perception updating and day-to-day travel choice dynamics in traffic

Perception updating and day-to-day travel choice dynamics intra�c networks with information provision

Mithilesh Jhaa,*, Samer Madanat b,1, Srinivas Peeta c,2

aThe center for Transportation Studies, Massachusetts Institute of Technology, 3 Cambridge Center, Room 208,

Cambridge, MA 02139, USAbDepartment of Civil and Environmental Engineering, University of California at Berkeley, CA 94720, USA

cSchool of Civil Engineering, Purdue University, West Lafayette, IN 47907, USA

Abstract

A Bayesian updating model is developed to capture the mechanism by which travelers update their traveltime perceptions from one day to the next in light of information provided by Advanced Traveler Infor-mation Systems (ATIS) and their previous experience. The availability and perceived quality of tra�cinformation are explicitly modeled within the proposed framework. The uncertainty associated with adriver's travel time estimate is modeled in a stochastic dynamic framework and is incorporated in a travelchoice model. Each driver uses a disutility function of perceived travel time and perceived schedule delay toevaluate the alternative travel choices, then selects an alternative based on the utility maximization princi-ple. The perception updating model and the choice model are integrated with a dynamic tra�c simulator(DYNASMART). Empirical results from the simulation experiments and their implications are also pre-sented. # 1998 Elsevier Science Ltd. All rights reserved.

Keywords: ATIS; ITS; Day-to-day dynamics; Driver behavior; Drivers' perception updating; Drivers' learning;

Dynamic network modeling

1. Introduction

Recent developments in communication technologies o�er opportunities for providing tra�cinformation to drivers and thereby improving system performance. In order to evaluate thee�ectiveness of tra�c information, it is important to model driver response to ATIS. Traditionaldriver behavior models for route and departure time selection need to be modi®ed (Kaysi, 1991)

TRC 104m

0968-090X/98/$Ðsee front matter # 1998 Elsevier Science Ltd. All rights reserved.

PII: S0968-090X(98)00015-1

TRANSPORTATION

RESEARCH

PART C

Transportation Research Part C 6 (1998) 189±212

* Corresponding author. Tel.: 001 617 252 1112; fax: 001 617 252 1130; e-mail: [email protected] E-mail: [email protected]{ E-mail: [email protected]

to be applicable in the context of dynamic tra�c conditions and ATIS. Drivers' responses dependon their perceptions of travel times in the network. Therefore, it is important to model the processby which drivers update their perceptions of travel times. The purpose of this paper is to present aperception updating model, its integration with a route and departure time choice model and todemonstrate its applicability in the context of day-to-day dynamics under information provision.Drivers' perception updating is a function of both experience and information travel times. Themodel explicitly accounts for variations in perceptions across drivers.The driver behavior model proposed in this paper integrates a perception updating model and a

travel choice model. The behavioral model is embedded in a simulation framework to study day-to-day dynamics. DYNASMART, (Jayakrishnan et al., 1994) a mesoscopic simulation model fordynamic network tra�c analysis, is used for performing experiments in a hypothetical network.The remainder of this paper is organized as follows: in Section 2 we summarize previous rele-

vant research in the area of dynamic network modeling. The framework and driver behaviormodel are presented in Section 3. The simulation framework and the results from the empiricalstudy are described in Section 4. Finally, in Section 5, the contributions of this paper and thescope of future research in this area are discussed.

2. Literature review

Several models have been proposed in the past to describe drivers' travel choice behavior in anetwork. The assumption that drivers select paths to minimize their perceived travel times(Daganzo and She�, 1977) gives rise to the stochastic user equilibrium (SUE). The formulationby Daganzo and She� was used to develop a stochastic equilibrium model for departure timeselections (de Palma et al., 1983). This model was extended (Ben-Akiva et al., 1984) to includeday-to-day variations. A further extension of the model (Ben-Akiva et al., 1986) included routechoice and the option of not making the trip. The evolution of tra�c pattern from one day to thenext is derived from a Markovian model, assuming that the system reaches a unique steady-state.The framework used by de Palma et al. (1983) and Ben-Akiva et al. (1986) was extended(Vythoulkas, 1990) to develop a dynamic stochastic equilibrium assignment model for a generalnetwork.Daganzo and She� (1977) were among the ®rst researchers to model the evolution of the tra�c

pattern under a Markovian assumption. A stochastic process approach was proposed (Cascetta,1989) to analyze day-to-day dynamics in a transportation network. The framework precludes thebasic assumptions needed for the equilibrium approach and accounts for the random ¯uctuationsin the number of users actually choosing each path under the assumption that O±D ¯ows arelarge enough. A key assumption for proving the stationarity and ergodicity of the stochasticprocess is that the path choice probabilities are time homogeneous. The time homogeneity of pathchoice probabilities implies that if the states occupied by the system on any m continuous daysare repeated, then the choice probabilities of the user will also be repeated, which means thatthere are no time-dependent variations in drivers' perceptions. Similar stochastic process modelfor route choice was proposed by Watling (1996). The stochastic process approach for modelingday-to-day dynamics was further extended (Cascetta and Cantarella, 1991) to include within daydynamics. More recently, various dynamic process models for probabilistic assignment were

190 M. Jha et al./Transportation Research Part C 6 (1998) 189±212

proposed and conditions for the existence and uniqueness of an equilibrium state were derived(Cantarella and Cascetta, 1995). The theories of optimal control and variational inequality havealso been used to demonstrate the similarity between the day-to-day adjustment processes (Frieszet al., 1994) of drivers and the demand/supply adjustment in a typical commodity market. How-ever, no dynamic driver behavior model was embedded in the dynamic equations which describethe evolution of tra�c patterns.

2.1. Dynamic behavior modeling and driver response to ATIS

An experimental approach for studying dynamic interactions within a transportation systemwas proposed by Mahmassani et al. (1986). A simulation system was developed for dynamicnetwork analysis under various information strategies. The simulation system, known asDYNASMART (Jayakrishnan et al., 1994), has been extensively used for experimental studies inthis area. Mahmassani (1990) presented a number of experimental results on day-to-day dynam-ics and discussed their implications. Hu and Mahmassani (1995) proposed a simulation frame-work for studying day-to-day dynamics under real-time information. However, drivers' pre-tripdecisions in this framework are based on the previous day's experience and hence are independentof current network conditions. The concepts of approximate reasoning and fuzzy theory havealso been used to develop a model (Lotan and Koutsopoulos, 1993) for route choice in the pre-sence of information. Cascetta (1992) proposed a logit model for departure time and route choicefor home to work trips. The variations in perceptions of schedule delay across drivers were foundsigni®cant, which indicates the need for a detailed, individual level, driver behavior model whichwould explicitly account for drivers' perceptions. Dynamic behavior models were proposed(Kaysi, 1991) to predict drivers' responses to ATIS. Kaysi (1991) has also discussed the variousissues which need to be addressed in order to predict drivers' travel choices from a human factorpoint of view. The issues include drivers' perception updating, their attitude towards informationand so on. To the best of our knowledge, there exists only one study (Mirchandani and Soroush,1987) in which drivers' perceptions have been given adequate importance in network assignment.Mirchandani and Soroush (1987) proposed a generalized static tra�c equilibrium model whereeach traveler perceives a probability distribution function for a route travel time and makes travelchoices to minimize his/her expected disutility.More recently, drivers' route choice behavior has been modeled Abdel-Aty et al. (1995)

with stated preference data. The results of this study indicate that the extent of travel timevariations and tra�c information can have a signi®cant impact on route choices. Khattak et al.(1995) proposed a combined revealed and stated preference model to explore how drivers'pre-trip decisions are a�ected by ATIS in the context of unexpected congestion. Most of thestudies described above ignore the information processing mechanism on the drivers' part.However, an important factor in modeling drivers' responses to ATIS is how they deal withinformation. More speci®cally, how information in¯uences their perceptions of travel times in thenetwork.The potential of information systems in a case of a non-recurring congestion has been studied

by Emmerink et al. (1995) using a simulation framework. Emmerink et al. (1995) studied theimpact of information quality and market penetration level on a tra�c network. Using aboundedly rational approach for modeling driver behavior they found that drivers were not able

M. Jha et al./Transportation Research Part C 6 (1998) 189±212 191

to e�ciently select their route in case of a non-recurring congestion. However, the quality ofinformation was re¯ected by the updating frequency that is, how often travel times are updatedby ATIS and the delay in transmitting information.It is expected that an e�ective ATIS will help drivers make e�cient travel decisions. Interaction

between drivers and information plays an important role in the e�ectiveness of information sys-tem. Various issues regarding the impact of ATIS on network performance have been studiedbefore (Ben-Akiva et al., 1991; Bonsall, 1994). However, Watling (1994) concluded that theexisting models for describing the potential of ATIS in improving network performance may notbe able to fully capture all the system characteristics.

2.2. Drivers' perception updating

The dynamic network models described above assume drivers' perceptions as given. In thecontext of day-to-day dynamics, especially in presence of information, it is highly likely that dri-vers' knowledge/perception of network performance will vary depending on their past experience,accessibility to ATIS and personal attributes, etc. Therefore, assumptions on drivers' perceptionsin the above network models are restrictive and unrealistic. Modeling the process by which tra-velers combine their experience and other information received through various sources (on-linecomputer, radio, newspaper, TV etc.) is important.The role of information integration andlearning in the decision making process has been studied (Einhorn and Hogarth, 1981; Bagozziand Silk, 1983) in the context of marketing. However, applications of such information integra-tion models to transportation seem to be limited. Horowitz (1984) proposed a learning model forroute choice selection whereas Lerman and Manski (1982) developed a learning model for modechoice.

2.2.1. Weighted average of measured travel timesHorowitz (1984) suggested an equilibrium model in which the travel choices on each day are

based on weighted averages of measured travel times on previous days. A simple extension of theabove approach would be to allow the weights to vary across individuals. This extension was usedin the experimental study by Mahmassani and Chang (1986).

2.2.2. A myopic adjustment modelUnder the myopic adjustment approach, (Mahmassani and Chang, 1986) drivers' travel choi-

ces are based on the previous day's experience. The updating process is given by the followingequation:

�i;t � T�i;tÿ1 � ai ci;tÿ1E

�i;tÿ1 � bi

li;tÿ1E

�i;tÿ1; �1�

where �i;t is the updated travel time estimate by driver i; T�i;tÿ1 is the experienced trip timeby driver i on day �tÿ 1�; E�i;tÿ1 is the schedule delay of driver i on day �tÿ 1�; ci;tÿ1 is thebinary variable such that ci;tÿ1 � ÿ1 for early arrival and ci;tÿ1 � 0 otherwise; li;tÿ1 is also abinary variable such that li;tÿ1 � 1 for late arrival and li;tÿ1 � 0 otherwise; ai and bi areparameters such that 04a; b41; thus, these parameters re¯ect the relative weights of earlinessand lateness.


2.2.3. A weighted average of historical and current informationBen-Akiva et al. (1991) proposed a convex combinations approach for information integration.

According to this approach, the updated estimation of travel time on a path is given by

Iu � �Ih � �1ÿ ��Ii; 04�4 1; �2�

where Ih and Ii are the historical perception and the travel time provided by ATIS, respectively. �indicates the relative importance of information and historical perception in the updating process.For example, a high value of � (equal or close to 1) suggests that the driver gives more impor-tance to the historical perception and less to the information travel time.

Based on the literature review, the following comments can be made:

. The focus of the existing dynamic network models is on the ®nal state of the networkassuming equilibrium or steady state conditions. Such assumptions are restrictive and pro-hibit realistic and detailed modeling of dynamic driver behavior.

. In the context of ATIS, it is reasonable to expect that drivers' perceptions would change withtime. Some studies have been directed towards modeling the information integration processof drivers. However, these studies consider drivers' perceptions as deterministic and hencedo not account for drivers' con®dence in the perceived travel time.

In the next section, we propose a Bayesian model for the driver's perception updating from oneday to the next in light of experienced travel time and available information.

3. Framework and models

In order to explain day-to-day travel choice dynamics, it is necessary to explicitly model dri-vers' perceptions and the change in drivers' perceptions over time. A key drawback of all theperception updating models described in the previous section is that they do not capture drivers'con®dence in their travel time estimation. Both the mean and the variance of travel time deter-mine the extent of the attractiveness of an alternative. This cannot be captured by a model whichrepresents drivers' perceptions through the mean travel time only. In this section, a Bayesianmodel is described for updating driver's perception in the context of day-to-day dynamics and inthe presence of information. In the Bayesian framework, perceived experienced time and per-ceived information travel time are represented by random variables, whose variances are indica-tors of drivers' con®dence. Thus, the Bayesian approach systematically accounts for drivers'con®dence in their estimation. In our driver behavior model, drivers' travel choices are based ontheir time varying perceptions of travel times in the network. Thus, we account for both drivers'travel time estimates and the associated uncertainty. To the best of our knowledge, this is the ®rstattempt to dynamically model the uncertainty associated with drivers' estimates of travel times inthe network and use this information in a travel choice model. Next we describe the variouscomponents in the overall framework of this research.


3.1. Components of the framework for day-to-day dynamics

The proposed framework for studying day-to-day dynamics, described later in this section,integrates the following components:Network performance model: An appropriate network performance model is needed to obtain

the experienced travel time of each vehicle every day. We adopted a simulation model for thispurpose. Currently, there are three well known simulation models available for dynamic tra�canalysis: DYNASMART, CONTRAM and INTEGRATION. DYNASMART has been usedextensively for tra�c simulation for a variety of applications (Mahmassani, 1990), some of whichare very similar to this study (Hu and Mahmassani, 1995). Therefore, DYNASMART wasselected for the present research. DYNASMART uses macroscopic tra�c ¯ow relations to movethe vehicles. However, it keeps track of each individual vehicle's statistics (such as path, trip time,stop time etc.). The macroscopic approach allows a fast execution, whereas the microscopicfeatures enable the analyst to collect data at the individual vehicle's level and with highaccuracy. Thus, DYNASMART is a mesoscopic simulation model for dynamic network tra�canalysis.Driver's day-to-day travel choice dynamics: There are two sub-components within this compo-

nent: a perception updating model and a travel choice model. The ®rst component combinestra�c information and drivers' historical perceptions to update drivers' perceptions of networkperformance. On the basis of updated perceptions, the travel choice model predicts the route anddeparture time of the driver. The two sub-components will be described in detail in the nextsection.Updating O±D matrices: Route and departure time selections (by drivers) are made before the

intended departure time. In our framework, it is assumed that drivers make the travel choicedecisions 5 min before their intended departure interval. The intended departure interval for adriver on day �wÿ 1� is same as the actual departure interval for that driver on day w. It is worthmentioning here that a real time O±D updating may give rise to the inconsistency between thetra�c pattern observed in the network and that on the basis of which information travel time isestimated. A discussion of this ®xed point problem is beyond the scope of this paper and isaddressed in the literature (Peeta, 1994; Jha, 1996) in the area of Dynamic Tra�c Assignment(DTA).

3.2. Decision making at the individual level

The proposed framework uses disaggregate choice models for route and departure time selec-tion and for perception updating. Figure 1 describes the conceptual framework for our travelbehavior model. As indicated in the ®gure, drivers are assumed to update their perceptions in twostages. The ®rst stage corresponds to pre-trip updating when drivers combine historical percep-tions with real time information. Drivers' travel choices are based on the updated travel timeperceptions. The second stage corresponds to the post-trip updating. Thus, in post-trip updating,perceptions before the trip are combined with the perceived experienced travel time on the currentday. The updated perceptions become the historical perceptions for the next day. The modeldescribed in Fig. 1 is the backbone of the simulation framework for studying day-to-daydynamics in the present research.


3.3. The simulation framework for day-to-day dynamics

The simulation framework, shown in Fig. 2, describes the embedding of various componentsdescribed in Section 3.1. A tra�c simulator is used as a network performance model. The simu-lator provides the current network conditions, based on which information is provided to drivers.The real time information is integrated with the historic perception during the pre-trip stage. Theupdated perception is used in route and departure time choice. Based on the selected alternative,drivers are assigned a departure time and a route and consequently the future O±D matrices aremodi®ed. Thus, at each departure interval O±D matrices in the future are modi®ed. At the end ofthe day, the experienced time for each traveler is obtained from the tra�c simulator. The experi-enced time is then combined with the previous perception which was obtained from pre-tripupdating on the same day. This stage is called the post-trip updating and the updated perceptionforms the historic perception for the next day. The updating process is continued in the samemanner for the next day and so on. The proposed simulation framework can incorporate anykind of learning model and travel choice model.

3.4. The perception updating model

The proposed framework is based on Bayesian updating. Rasmussen (1986) proposed aninformation integration model based on Bayesian updating. Later, the possibility of using Bayesian

Fig. 1. Individual decision making process.


updating for representing drivers' dynamic behavior in the context of day-to-day dynamics wasalso suggested by Kaysi (1991). In our Bayesian updating model, the travel time perceptions ofdrivers as well as travel time information provided by ATIS are described by probability dis-tribution functions. The variance of each distribution function indicates a driver's level of con-®dence in that source of information. It is hypothesized that, on any given day, a driver has acertain level of knowledge about travel times in the network and updates his/her knowledge inlight of the travel time provided by ATIS. The updated perception of travel time is obtained byBayesian updating, where the perception before receiving information corresponds to the priorand the updated perception corresponds to the posterior. Drivers use their posterior perceptionsto select a combination of route and departure time on that day. After making the trip, driversupdate their posterior perception in light of the experienced travel time and obtain the priorperception for the next day. Thus, there are two stages of updating on a given day: before thetrip, pre-trip updating, and after the trip, post-trip updating. Pre-trip updating incorporates bothinformation availability and perceived quality whereas post-trip updating accounts for the tripexperience. The no-information case, i.e. if a user does not have access to information, can beeasily modeled. Therefore, the model can also be used to determine an extent of informationdissemination which satis®es some system-wide goals such as reduction of total travel time in thenetwork. Only pre-trip information is modeled in the proposed framework. However, the frame-work can be extended to en-route perception updating by using multi-stage Bayesian updatingwhere each decision node (nodes where path switching is possible) would correspond to anupdating stage. Next, we describe the mathematical formulation of the perception updatingmodel.

Fig. 2. Simulation framework for day-to-day dynamics.


3.4.1. Model formulationWe ®rst de®ne some variables which are used in this model:

T1;i;wk;r mean perceived travel time by individual i on day w before receiving information

and before the trip for kth path between his/her origin and destination (since ori-gin and destination do not vary for a given user, no subscripts or superscripts areused for origins and destinations) and rth departure interval. Also, this is theparameter which is updated in our Bayesian model.

T2;i;wk;r updated distribution T1;i;w

k;r in light of information (i.e. after pre-trip updating).T3;i;wk;r updated distribution of T1;i;w

k;r in light of experience (i.e. after post-trip updating)�j;i;wk;r Mean of of Tj;i;w

k;r where j � 1; 2; 3 which is viewed as the driver's best estimate ofthe travel time.

The mean perceived travel time is the mean of a traveler's perceived travel time. It is worthmentioning that in this paper we consider only the mean perceived travel time and propose amodel for updating its distribution. We do not model the overall perceived travel time of a driverwhich may be obtained by adding an error term to the mean perceived travel time (this issue hasbeen discussed in Appendix A). Let the mean perceived travel time for individual i on day wbefore receiving information and before the trip be expressed by

T1;i;wk;r � �1;i;wk;r � �i;w1;k;r; �3�

where �1;i;wk;r is the best estimate of travel time on day w by driver i for kth path and rth departureinterval, �i;w1;k;r is the perception error; its distribution is assumed as N�0; �i;w1;k;r�.Thus, a driver's mean perceived travel time is modeled as a random variable whose mean is his/

her best estimate of travel time. Note the similarity between this representation and the one in atraditional stochastic network assignment (Daganzo and She�, 1977). Also, if we interpret per-ceived travel times in the Daganzo and She� model as mean perceived travel times, then theirsolution method for SUE can be applied to ®nd an equilibrium solution (Mirchandani and Sor-oush, 1987) in the current context. However, for reasons described earlier we prefer not to imposeany restriction on the ®nal state of the network and hence do not attempt to ®nd an equilibrium¯ow pattern. �1;i;wk;r is viewed as the best travel time estimate because of the following two reasons:

1. In the context of day-to-day dynamics, the true travel time can be known only after it isexperienced because of the randomness in network performance and ¯uctuation in thedemand pattern from one day to the next; the travel time experienced by a user can beviewed as the outcome of a random process.

2. Considering the stochasticity and dynamics in the estimation of travel time, it is reasonableto assume that the mean perceived travel time for a user will be distributed around his/herbest estimate.

In the proposed modeling framework we consider two driver classes: informed and non-informed.Non-informed drivers do not have access to information system but they systematically use theirexperienced travel times to update their perceptions. On the other hand, informed drivers useboth information travel times and experienced travel times to update their perceptions.


3.4.2. Pre-trip updatingLet the travel time provided by ATIS on day w on kth path for rth departure interval be tiwk;r.

Without loss of generality, it is assumed that the travel times provided by ATIS do not varyacross individuals. It is hypothesized that when users receive tra�c information, they modify it basedon their perceptions of information. The modi®ed information travel time can be expressed as:

tpi;wk;r � tiwk;r � �i;w2;k;r; �4�where tpi;wk;r is the perceived value of information travel time by individual i for kth path and rthdeparture interval on day w. �i;w2;k;r is the perception error, which is due to the user's past experi-ence with tra�c information, his/her attitude towards the information system, etc. The distribu-tion of �i;w2;k;r is assumed N�0; �i;w2;k;r�.The updated best estimate is given by the following Bayesian formula (Ang and Tang, 1975):

�2;i;wk;r � E�T2;i;wk;r � �

tpi;wk;r �Var�T1;i;wk;r � � T1;i;w

k;r �Var�tpi;wk;r�Var�T1;i;w

k;r � � Var�tpi;wk;r�: �5�

The updated variance of the mean perceived travel time is given by

Var�T2;i;wk;r � �

Var�T1;i;wk;r ��Var�tpi;wk;r�

Var�T1;i;wk;r � � Var�tpi;wk;r�

: �6�

The updated mean perceived travel time is given by

T2;i;wk;r � �2;i;wk;r � �i;w3;k;r: �7�

In Eq. (7), �2;i;wk;r is given by Eq. (5), and the variance of �i;w3;k;r is given by Eq. (6). The pre-tripupdating described above is done for all the paths and departure times for which information isavailable. Note that the variance of tpi;wk;r in Eq. (5) determines the relative importance of infor-mation in the estimation of the updated mean perceived travel time. The variance of tpi;wk;r indi-cates driver's con®dence in ATIS. Thus, drivers' perceived quality of information is modeledthrough �i;w2;k;r. Another important characteristic of our model, which is directly implied by theBayesian updating approach, is that the variance of T2;i;w

k;r will always be less than the variance ofT1;i;wk;r , indicating that the variance of the mean perceived travel time decreases as users receive

more information.

3.4.3. Post-trip updatingThe following updating is done in light of the travel time experienced by individual users.

Therefore, post-trip updating takes place only for that route and departure time combinationwhich a user selects for the trip. Since experience for a particular departure interval may havesome e�ect on the perceptions for neighboring departure intervals, an extension is to includeupdating of travel time perceptions for neighboring departure intervals. If an individual i selectspath k and departure time r then we denote the travel time experienced by individual i by tei;wk;r. Asin Eq. (4), the experienced travel time as perceived by individual i, can be written as:

txi;wk;r � tei;wk;r � �i;w4;k;r; �8�


where �i;w4;k;r is due to perception and measurement error, whose variance determines the relativeimportance of experienced travel time in perception updating. The distribution is assumed Nor-mal with mean 0 and standard deviation �i;w4;k;r.As mentioned earlier, the experienced travel time can be viewed as the outcome of a random

process. The randomness comes from the network performance as well as the travel choicedynamics. The best travel time estimate is further updated and is given by

�3;i;wk;r �T2;i;wk;r �Var�txi;wk;r� � txi;wk;r �Var�T2;i;w

k;r �Var�txi;wk;r��Var�T2;i;w

k;r �: �9�

The updated variance of the mean perceived travel time is given by

Var�T3;i;wk;r � �

Var�T2;i;wk;r ��Var�txi;wk;r�

Var�T2;i;wk;r � � Var�txi;wk;r�

: �10�

As in Eq. (6) the posterior mean perceived travel time is given by

T3;i;wk;r � �3;i;wk;r � �i;w5;k;r: �11�

Note that Eq. (11) describes the distribution of mean perceived travel time of user i on day w+1.Thus, on day w+1, the parameters of Eq. (3) are obtained by

�1;i;w�1k;r � �3;i;wk;r ;

Var�T1;i;w�1k;r � � Var�T3;i;w

k;r �:On day w� 1, updating is done in a similar way. Updating is repeated for the desired number ofdays. The variance of the mean perceived travel time for that combination of departure time androute selected on day w decreases further.Following are the assumptions and limitations of the proposed Bayesian model:

1. The variance of the mean perceived travel time always decreases from one day to the nextand it is the characteristic of the Bayesian model. However, under an assumption of con-sistent information it is reasonable to expect that the variance of mean perceived travel timewill decrease with time. Also, even though the variance of mean perceived travel time isdecreasing, the overall variance of perceived travel time could still increase.

2. It is assumed that the underlying travel time has a steady mean value. Therefore, the model,in its present form, will not be applicable if there is any drastic change in tra�c ¯ow patterndue to some external reasons, for example an incident.

3.5. Route and departure time choice model

In this section, we present a modi®ed form of existing models for route and departure timechoice. Our demand model is a random utility based model and we follow the hypothesis pro-posed by Ben-Akiva et al. (1986), namely that travelers de®ne a hierarchy between departure timeand route choice. Based on this hypothesis, the probability of selecting path k and departureinterval r on day w by individual i is expressed as


Pi;wk;r � Pi;w

r �Pi;wk=r; �12�

where Pi;wr is the probability of selecting departure interval r by individual i on day w. Pi;w

k=r is theprobability (on day w) of selecting route k given that the user has selected departure interval r.Both probabilities in Eq. (12) are represented by logit models. The probability that an indivi-

dual i selects path k and departure interval r is obtained by the following nested logit model:

Pi;wk;r �

e�r�Vi;w�rPr�Ti;w�l

r�Ti;w e�r�Vi;w�r

� e�k�Vi;wk;rP

k2Ki;wr

e�k�Vi;wk;r

; �13�

where Vi;wk;r is the measured utility corresponding to path k and departure interval r for individual i

on day w. �r and �k are the scale parameters for departure time and route, respectively. �k is setto 1 for normalization. Vi;w�

r is the maximum expected utility among the available alternate routesduring departure interval r. It is expressed by

Vi;w�r �

Xk2Ki;w

r

e�kVi;wk;r ; �14�

where Ti;w is the earliest departure interval and Ti;w � l is the latest. Ki;wr is the set of all the routes

which are considered reasonable by user i on day w for departure interval r.Expressions for choice probabilities are derived from previous studies (Ben-Akiva et al., 1986;

Ben-Akiva and Lerman, 1985) in this area.The structure of the nested logit model based on the above description is presented in Fig. 3. It

suggests a higher variation for route choice and hence a larger number of route switching should beexpected. However, an opposite structure of the nest which would give rise to a larger variation fordeparture time selection and hence a larger number of departure time switching could also beadopted for this study. In other words, this structure of the nest is adopted solely for demonstratingthe applicability of our model and could easily be reverted if empirical observations so suggested.The main sources of disutility that a�ect drivers' route and departure time selections are: (1)

travel time and (2) schedule delay. The schedule delay is de®ned as the di�erence between pre-ferred and actual arrival time. Thus, it captures the impact of early or late arrivals. The basicform of the utility function used in our model is adopted from a previous study (de Palma et al.,1983). Following this study, the systematic part of the total utility can be expressed as:

Fig. 3. Structure of the nested logit model.


Vi;wk;r � ÿ��tti;wk;r� ÿ ��i;w1;k;r�Di;w

k;r� ÿ � �i;w2;k;r�Ei;wk;r�; �15�

where Vi;wk;r is the systematic part of the utility for individual i on day w for path k and departure

interval r. tti;wk;r is the perceived travel time of user i on day w for path k and departure interval r.Di;w

k;r is the anticipated schedule delay by user i on day w for late arrival for path k and departureinterval r. If A is the preferred arrival time then D is given by: Di;w

k;r � Tr � tti;wk;r ÿ A, where Tr is themid-point of interval r. Ei;w

k;r is the anticipated penalty for early arrival. � and � are the parameters ofthe utility function, indicating the relative importance of travel time and schedule delay respectively.The �s are indicator variables and are de®ned as:

�i;w1;k;r � 1 if Di;wk;r > 0; 0 otherwise;

�i;w2;k;r � 1 if Ei;wk;r > 0; 0 otherwise:

The di�erence between our utility function and the existing ones is that we use mean perceivedtravel time instead of the travel time actually observed or calculated from the tra�c ¯ow model.This has two advantages: (1) using an individual's perceived travel time in the utility function ismore realistic; and (2) it overcomes the mathematical tractability issues (Vythoulkas, 1990)involved in the derivation of an analytical solution for departure rate using the complex rela-tionships involved in a demand model and a travel time model.

3.6. Theoretical properties of the model: a conjecture

A disaggregate perception updating model was proposed in this section in which drivers com-bine their experience with travel time provided by ATIS. The network tra�c ¯ow pattern is anaggregate result of driver's route and departure time choice behaviour which is based on theirperceptions of network performance. It is assumed that the underlying travel time has a truesteady mean and the information provided by ATIS on average is correct. Drivers systematicallycombine experience and/or information to update their perceptions. Thus, we do not model theevolution of the system itself. Nevertheless, the following conjecture regarding the evolution ofthe network tra�c ¯ow pattern can be made based on the proposed perception updating model:

It is expected that after a su�ciently large number of trips, drivers' mean perceived travel timeswill converge to the true steady mean. Since drivers are modeled as rational decision makingindividuals, the convergence of drivers' mean perceived travel time to a true value may suggestsome sort of steady state tra�c ¯ow pattern on the network.

The validity of the above conjecture is tested using simulation experiments. If the time dependent¯ow pattern for links in the network do not change from one day to the next then we concludethat the network is in a steady state.

4. Empirical study

This section describes an empirical study and discusses the implications from the results. Thenetwork used for the empirical study is shown in Fig. 4. This network has also been used beforefor studying day-to-day dynamics (Hu and Mahmassani, 1995). It consists of 50 nodes and 168


links and is divided into 10 zones. The analysis period is 35 minutes. The total number of driverssimulated in the network is 15044. Drivers' perceptions are initialized on ®ve shortest paths forthe initial desired departure interval.The departure pattern on the ®rst day is the same for all the experiments and is shown in Fig. 5.

The granularity of the simulator is ®xed as 6 seconds which is a compromise between desiredaccuracy and computational requirements. Departure time is descretized into two-minute inter-vals. The statistics reported in this paper are based on the vehicles generated between the ®ve

Fig. 4. The test network.


minute and the twenty minute simulation clock time. This was done in order to ensure that theresults are not in¯uenced by vehicles in the warm-up period or the end period of the simulation.The information provided in this study is based on the experienced travel time on the previousday. Drivers are provided with travel time information on ®ve shortest paths for ®ve departureintervals. These ®ve intervals consist of the departure interval that was selected on the previousday, two earlier and two later intervals. Thus, it is assumed that travelers do not shift theirdeparture intervals by more than two intervals with respect to their previous departure intervals.Predictive information was not used because of the unavailability of a reliable method to predicttravel times. Variances for information updating and experience updating are equal for allexperiments, implying that drivers assign equal importance to experience and information. Dri-vers are divided into two categories in terms of their preferred arrival times. One group of drivershave preferred arrival times as simulation clock 15 minute and the other group as simulationclock 25 min.

4.1. Simulation experiments

The empirical study focuses on evaluating the impact of various updating strategies used bydrivers and the parameters in the utility function.(1) Updating: Two cases of updating are considered: In Case 1, drivers do not use information

but they still use experienced travel time to systematically update their perceptions; Case 2assumes that drivers use both information and experience to update their perceptions.

Fig. 5. Departure pattern on the ®rst day.


(2) Utility function: Two sets of parameters for the utility functions are considered. Both arebased on previous studies. The ®rst set of parameters in the utility function is based on the cali-bration of Hendrickson and Plank's model (1984). Some of the terms used by Hendrickson andPlank are not relevant here. The modi®ed form of the utility function has three attributes: traveltime, earliness and lateness. The form of the utility function is described in Section 3.5. Thecoe�cients of travel time, earliness and lateness in the ®rst set are 0.021, 0.148 and 0.00042,respectively. The utility function with this set of parameters is called UT1. The second set ofcoe�cients is based on the survey carried out by Small (1982). Based on this survey, the secondset of coe�cients in the utility function for travel time, earliness and lateness are 6.4, 3.59 and1.52, respectively. We refer to the second utility function as UT2.

4.1.1. Simulation resultsIn the ®rst set of experiments drivers only use their experienced travel times to update their

perceptions. The average travel times, number of switchings and the evolution of tra�c patternfor UT1 for this set are presented in Figs 6±9. In our next set of experiments, the drivers use thesame utility function as in the previous set of experiments to make route and departure timedecisions but also use information travel time to update their perceptions. The average travel timefrom this scenario is also presented in Fig. 6 for comparison. Figs 10±13 presents the results fromthe scenario with UT2 and without information.

Fig. 6. Average travel time with UT1.


Fig. 7. Number of switchings with UT1 and without information.

Fig. 8. Evolution of departure pattern for Day 1±5 with UT1 and without information.



Fig. 10. Average travel time with UT2 and without information.


Fig. 11. Number of switchings with UT2 and without information.



4.1.2. Implications from the resultsImpact of historical information: Fig. 6 shows the impact of providing historical information on

average travel times. It is clear from this ®gure that historical information had practically noimpact on average travel time in the network. Also, it was found (Jha, 1996) that the historicalinformation does not have much impact on either network performance or tra�c pattern.Therefore, we do not present further results for those scenarios in which travelers use informationtravel time to update their perceptions. It should be noted that in no information case, eventhough drivers were not receiving information, they used their experience to update perceptions.In other words, under no information scenario, drivers updated their perceptions of one combi-nation of route and departure time, whereas in the other case, drivers updated their perceptionsfor all the combinations of route and departure times for which information was available. Theconclusion that can be drawn from this is that the marginal advantage of providing historicalinformation on more alternatives may not be signi®cant if drivers systematically learn from theirexperience.Average travel time: In all the cases, the average travel time increased with time and stabilized

after about 10 days. This is because the departure pattern on the ®rst day in all the cases (Fig. 5)was very spread out. Drivers' adjustments in their departure times, in order to avoid a largeschedule delay, gave rise to peaking, which in turn increased the average travel time in the system.Switching activities: In all the experiments, the number of route and departure time switchings

became stable after about ten days. The switchings pattern was also almost the same in all theexperiments. Departure time switchings decreased with time, implying that drivers tend to settleto a single travel choice after some time. Also, departure time switchings were consistently less



than the route switchings. This follows directly from our hypothesis on the hierarchy of travelchoice. However, the large number of switching is probably due to the fact that we are notimposing any constraint or penalty for switching.Evolution of departure pattern: Departure rates in all the cases stabilized after a few days. The

evolution of the departure pattern for UT1 is presented in Figs 8 and 9. The same trend wasfound for the second utility function. This suggests that there may be a unique steady statedeparture rate which is independent of the initial state of the network as well as the informationstrategy. Thus, the results support the assumption of some of the previous studies (Cascetta andCantarella, 1991; Ben-Akiva et al., 1986; Vythoulkas, 1990; Friesz et al., 1994).In order to further investigate this issue, the time varying tra�c pattern on speci®c links was

also examined. It was not practical to study the time varying tra�c pattern on each link in the testnetwork. Therefore, two links were selected for this purpose. These two links were link numbers142 and 156. The upstream and downstream nodes of link 142 are 38 and 37, respectively whereasupstream node of link 156 is 44 and downstream node is 43 (Fig. 4). The results, which are notshown here, clearly indicated (Jha, 1996; Jha et al., 1997) that the variations in the number ofvehicles entering a link (for a particular departure interval) from one day to the next were highfor the ®rst 10 days. After that, these variations gradually decreased. We did not perform a rig-orous convergence analysis of a possible steady-state. However, a visual inspection of the evolu-tion of time dependent link tra�c pattern indicated that there was almost no variation in thenumber of entering vehicles after Day 17. This implies that the time dependent tra�c pattern onthese two links reached a stable state after about 15 days. Although an appropriate convergencecriteria is necessary for a statistical analysis, the evolution of tra�c patterns on the two linksseems to substantiate the assumption of the existence of a steady-state.Impact of the utility function: Among all the factors studied in the empirical study, the para-

meters of the utility function were found to have the largest impact on network performance andtra�c pattern. Results presented in Fig. 6 through 9 are based on the ®rst utility function andthose in Figs 10±13 are based on the second one. Based on these ®gures, the following observa-tions can be made regarding the impacts of the coe�cients of the utility function:

1. Average travel times in all the cases using the second utility function were consistently andsigni®cantly lower than those using the ®rst utility function.

2. The number of switchings corresponding to the second utility function was higher than the®rst. In particular, the number of departure time switchings was signi®cantly higher for thesecond utility function. Since there was no penalty for switching, a large number of switch-ing was observed. This can be accounted for by including a penalty for switching in the uti-lity function.

3. The evolution of departure rates for both utility functions are similar. However, the peakingcharacteristic for the ®rst utility function was more pronounced. For example, compareFigs 8 and 9 with Figs 12 and 13, respectively. Note that Figs 8 and 9 indicate a higherdemand level during the peak time which is at ten minutes (simulation clock).

The ratio of the coe�cient for a late arrival to that for travel time in the ®rst utility functionwas almost equal to 7. In the second utility function this ratio was close to 0.55. This implies thataccording to the ®rst utility function, one unit of the delay causes a disutility which is equivalentto that caused by 7 units of travel time. Similarly, 0.55 unit of delay is equivalent to one unit of


travel time in the second utility function. Clearly, the relative penalty of a late arrival to traveltime is much higher in the ®rst utility function, implying that a driver would rather spend moretime traveling than being late. In the second case, the said ratio is lower and therefore drivers mayprefer a late arrival in order to avoid congestion. This variation in relative importance of scheduledelay (early arrival has a similar impact) and travel time led to a higher travel time when the ®rstutility function was used, because drivers did not tend to avoid congestion at the cost of beinglate (or early). The second observation can be explained similarly: when the penalty for beinglate or early (compared to travel time) was low, drivers switched their departure times androutes more often. The third observation indicates the di�erence in peaking characteristic. Sincethe relative penalty for a late arrival was higher in the ®rst utility function, a more pronouncedpeak was observed implying that drivers were more willing to spend extra travel time than to belate.

5. Summary and conclusions

A simulation framework for studying day-to-day dynamics was presented. Various compo-nents embedded in this framework were also discussed. In this paper, the emphasis was on mod-eling driver behaviour at the micro level and on capturing the learning process of drivers in a day-to-day dynamics scenario in the presence of tra�c information. This non equilibrium approachallowed us the ¯exibility to model drivers' decision making process in a less restrictive manner.Travel times (experienced and information) were represented by probability distribution func-tions in the model. Each driver has his/her own travel time distribution which indicates perceivedtravel times. This representation was crucial for modeling the variations in uncertainty associatedwith perceived travel times across drivers. The variance associated with the travel time functionsindicates drivers' con®dence in that source of information. Thus, drivers' con®dence in varioussources of information and their perceived quality of information are explicitly modeled in adynamic stochastic framework and are accounted for in our travel choice model. We believe thatusing perceived travel time as an attribute (as compared to measured travel time or that predictedby a tra�c ¯ow model) of the utility function is more realistic.Drivers' perception updating mechanism in the context of day-to-day dynamics in the presence

of advanced information system needs more attention. Many questions need to be answered inorder to predict travelers' response to ATIS. One of the key questions is how does the quality ofinformation change drivers' reliance on the information system. The proposed model assumesthat drivers' reliance on ATIS does not change with time; we are currently working on relaxingthis assumption.

Acknowledgements

The authors would like to thank Kumares Sinha for his contribution to the initiation of thisresearch. This research is a part of the doctoral dissertation of the ®rst author which was fundedby the Indiana Department of Transportation (INDOT) and the Federal Highway Administra-tion (FHWA).


Appendix A

In a conventional static stochastic assignment, a driver's perceived travel time is represented by

Tip � t� �; �A:1�

where t is the true travel time in the network and � represents perception error on the driver's partwhich is assumed to be normally distributed across drivers. This representation assumes a deter-ministic network in the sense that each driver's mean perceived travel time is equal to the truetravel time in the network. In the proposed framework, the mean perceived travel time t is mod-eled as a random variable whose distribution is updated from one day to the next using a Baye-sian approach. Drivers' travel choices are based on the distribution of their mean perceived traveltimes. The error term in Eq. (A.1) is not used in this study. It is clear from Eq. (A.1) that adecrease in the variance of mean perceived travel time does not necessarily imply a decrease in thevariance of overall perceived travel time given by Eq. (A.1).

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