns will result. The feasible routes set consist of acyclic routes which not only exclude cyclic routes, but also ensure that all available routes can be chosen by travelers. From both theoretical and practical points

of view, all available acyclic routes are selected as feasible routes in this paper. The feasible route sets can be determined through a layered algorithm[14]. The layered algorithm is a traversal algorithm, in which each link and node of the network will be accessed at least once, so all the feasible routes will be enumerated and the link-route incidence matrix Δw for each OD pair will eventually be obtained.

When the perception is updating, the variance of the perception error is decreasing. But the error cannot be vanished because the trip-making decision by travelers still involves stochasticities, and the traffic flow aggregated by numerous travelers is also random. Thus the network system can be treated as a stochastic dynamic system. For each OD pair w, travelers choose a feasible route among the feasible route sets according to their disutility. By aggregating travelers’ choices, the probability distribution of random flow on route k on day n is actually the distribution condition on the flow on route k in the past m days, which can be approximated as a conditional multinomial distribution parameterized with a OD demand dw and choice probability vector ( )n

wP . ( )n

w%F | ( ) ( ){ }1 , ,n n m

w w− −Lf f ~ ( )( )MN , n

w wd P w W∈ (5)

where ( )nwf is a realization of the random vector ( )n

w%F . Thus,

the stochastic process ( ) ( ) ( ){ }0 1, , , Mw w w w=% % % %LF F F F describes the

stochastic dynamics of route flow of the OD pair w.

4 Stochastic dynamic assignment model

From the above analysis and referring to work[7], a perception-updating-based stochastic dynamic assignment model is formulated below:

( ) ( ) ( ) ( )1 11

w

n n n na a ak k a

w W k Kx x F x

nδ− −

∈ ∈

⎛ ⎞= + −⎜ ⎟

⎝ ⎠∑ ∑ % a A∈ (6)

( )nw%F ~ ( )( )( )MN , n

w w wd P V w W∈ (7)

( )( )( )1

1

11

,m n iiak a a ai a An

k m ii

x cV

γ δ τ

γ

−−= ∈

−=

⎡ ⎤⎣ ⎦=

∑ ∑∑

wk K∈ (8)

Proposition 1. The stochastic process ( ) ( ) ( ){ }0 1, , , Mw w w w=% % % %LF F F F

has one unique stationary probability distribution. Proof. In the research time domain, the route choice

probability satisfies: (1) time-homogeneity; (2) non-zero route choice probability ensured by the Logit choice probability function; (3) the present route choice probability dependant on past finite states. Hence, according to proposition A and proposition B of Ref. [7], this proposition can be proved.

Proposition 1 shows that the distribution of route flows of a road network will converge to a unique stationary probability distribution over time. Eq. (6) means that the link flow on day n is actually the average value of all previous link flows. Thus the flow loading process of the stochastic dynamic assignment model is similar to the steps in the method of successive averages (MSA). Thus, the assignment steps are given.

ZHANG Xi et al. / J Transpn Sys Eng & IT, 2013, 13(1), 118−123, 178

1 2 3

4 5 6

7 8 9

O

D Fig. 1 Test network

10 20 30 40 50 60 70 80 90 100 0

50

100

150

200

250

300

350

Day

Rou

te fl

ow (v

eh/h

)

Route 1-4-7-8-9 (with ATIS) Route 1 -4-7-8-9 (without ATIS) Route 1 -2-5-6-9 (with ATIS) Route 1 -2-5-6-9 (without ATIS)

Fig. 2 Variation of route flow

Step 1: Initialization. Set parameters γ, m, M, φ and σ(0). Let n=0 and

( )0k ak aa A

V tδ∈

=∑

For each OD pair, generate vector ( )0wf from the

multinomial distribution, ( )0w%F ~ ( )( )( )0MN ,w w wd P V , through the

random vector generator. Set ( ) ( )0 0w w= %x Δ F according to the

link-route incidence, which yields the initialized link flow vector.

Step 2: Updating the variance of perception error. Let n=n+1. Update σ(n) through Eq. (3). Then set the dispersion parameter as ( ) ( )6n nθ π σ= .

Step 3: Updating the perceived route cost. Calculate the link cost through ( ) ( )( )1 1 ,n n

a a a at x cτ− −= . If n≤m, set the perceived route cost as ( ) ( )1n n

k ak aa AV tδ −

∈=∑ ; otherwise, update the perceived

route cost through Eq. (1). Step 4: Generating random route flow. For each OD pair w,

substitute ( )nkV to Eq. (4), which yields a route choice

probability ( )nkP . Then generate vector ( )0

wf from the multinomial distribution, ( )n

w%F ~ ( )( )MN , n

w wd P , through the random vector generator.

Step 5: Updating link flow. Substitute ( )nwf and ( )1n

ax − into Eq. (6), this yields a new link flow, ( )n

ax . Step 6: Checking stop criterion. If n=M, stop assignment

process; otherwise go to Step 2.

5 Numerical example

The test network[4] has 9 nodes, 12 links, and one OD pair,

as shown in Fig. 1. OD demand is set as dOD=500 veh/h. Other network parameters such as the link capacity and the free flow travel time are referred to in work[4]. The link cost function is the common BPR function. The remaining parameters are set as follows: the initial standard deviation of the perception error is ( )0 0.33σ = h, the time-independent standard deviation of the information error is 0.08ϕ = h, the memory length is m=3 d, the learning weights parameter is γ=0.3, and the total days are M=100 d.

For the purpose of observing influences of the provided information on travelers’ perception updating, the stochastic dynamics of the network route flow is considered under two scenarios, with and without ATIS in this example. There are a total of six feasible routes in the test network. Now take route 1-4-7-8-9 and route 1-2-5-6-9 to observe their flow variation under scenarios with and without ATIS. Fig. 2 shows the sample trajectories of the two route flows.

Under both scenarios with and without ATIS, the flow distribution of the two routes seems to converge after ten days. For further observation of the mean and fluctuation of the route flows in the time domain, 1,000 stochastic dynamic assignments are carried out for each scenario with and without ATIS respectively. This assignment results in a total of 2,000 sample trajectories under two scenarios, and the expectation and standard deviation of the route flows are statistically calculated, as shown in Figs. 3 and 4. The expectation and the standard deviation of the route flows also seem to converge to fixed values after ten days.

10 20 30 40 50 60 70 80 90 100 0

50

100

150

200

250

300

350

Day

Expe

cted

rout

e flo

w (v

eh/h

)

Route 1-4-7-8- 9 (with ATIS) Route 1-4-7-8- 9 (without ATIS) Route 1-2-5-6- 9 (with ATIS) Route 1-2-5-6- 9 (without ATIS)

Fig. 3 Variation of the expected route flow

10 20 30 40 50 60 70 80 90 100 0

10

20

30

40

50

60

70

80

Day

Sta

ndar

d de

viat

ion

of ro

ute

flow

(veh

/h)

Route 1-4-7-8- 9(with ATIS) Route 1-4-7-8- 9(without ATIS) Route 1-2-5-6- 9(with ATIS) Route 1-2-5-6- 9(without ATIS)

Fig. 4 Variation of the standard deviation of route flow

ZHANG Xi et al. / J Transpn Sys Eng & IT, 2013, 13(1), 118−123, 178

Table Average route flows and travel times

Average flow (veh/h) Average travel time(h) Route

With ATIS Without ATIS With ATIS Without

ATIS 1-4-7-8-9 118 104 1.24 1.19

1-4-5-8-9 132 116 1.24 1.18

1-4-5-6-9 41 62 1.25 1.25

1-2-5-8-9 93 101 1.24 1.20

1-2-5-6-9 31 53 1.25 1.26

1-2-3-6-9 85 64 1.24 1.24

0 10 20 30 40 50 60 70 80 90 100 550

600

650

700

750

800

850

Day

Tota

l tra

vel t

ime

(h)

With ATIS

Without ATIS

Fig. 5 Network total travel time

Therefore, from the statistical point of view, it is argued that the first two order moments of the route flow probability distributions both converge stably after a certain period. This means that the distributions of the route flows converge to stationary probability distributions. For example, the flow distribution of route 1-4-7-8-9, under the scenario with ATIS, converges to a probability distribution with a mean of 117.8141veh/h and a standard deviation of 10.5510veh·h-1, and the one without ATIS converges to another probability distribution with a mean of 104.2911veh/h and a standard deviation of 9.1382veh/h.

For observations of the average levels of route flow and travel time in a stationary state, the average values are calculated by using the data from day ten to day 100. The results are collected in Table. It is found that under the scenario with ATIS the travel time of each route is about 1.24h, this means that the average flow approximates to stochastic user equilibrium (SUE) flow. Nonetheless, under the scenario without ATIS, the average travel times of route 1-4-7-8-9, route 1-4-5-8-9, and route 1-2-5-8-9 are about 0.04-0.06h less than the ones with ATIS. Yet the travel times of the remaining routes have apparently not changed, while the corresponding flows are more or less than the ones with ATIS. The total travel times of the network system under the two scenarios are shown in Fig. 4. The total travel time in the stationary state is about 620.83h under the scenario with ATIS, and is about 605.97h without ATIS. The former is nearly

14.86 h larger than the latter. It seems that in spite of the improved convenience of trip making and the increased fairness of acquiring traffic information for travelers benefited from ATIS, the trade-off is that the efficiency of the traffic system is degraded to some extent.

6 Conclusions

ATIS provides reliable and extensive traffic information which enhances travelers’ ability to perceive the state of a network, and therefore facilitates their trip-making decisions to fulfill their requirements. Due to the various attributes of traffic systems, the imprecise demand and supply generate uncertainties. It is unlikely for travelers to know the exact state of a road network even with ATIS. However, travelers may improve their knowledge of the network state through past experiences. Hence, an accumulating process of travelers’ experiences and the random route flow of a network are assumed in this paper. Then, a stochastic dynamic assignment model is proposed to describe the variation of network flows based on perception updating with ATIS. In the demonstrated numerical example, it is found that the route flow quickly converges to a stationary probability distribution under both scenarios, with and without ATIS. Specifically, the expected flow of the stationary probability distribution is similar to SUE route flow under the scenario with ATIS. And under the scenario without ATIS, the travel times of three routes are less than the case with ATIS, yet the travel times of the remaining routes apparently do not change when compared to the case with ATIS despite the route flow changes. As for the total travel time, the case with ATIS is higher than the case without ATIS. This means that even though traffic information from ATIS might increase the convenience of trip making and information acquiring, it somewhat satisfies the efficiency of the traffic system. This is important for the sustainable development of social economic systems, where both fairness and efficiency should be considered while introducing new traffic technologies or executing new traffic regulations.

Acknowledgements

This research was funded by the National Natural Science Foundation of China (No. 50908196 and No. 51278429) and the Fundamental Research Funds for the Central Universities of Ministry of Education of China (No. SWJTU09CX041).

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