The α-reliable mean-excess traffic equilibrium model with stochastic travel times

Transportation Research Part B 44 (2010) 493–513

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Transportation Research Part B

journal homepage: www.elsevier .com/ locate/ t rb

The a-reliable mean-excess traffic equilibrium model withstochastic travel times

Anthony Chen a,*, Zhong Zhou b

a Department of Civil and Environmental Engineering, Utah State University, Logan, UT 84322-4110, USAb Citilabs, 1040 Marina Village Parkway, Alameda, CA 94501, USA

a r t i c l e i n f o

Article history:Received 26 June 2008Received in revised form 20 November 2009Accepted 23 November 2009

Keywords:Travel time reliabilityTravel time budgetMean-excess travel timeUser equilibriumVariational inequality

0191-2615/$ - see front matter Published by Elseviedoi:10.1016/j.trb.2009.11.003

* Corresponding author. Tel.: +1 435 797 7109; faE-mail address: [email protected] (A. C

a b s t r a c t

In this paper, we propose a new model called the a-reliable mean-excess traffic equilib-rium (METE) model that explicitly considers both reliability and unreliability aspects oftravel time variability in the route choice decision process. In contrast to the travel timebudget (TTB) models that consider only the reliability aspect defined by TTB, this newmodel hypothesizes that travelers are willing to minimize their mean-excess travel times(METT) defined as the conditional expectation of travel times beyond the TTB. As a routechoice criterion, METT can be regarded as a combination of the buffer time measure thatensures the reliability aspect of on-time arrival at a confidence level a, and the tardy timemeasure that represents the unreliability aspect of encountering worst travel times beyondthe acceptable travel time allowed by TTB in the distribution tail of 1 � a. It addresses bothquestions of ‘‘how much time do I need to allow?” and ‘‘how bad should I expect from theworse cases?” Therefore, travelers’ route choice behavior can be considered in a more accu-rate and complete manner in a network equilibrium framework to reflect their risk prefer-ences under an uncertain environment. The METE model is formulated as a variationalinequality problem and solved by a route-based traffic assignment algorithm via theself-adaptive alternating direction method. Some qualitative properties of the model arerigorously proved. Illustrative examples are also presented to demonstrate the character-istics of the model as well as its differences compared to the recently proposed travel timebudget models.

Published by Elsevier Ltd.

1. Introduction

Uncertainty is unavoidable in real life. It surrounds all aspects of decision-making and affects our daily life as well as thesociety. In transportation, uncertainty is a critical and inseparable part of many problems. For example, the road network isone of the systems that serves the travel demands in order to connect people engaged in various activities (e.g., work, trav-eling, shopping, etc.) at different locations. The uncertainty of network travel times exists in both supply side (roadwaycapacity variation) and demand side (travel demand fluctuation). Fig. 1 provides an illustration of various sources of uncer-tainty that contribute to travel time variability. From the figure, we can observe that several exogenous sources of uncer-tainty exist in the supply side. Weather conditions refer to environmental conditions that can lead to changes in travelerbehavior. For example, travelers may lower their speeds or increase their headways (spacing between vehicles) due to re-duced visibility when fog, rain or snow is present. Traffic incidents, such as car crashes, breakdowns or debris in lanes, oftendisrupt the normal flow of traffic. Work zones are construction activities on the roadways that usually introduce physical

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Supply Side Uncertainty

Demand Side Uncertainty

Traffic Incidents

Work Zones

Weather Conditions

Demand Fluctuation

Traffic Management and Control

Special Events

Traffic Information

Non-recurrent Congestion

Travel Time Variability

Recurrent Congestion

Capacity Variation

Temporal Factors(Time of day, Day of

week, etc.)

Population Characteristics

Fig. 1. Selected sources of uncertainty inducing travel time variability (modified from van Lint et al., 2008).

494 A. Chen, Z. Zhou / Transportation Research Part B 44 (2010) 493–513

changes to the highway environment. The number or width of lanes may be changed, shoulders may be eliminated, or road-ways may be temporarily closed. Delays caused by work zones have been regarded as one of the most frustrating conditionsthat travelers encounter on their trips. Traffic control devices, such as signal timing and ramp metering, also contribute totravel time variability. The uncertainty introduced by these supply-side sources can be referred to as stochastic link capacityvariations, and typically lead to non-recurrent congestion (Chen et al., 2002b; Lo et al., 2006; Al-Deek and Emam, 2006). Onthe other hand, there are several sources of uncertainty exist in the demand side. Travel demand fluctuations can be intro-duced by temporal factors, such as time of day, day of week or seasonal effects. Special events are a special case of traveldemand fluctuations, where the traffic flow is significantly different from the typical pattern in the vicinity of the event. Pop-ulation characteristics, such as age, car ownership, and household income, also affect the propensity of travel demand. Trafficinformation provided by Advanced Traveler Information Systems (ATIS) can also influence the travelers’ trip decision, includ-ing their departure time, destination, mode, and route choice, which consequently affect the traffic flow pattern. These de-mand variations usually lead to recurrent congestion (Asakura and Kashiwadani, 1991; Clark and Watling, 2005). There arealso complex interactions between the supply-side and demand-side sources of uncertainty. For example, bad weather mayreduce roadway capacity in the network, and may at the same time change the spatial and temporal pattern of travel de-mand, because travelers may decide to change their departure time, choose a different route, or even cancel the trip. In short,these uncertain events result in the variation of traffic flow, which directly contributes to the spatial and temporal variabilityof network travel times. Such travel time variability introduces uncertainty for travelers such that they do not know exactlywhen they will arrive at the destination. Thus, it is considered as a risk to a traveler making a trip.

The effects of travel time variability on travelers’ route choice behaviors have been studied by several empirical surveys(Abdel-Aty et al., 1995; Small et al., 1999; Lam, 2000; Brownstone et al., 2003; Cambridge Systematics, 2003; Recker et al.,2005). Abdel-Aty et al. (1995) found that travel time reliability was either the most or second most important factor for mostcommuters. In the study by Small et al. (1999), they found that both individual travelers and freight carriers were stronglyaverse to scheduling mismatches. For this reason, they were willing to pay a premium to avoid congestion and to achievegreater reliability in travel times. From the two value-pricing projects in Southern California, Lam (2000) and Brownstoneet al. (2003) also consistently found that travelers were willing to pay a substantial amount to reduce variability in traveltime. Another study conducted by Recker et al. (2005) on the freeway system in Orange County, California observed that:(i) both travel time and travel time variability were higher in peak hours than non-peak hours; (ii) both travel time and traveltime variability were much higher in winter months than in other seasons; and (iii) travel time and travel time variabilitywere highly correlated. According to these observations, they suggested that commuters preferred departing earlier to avoidthe possible delays caused by travel time variability. These empirical studies revealed that travelers considered travel timevariability as a risk in their route choice decisions. They are interested in not only travel time saving but also the travel timevariability reduction to minimize risk. Thus, it is suffice to say travel time variability is a significant factor for travelers when

A. Chen, Z. Zhou / Transportation Research Part B 44 (2010) 493–513 495

making their route choice decisions under risk or circumstances where they do not know with certainty about the outcomeof their decisions. Furthermore, a recent empirical study conducted by van Lint et al. (2008) reveals that the travel time dis-tribution is not only very wide but also heavily skewed with a long fat tail. For example, it has been shown that about 5% ofthe ‘‘unlucky drivers” incur almost five times as much delay as the 50% of the ‘‘fortunate drivers” on the densely used freewaycorridors in the Netherlands. As for freight, the cost of unexpected delay for trucks is another 50–250% higher than the ex-pected delay (FHWA, 2006). These empirical studies show that travel time distribution is not only heavily skewed with a longfat tail, but also the consequences of these late trips may be much more serious than those of modest delays, and can have asignificant impact on both travelers and freight shippers and carriers. Another recent study related to the mean-excess traveltime is the modeling of the ‘‘mean lateness” factor used in benefit-cost analysis for the Stockholm roads by Franklin and Karl-strom (2009). They found that only a certain portion of the right-hand tail of the standardized travel time distribution needsto be same in order to estimate the ‘‘mean lateness” factor, which can be adopted into the departure time choice model ofFosgerau and Karlström (2010) by maximizing the utility function consisting of departure time, travel time, and expectedexcess delays beyond the traveler’s desired arrival time to determine the optimal departure time.

However, travel time variability is not considered in the traditional user equilibrium (UE) model, where travelers are allassumed to be risk-neutral and the route choice decisions are based solely on the expected travel time. To model the routechoice behavior under stochastic travel times, some of the previous works are briefly described as follows: One stream ofequilibrium-based models adopted the concept of utility (Emmerink et al., 1995; Mirchandani and Soroush, 1987; Noland,1999; Noland et al., 1998; van Berkum and van der Mede, 1999; Yin and Ieda, 2001; Chen et al., 2002a), where the disutilityfunction was constructed using a combination of attributes (e.g., expected travel time, travel time variance or standard devi-ation, late arrival penalty, etc.), and the network equilibrium conditions are similar to those of the traditional UE model ex-cept for the disutility function is replaced by the expected travel time. That is, travelers attempt to make a tradeoff betweenthe expected travel time and its uncertainty. Recently, Watling (2006) proposed a late arrival penalized user equilibrium(LAPUE) model based on the concept of schedule delay. The new disutility function consists of the expected generalized tra-vel cost plus a schedule delay term to penalize the late arrival under fixed departure times.

A game theory based approach to model travel time variation was proposed by Bell (2000). It assumes that the travelersare highly pessimistic about the travel time variability and behave in a very risk-averse manner. Based on this approach, Belland Cassir (2002) proposed a risk-averse traffic equilibrium model as a non-cooperative, mixed-strategy game to model thetravelers’ route choice process. In this game, travelers seek the best routes to avoid link failures, while the demons selectlinks to cause the maximum damage to the travelers. Szeto et al. (2006) further extended this approach to include elasticdemand and provided a nonlinear complementarity problem (NCP) formulation.

Uchida and Iida (1993) defined travel time variation as a risk and proposed two risk assignment models. Both models areestablished using the concept of effective travel time, which is defined as the mean travel time plus a safety margin. Lam andChan (2005) argued that travelers should consider both travel time and travel time reliability for their route choices. Thus, apath preference index (PI) that combines the path travel time index (TI) and the path travel time reliability index (RI) is pro-posed to be the criterion of route choices. Lo and Tung (2003) proposed a probabilistic user equilibrium (PUE) model, wheretravel time variability is introduced by day-to-day stochastic link degradation with predefined link capacity distributions. Loet al. (2006) extended this approach by incorporating the concept of travel time budget (TTB) to develop the within budgettime reliability (WBTR) model. Different from Uchida and Iida’s (1993) definition, TTB is defined by a travel time reliabilitychance constraint, such that the probability that travel time exceeds the budget is less than a predefined confidence level aspecified by the traveler to represent his/her risk preference. This definition is similar to that defined by Chen and Ji (2005),where the route with the minimum TTB is termed ‘‘a-reliable route”. Therefore, each commuter learns the travel time vari-ations through his/her daily commutes and chooses a route that minimizes his/her TTB. Later, Siu and Lo (2006) extended theWBTR model to take traveler’s perception variation into consideration. By assuming travel time variation is induced by dailytravel demand variation instead of capacity degradation, Shao et al. (2006a) extended the PUE model and proposed a demanddriven travel time reliability-based user equilibrium (DRUE) model. The DRUE model was further extended to the reliability-based stochastic user equilibrium (RSUE) model to account for both perception error and multiple user classes (Shao et al.,2006b). The core idea behind the travelers’ route choice behavior of the above models (Lo et al., 2006; Shao et al., 2006a; Siuand Lo, 2006) is based on the concept of TTB, which is defined as the average travel time plus an extra buffer time as anacceptable travel time, such that the probability of completing the trip within the TTB is no less than a predefined reliabilitythreshold (or a confidence level a). In fact, the concept of TTB is analogous to the Value-at-Risk (VaR), which is by far themost widely applied risk measure in the finance area (Szego, 2005). However, it has been determined that VaR is not evena weakly coherent measure of risk (Artzner et al., 1997, 1999). Models using VaR is unable to deal with the possibility thatthe losses associated with the worst scenarios are excessively higher than the VaR, and reduction of VaR may lead to stretchof tail exceeding VaR (Larsen et al., 2002; Yamai and Yoshiba, 2001). In the same spirit, TTB may also be an inadequate riskmeasure, which could introduce overwhelmingly high trip times (i.e., unreliability aspect of travel time variability) to trav-elers if it is used solely as a route choice criterion in the network equilibrium based approach.

Furthermore, to describe travelers’ route choice decision process under travel time variability, considering only the reli-ability aspect (i.e., buffer time or acceptable travel time defined by TTB) may not be adequate to describe travelers’ risk pref-erences. On the one hand, the reports issued by the Federal Highway Administration (FHWA, 2004, 2006) documented thattravelers, especially commuters, do add a ‘buffer time’ (or safety margin) to their expected travel time to ensure morefrequent on-time arrivals when planning a trip. It represents the reliability aspect in the travelers’ route choice decision


process. On the other hand, the impacts of late arrival and its explicit link to the travelers’ preferred arrival time were alsoexamined in the literature (e.g., Noland et al., 1998; Noland, 1999; Hall, 1993; Porter et al., 1996) and appreciated as the‘value’ of unreliability (Watling, 2006). It represents travelers’ concern of the unreliability aspect of travel time variabilityin their route choice decision process, where trip times longer than they expected would be considered as ‘unreliable’ or‘unacceptable’ (Cambridge Systematics, 2003). Therefore, it is reasonable to incorporate both reliability and unreliability as-pects of travel time variability into the network equilibrium model to describe travelers’ route choice behavior under sto-chastic travel times.

In this paper, we present a new a-reliable mean-excess traffic equilibrium model or mean-excess traffic equilibrium(METE) model for short that explicitly considers both reliability and unreliability aspects of travel time variability in the routechoice decision process. In contrast to the TTB models (Lo et al., 2006; Shao et al., 2006a; Siu and Lo, 2006) that consider onlythe reliability aspect defined by TTB, this new model hypothesizes that travelers are willing to minimize their mean-excesstravel time (METT) defined as the conditional expectation of travel times beyond the TTB. As a route choice criterion, METT canbe regarded as a combination of the buffer time measure that ensures the reliability of on-time arrival, and the tardy timemeasure that represents the unreliability impacts of excessively late trips (Cambridge Systematics, 2003). It incorporates bothreliability and unreliability aspects of travel time variability to simultaneously address both questions of ‘‘how much time do Ineed to allow?” and ‘‘how bad should I expect from the worse cases?” Therefore, travelers’ route choice behavior can be consid-ered in a more accurate and complete manner in a network equilibrium framework to reflect their risk preferences under anuncertain environment. Furthermore, the definition of METT is consistent with the conditional Value-at-Risk (CVaR) measurein the literature (Rockafellar and Uryasev, 2000, 2002) for risk optimization. CVaR has been proofed as a more consistent mea-sure of risk since it is sub-additive and convex (Artzner et al., 1999). Recently, Pflug (2000) further proved that CVaR is a coher-ent risk measure. Therefore, CVaR has been recognized as a good alternative measure of risk, and it is suitable for modelingflexible travel time distributions, including non-symmetric and heavy tailed distributions. Thus, METT is indeed a reasonableand meaningful alternative to the TTB, which not only has nice mathematical properties, but also allows for a more completeevaluation of the travel time variability in the route choice decision process. In addition, CVaR has been applied in variousfields, such as portfolio optimization (Rockafellar and Uryasev, 2002), facility location (Chen et al., 2006) and fleet allocation(Yin, 2008), albeit not in the context of traffic equilibrium problem under uncertainty.

The remainder of the paper is organized as follows: In Section 2, the concept of mean-excess travel time in a stochasticnetwork is introduced and the mean-excess traffic equilibrium model is proposed, where the model is formulated as a var-iational inequality (VI) problem. Some properties and analytical results of the model and formulation are also presented. InSection 3, a route-based traffic assignment algorithm based on the self-adaptive alternating direction method is developed todetermine the equilibrium flow pattern. In Section 4, numerical examples are presented to illustrate the features of the pro-posed model, to compare with other related user equilibrium models, and to demonstrate the applicability of the solutionprocedure. Finally, conclusions and recommendations for future research are given in Section 5.

2. The mean-excess traffic equilibrium model

This section describes the METE model for determining the equilibrium flow pattern under stochastic travel times.Notation is provided first for convenience, followed by the definition of mean-excess travel time, an illustrative exampleto highlight the differences of using the expected travel time, the TTB, and the METT as a route choice criterion, the varia-tional inequality formulation for the METE model and its qualitative properties, and the derivation of travel time distributionunder different sources of uncertainty.

2.1. Notation

GðN;AÞ a stochastic network composed by nodes and links
N set of nodes A set of links R set of origins S set of destinations Prs set of routes between origin r and destination s a link index r origin index s destination index p route index a reliability threshold (or confidence level) that represents the traveler’s risk preference Trs
p
random travel time on route p between origin r and destination s
EðTrsp Þ
expected travel time on route p between origin r and destination s
crsp ðaÞ
buffer time on route p between origin r and destination s required to ensure on-time arrival at a confidence
level a


nrsp ðaÞ
travel time budget (i.e., expected travel time + buffer time) on route p between origin r and destination s
required to ensure on-time arrival at a confidence level a
grs
p
mean-excess travel time on route p between origin r and destination s g vector of mean-excess route travel times ð. . . ;grs
p ; . . . ÞT

prs
minimal mean-excess travel time between origin r and destination s Ta random travel time on link a �ta ¼ E½Ta� mean travel time on link a
r2a ¼ Var½Ta�
variance of travel time on link a
Qrs
random O–D demand between origin r and destination s qrs mean O–D demand between origin r and destination s q vector of mean O–D demands ð. . . ; qrs; . . . ÞT
f rsp
flow on route p between origin r and destination s
f
vector of route flows ð. . . ; f rsp ; . . . ÞT
K
OD-route incidence matrix va flow on link a D ¼ ½drs
pa�
route–link incidence matrix, where drspa ¼ 1 if route p from origin r to destination s uses link a, and 0 otherwise
2.2. Definition of mean-excess travel time

In this study, travelers are assumed to have knowledge of the distribution of travel time variability. In practice, this kindof knowledge can be acquired from various sources, such as their past commuting experiences or Advanced Traveler Infor-mation Systems (ATIS). Travelers then incorporate this information into their route choice decisions along with their ownrisk-preferences to reach a long-term habitual equilibrium flow pattern. Therefore, to study the user equilibrium problem,the key factor here is to understand the travelers’ route choice behavior under travel time variability. In this section, we willillustrate the concept of METT (i.e., analogous to the Conditional Value-at-Risk (CVaR) measure used in financial engineering)adopted as a route choice criterion in a traffic equilibrium framework and demonstrate its differences with the travel timebudget (TTB) criterion adopted in many recent developed traffic equilibrium models to consider the reliability aspect of tra-vel time variability in the route choice decision process.

As mentioned above, travelers are unable to accurately estimate the travel time from their origin to destination due totravel time variability. This is considered as a risk associated with their route choice decisions. According to Mirchandaniand Soroush (1987), travelers making route choice decisions under an uncertain environment can be categorized into threegroups according to their attitudes toward risk (e.g., risk-prone, risk-neutral and risk-averse). In the traditional UE model,travelers are assumed to be risk-neutral since they make their route choice decisions solely based on the expected traveltime EðTrs

p Þ, where Trsp is the random travel time on route p between origin r and destination s. However, recent empirical

studies (Small et al., 1999; Lam, 2000; Brownstone et al., 2003; Liu et al., 2004; de Palma and Picard, 2005; Cambridge Sys-tematics, 2003; FHWA, 2004, 2006) revealed that most travelers are actually risk-averse. They are willing to pay a premiumto avoid congestion and minimize the associated risk.

By considering the travel time reliability requirement, travelers are searching for a route such that the corresponding TTBallows for on-time arrival with a predefined confidence level a (Lo et al., 2006; Siu and Lo, 2006; Shao et al., 2006a). Mean-while, they are also considering the impacts of excessively late arrival (i.e., the unreliable aspect of travel time variability)and its explicit link to the travelers’ preferred arrival time in the route choice decision process (Noland et al., 1998; Noland,1999; Watling, 2006). Therefore, it is reasonable for travelers to choose a route such that the travel time reliability (i.e.,acceptable travel time defined by TTB) is ensured most of the time and the expected unreliability impact (i.e., unacceptabletravel time exceeding TTB) is minimized. This tradeoff between the reliable and unreliable aspects of travel time variabilityin travelers’ route choice decision process can be represented by the mean-excess travel time (METT) defined as follows.

Definition 1. The mean-excess travel time grsp ðaÞ for a route p 2 Prs between origin r to destination s with a predefined

confidence level a is equal to the conditional expectation of the travel time exceeding the corresponding route TTB nrsp ðaÞ, i.e.,

h i
grs
p ðaÞ ¼ E Trsp jT

rsp P nrs

p ðaÞ ; 8p 2 Prs; r 2 R; s 2 S; ð1Þ

where Trsp is the random travel time on route p from origin r to destination s, E[�] is the expectation operator, and nrs

p ðaÞ is theTTB on route p from origin r to destination s defined by the travel time reliability chance constraint at a confidence level a inEq. (2) (Chen and Ji, 2005; Lo et al., 2006):

nrsp ðaÞ ¼minfnjPrðTrs

p 6 nÞP ag; ð2Þ¼ EðTrs

p Þ þ crsp ðaÞ; 8p 2 Prs; r 2 R; s 2 S; ð3Þ

where n is the travel time budget (a threshold value in the probabilistic constraint) required to ensure on-time arrival at aconfidence level a, and crs

p ðaÞ is the extra time added to the mean travel time as a ‘buffer time’ to ensure more frequent


on-time arrivals at the destination under the travel time reliability requirement at a confidence level a. It should be notedthat Eq. (3) is exactly the definition of the TTB used in the a-reliable route finding model of Chen and Ji (2005), the DRUEmodel of Shao et al. (2006a), the RSUE model of Shao et al. (2006b), and the WBTR model of Lo et al. (2006).

If the route travel time distribution f ðTrsp Þ is known, the METT definition given in Eq. (1) as a conditional expectation can

be rewritten as:

grsp ðaÞ ¼ E Trs

p jTrsp P nrs

p ðaÞh i

¼Z

Trsp Pnrs

p ðaÞTrs

p

f ðTrsp Þ

Pr½Trsp P nrs

p �dðTrs

p Þ ¼1

1� a

ZTrs

p Pnrsp ðaÞ

Trsp f ðTrs

p ÞdðTrsp Þ; ð4Þ

which is the expected value of a random variable with respect to a conditional probability distribution. Moreover, Eq. (1) canbe restated as:

grsp ðaÞ ¼ nrs

p ðaÞ þ E Trsp � nrs

p ðaÞjTrsp P nrs

p ðaÞh i

: ð5Þ

Therefore, the METT can be decomposed into two individual components. The first component is exactly the TTB of routep, which reflects the reliability aspect of travel time variability (or acceptable risk) allowed by the travelers at a confidencelevel a. The second component is the expected value of the late trips with respect to the TTB, which can be regarded as a kindof ‘‘expected excess delay” or ‘‘conditional expected regret” for choosing the current route, to reflect the unreliable aspect oftravel time variability (or unacceptable risk with trip times exceeding the acceptable travel time defined by TTB) in the dis-tribution tail of 1 � a.

In the literature, the expected excess delay has been adopted to reflect travelers’ utility under travel time uncertainty. Fos-gerau and Karlström (2010) proposed a departure time choice model based on the schedule delay model of Noland and Small(1995) by maximizing the expected utility, which includes the expected excess delay beyond the desired arrival time as onecomponent of the utility function. Watling (2006) proposed a late arrival penalised user equilibrium model (LAPUE), whereroute choice criterion was assumed to be the route disutility that consists of the generalized travel time and the penalizedexpected excess delay. Note that the expected arrival time used in these two models for computing the expected excess delaycan be regarded as the travel time budget. However, in our model, the travel time budget is endogenously computed accord-ing to traveler’s confidence level of on-time arrival, while it was determined by utility maximization in Fosgerau and Kar-lström (2010) and adopted as a predefined constant in Watling (2006). In our model, both travel time budget andexpected excess delay (i.e., both reliability and unreliability of travel time variability) are derived under a unified frameworkbased on travelers’ confidence level. It explicitly considers the effects of mean travel time, buffer time, and expected excessdelay (see Eqs. (3) and (5)), while the other two models only consider the effects of mean travel time and expected excessdelay. Clearly, as a new route choice decision criterion, the METT incorporates both reliable and unreliable factors into theroute choice decision process. The METT simultaneously addresses both questions of ‘‘how much time do I need to allow?” and‘‘how bad should I expect from the worse cases?” Both questions relate particularly well to the way travelers make decisions(Cambridge Systematics, 2003).

To illustrate the definition of METT and its relation to TTB, a hypothetical route travel time distribution shown in Fig. 2 isadopted. The dotted line represents the probability distribution function (PDF), while the solid line represents the cumula-tive distribution function (CDF). Given a confidence level a, the TTB is the minimum travel time threshold (i.e., mean traveltime + safety margin) allowed by the travelers such that the corresponding cumulative probability of actual travel time lessthan this threshold is at least a. The shaded area (i.e., tail) represents all possible worse situations (late trips) that the actual

0 5 10 15 20 25 30

P r

o b

a b

i l i

t y

D e

n s

i t

y

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

R o u t e T r a v e l T i m e

T r

a v

e l

T i

m e

R e

l i a

b i

l i t

y

C u m u l a t i v e P r o b a b i l i t y

P r o b a b i l i t y D e n s i t y

C o n f i d e n c e L e v e l

T r a v e l T i m e B u d g e t

M e a n E x c e s s T r a v e l T i m e

Fig. 2. Illustration of the travel time budget and mean-excess travel time.


travel time is higher than the TTB, and the METT is the conditional expectation of the late trips. Clearly, from the figure, wecan observe that the TTB does not assess the magnitude of the possible travel times associated with the worse situations andis unable to distinguish the situations where the actual travel times are only a little bit higher than the TTB from those inwhich the actual travel times are extremely higher. In other words, it does not address the question that concerns withthe unreliability aspect, such as ‘‘how bad should I expect from the worse case?”. Therefore, the TTB only ensures the reliabilityaspect of on-time arrival for a given confidence level a, while the METT accounts for both the reliability aspect (i.e., TTB re-quired to ensure on-time arrival at a confidence level a) and the unreliability aspect of travel time variability (i.e., encoun-tering worse travel times beyond the TTB in the right tail).

2.3. Illustrative example

The following illustrative example shows the differences of using the expected travel time, TTB, and METT as the routechoice criterion. A small hypothetical network with three parallel routes connecting origin r and destination s is adoptedin this demonstration (see Fig. 3). In this example, all travelers are assumed to have a confidence level of a = 90%. In orderto facilitate the presentation of the essential ideas, the travel time distributions of the three routes are assumed to follow alog-normal distribution Lognðl;rÞ, whose PDF is shown as below:

f ðnjl;rÞ ¼ 1nr

ffiffiffiffiffiffiffi2pp exp

�ðln n� lÞ2

2r2

!; 8n > 0: ð6Þ

The log-normal distribution is closely related to the normal distribution and has been commonly adopted in practice tomodel a broad range of random processes. The parameters lp and rp of the log-normal distribution Lognðl;rÞ for each routep (p = 1, 2, 3) are shown in Fig. 3.

In this simple network where the route is equal to the link, the solution for the expected travel time criterion can be de-rived analytically as follows:

pp ¼ EðTpÞ ¼ expðlp þ r2p=2Þ; ð7Þ

where Tp represents the random travel time of route p (p = 1, 2, 3).If the travelers are concerned with travel time reliability and want to minimize their corresponding TTB at the same time,

the following minimization problem should be considered (Chen and Ji, 2005):

mincp

np ¼ EðTpÞ þ cp; ð8Þ

s:t: PrðTp 6 npÞP 90%: ð9Þ

Under the assumption of the log-normal distributed route travel time, the TTB for each route can be analytically com-puted (Aitchison and Brown, 1957) as follows:

np ¼ expffiffiffi2p

rperf�1ð2a� 1Þ þ lp

� �; ð10Þ

where erf�1ð�Þ is the inverse of the Gauss error function defined as:

erf ðxÞ ¼ 2ffiffiffiffipp

Z x

0expð�t2Þdt: ð11Þ

Now, suppose travelers are concerned with not only the reliability of on-time arrival, but also the impact of late arrival(i.e., actual travel times are higher than the TTB). Then, it is meaningful for them to minimize the METT, while still ensuringthe reliability requirement. Following the METT definition in Eq. (1), we consider the following minimization problem:

min gp ¼ E½TpjTp P np�; ð12Þs:t: PrðTp 6 npÞP 90%: ð13Þ

r s

Path 1: Logn(2.28,0.20)

Path 2: Logn(2.00, 0.39)

Path 3: Logn(1.77, 0.59)

Origin Destination

Fig. 3. Hypothetical network used to illustrate the different route choice criteria.


Under the assumption of log-normal distribution, the minimization problem (12) and (13) can be rewritten as:

min1

ð1� 0:9Þ

Z 1

np

Tp �1ffiffiffiffiffiffiffi

2pp

� Tprp

exp�ðln Tp � lpÞ

2

2r2p

!dTp; ð14Þ

s:t: PrðTp 6 npÞP 90%: ð15Þ

By performing some calculus manipulations on the minimization problem (14) and (15), the METT of route p can be ana-lytically derived as follows:

gp ¼ np þ1

1� a

Z 1

0½Tp � np�þ �

1ffiffiffiffiffiffiffi2pp

� Tprp

exp�ðln Tp � lpÞ

2

2r2p

!dTp

¼ expðlp þ r2p=2Þ �Uð�

ffiffiffi2p� erf�1ð2a� 1Þ þ rpÞ=ð1� aÞ; ð16Þ

where Uð�Þ is the standard normal CDF, ½a�þ ¼ a if a > 0, and ½a�þ ¼ 0 otherwise.The analytically derived results for all three routes are shown in Fig. 4, where the x-axis represents the different route

choice criteria and the y-axis represents the corresponding measurement value (in minutes) for a given criterion. For sim-plicity, the mean travel time, travel time budget and mean-excess travel time are abbreviated by MTT, TTB and METT,respectively.

From Fig. 4, it is clear that different route choice criteria provide different optimal routes, which reflect various risk pref-erences and considerations of the travelers toward travel time variability. If the travelers are all risk-neutral, they only con-sider the expected route travel time during their route choice decisions (i.e., the traditional UE model). Therefore, based onthe first group of bars in the figure, they should choose route 3, which has the minimum expected travel time of 7. However,to ensure a 90% confidence level of on-time arrival, choosing route 3 will no longer be the optimal decision. If the travelersare risk-averse and concern more about the travel time reliability, they may choose a route that gives them the minimumTTB. This can be observed in Fig. 5, where the TTB for each route is represented by the summation of two parts, i.e., meantravel time plus an extra buffer time. This is consistent with the definition of TTB given in Eq. (3), where the buffer timedescribes the impacts from the reliability aspect of travel time variability on travelers’ route choice decisions. From thefigure, we can see that to ensure a 90% travel time reliability, route 3 has to add the largest buffer time (5.51 min), whichmakes its TTB higher than that of route 2. Therefore, by considering the impacts of the travel time reliability aspect, travelerswill prefer route 2 due to its lowest TTB.

However, as discussed in the previous section, TTB is unable to account for the magnitude of the worse travel times in thedistribution tail. That is, it is unable to assess the impacts related to the unreliable situations where the actual travel timesmay be overwhelmingly higher than the TTB. Therefore, routes with a lower TTB may have a higher METT, and travelers whochoose these routes will have a 10% probability of encountering trip times much greater than the allowable TTB. This can bedemonstrated more clearly in Fig. 6, where the METT for each route is represented by the summation of three parts,

MTT

TTB

METT

0

4

8

12

16

18

Min

utes

Ro u t e 1 R o u t e 2R o u t e 3

10

7

8

13.9314.80

17.08

12.1712.51

12.64

Fig. 4. Analytical results for all three routes with different route choice criteria.

0

5

10

15

Route 1

Route 2

Route 3

Buffer Time+

MTT=

TTB

M i n u t e s

5.51

7

4.17

812.51

12.17

12.64

10

2.64

Fig. 5. Analysis of the impacts of travel time reliability.

0

5

10

15

Route 1

Route 2

Route 3=

+

+Buffer Time

Expected ExcessDelay

MTT

METT

4.57

5.51

72.63

4.17

8

1.292.64

10

17.08

14.80

13.93

M i n u t e s

Fig. 6. Analysis of the impacts of travel time unreliability.


i.e., mean travel time, buffer time and ‘‘expected excess delay”. This representation is consistent with the definition of METT bycombining Eqs. (3) and (5), where the ‘‘expected excess delay” describes the impacts from the unreliability aspect of traveltime variability on travelers’ route choice decisions, and the summation of the first two components (mean travel timeand buffer time) gives the TTB that represents the travel time reliability requirement specified by the travelers.

From the figure, we can observe that travelers on route 2 or route 3 may experience a higher expected excess delay thanthose on route 1 when the 10% worse cases happened, even though they have a lower TTB. These worse cases may be due tovarious sources, such as severe incidents, bad weather conditions and special events. Therefore, for travelers who are con-cerned with not only the travel time reliability, but also the unreliability of encountering worse travel times, they may preferto choose route 1, which has the lowest METT. Though, by doing that, the corresponding TTB is not the minimum, the expec-tation of the unacceptable travel times greater than the allowable TTB is significantly reduced compared to other choices. Inaddition, they can still enjoy at least a 90% reliability of punctual arrival. Note that the METT is always higher than the cor-responding TTB. This implies that the METT criterion can be regarded as a more conservative measure of risk, which reflectsthe behavior of travelers who have a more negative attitude toward schedule delay for late arrivals and a higher degree ofrisk-averseness in hedging against travel time variability.


2.4. Equilibrium conditions and variational inequality formulation

Consider a strongly connected transportation network [N, A], where N and A denote the sets of nodes and links, respec-tively. Let R and S denote a subset of N for which travel demand qrs is generated from origin r 2 R to destination s 2 S, and letf rsp denote the flow on route p 2 Prs; where Prs is a set of routes from origin r to destination s. We assume that the link travel

time is stochastic, which is represented by a random vector T ¼ fTag, where Ta represents the random travel time on linka 2 A.

Let D ¼ ½drspa� denote the route–link incidence matrix, where drs

pa ¼ 1 if route p from origin r to destination s uses link a, and0, otherwise. Then, the feasible flow set X can be described as below:

qrs ¼Xp2Prs

f rsp ; 8r 2 R; s 2 S; ð17Þ

va ¼Xr2R

Xs2S

Xp2Prs

f rsp drs

pa; 8a 2 A; ð18Þ

f rsp P 0; 8p 2 Prs; r 2 R; s 2 S; ð19Þ

where (17) is the travel demand conservation constraint, (18) is a definitional constraint that sums up all route flows thatpass through a given link a, and (19) is a non-negativity constraint on the route flows.

As discussed in the previous section, it is reasonable to assume that travelers are willing to minimize their METT whentraveling from an origin to a destination under an uncertain environment. Consequently, a long-term habitual traffic equi-librium can be reached. It is termed the mean-excess traffic equilibrium (METE) model. Let g denote the METT vectorð. . . ;grs

p ; . . . ÞT , prs denote the minimal METT between O–D pair (r, s), and f denote the route-flow vector ð. . . ; f rsp ; . . . ÞT . The

conditions of METE state can be characterized as follows.

Definition 2. The METE state is reached by allocating the O–D demands to the network such that no traveler can improve his/her METT by unilaterally changing routes. In other words, all used routes between each O–D pair have equal METT, and no unusedroute has a lower METT, i.e. the following conditions hold:

grsp ðf �Þ � prs

¼ 0 if ðf rsp Þ�> 0

P 0 if ðf rsp Þ� ¼ 0

(; 8p 2 Prs; r 2 R; s 2 S: ð20Þ

Such an equilibrium state is what results if each and every traveler simultaneously attempts to minimize his/her METT.Then the METE model can be formulated as a variational inequality problem VI(f, X) as follows.

Find a vector f � 2 X, such that

gðf �ÞTðf � f �ÞP 0; 8f 2 X: ð21Þ

The following two Propositions give the equivalence of the VI formulation and the METE model as well as the existence ofan equilibrium solution.

Proposition 1. Assume the mean-excess route travel time function gðf Þ is positive, the solution of the VI problem (21) is equivalentto the equilibrium solution of the METE model.

Proof. Note that f � is a solution of the VI problem (21) if and only if it is a solution of the following linear program:

minf2X

gðf �ÞT f ð22Þ

By considering the primal-dual optimality conditions of (22), we have

f rs�p � grs

p ðf �Þ � prs� �

¼ 0; 8p 2 Prs; r 2 R; s 2 S; ð23Þ

grsp ðf �Þ � prs P 0; 8p 2 Prs; r 2 R; s 2 S; ð24Þ

and Eq. (19). It is easy to see the METE condition (20) is satisfied. This completes the proof. h

Proposition 2. Assume the mean-excess route travel time function gðf Þ is positive and continuous, the METE model has at leastone solution.

Proof. According to Proposition 1, we only need to consider the equivalent VI formulation. Note that the feasible set X isnonempty and convex. Furthermore, the mapping gðf Þ is continuous according to the assumption. Thus, the VI problem(21) has at least one solution (e.g., see Nagurney, 1993). This completes the proof. h


Note that, in this study, the travelers are assumed to be risk-averse and concerned with both reliability and unreliabilityaspects of travel time variability, where the link travel time distribution is assumed to have a continuous CDF by fitting thereal surveillance data. Consider the link/route relationship

Trsp ¼

Xa2A

Tadrsap; 8p 2 Prs; r 2 R; s 2 S; ð25Þ

and the definition of METT in Eq. (1), it is reasonable to give the positive and continuous assumption of the function gðf Þ as inthe above propositions. Consequently, the validity of the VI formulation and the existence of the solution are ensured.

2.5. Stochastic travel times under different sources of uncertainty

The METE model and its VI formulation proposed above are ‘generic’ in the sense that the link/route travel time variabilityis characterized by a known PDF. In practice, the travel time variability could come from various sources of uncertainty asdescribed in Fig. 1. This section reviews the commonly studied sources of uncertainty and their corresponding derivations ofstochastic travel time in the literature in order to better understand the implications of our proposed modeling approach.

Let us consider the widely adopted Bureau of Public Road (BPR) link performance function

ta ¼ t0a 1þ b

va

ca

� �n� �; 8a 2 A; ð26Þ

where ta, t0a , va, and ca are the travel time, free-flow travel time, flow, and capacity on link a; b and n are the deterministic

parameters. The variability of travel time could come from the free-flow travel time, the link capacity, or the link flow asdescribed below.

2.5.1. Free-flow travel time variationTo describe the travel time variability due to various non-routine events such as weather, road conditions, or traffic de-

lays, Mirchandani and Soroush (1987) suggested a nonnegative random free-flow link travel time T0a , which follows a Gamma

distribution with shape parameter k and scale parameter h, i.e., T0a � Cðk; hÞ. Then, the mean and variance of Ta can be rep-

resented as below:

EðTaÞ ¼ kh 1þ bva

ca

� �n� �; ð27Þ

VarðTaÞ ¼ kh2 1þ bva

ca

� �n� �2

: ð28Þ

2.5.2. Capacity variationLo et al. (2006) considered the stochastic link capacity degradation, which is one of the main sources of travel time var-

iability. Under the relatively minor day-to-day events, such as vehicle breakdown and accident, the link capacity is subject tostochastic degradation to different degrees. By assuming the capacity degradation random variable Ca (we use capital lettersto represent random variables) is independent of the amount of traffic (va) on it, and follows a uniform distribution definedby an upper bound (the design capacity �ca) and a lower bound (the worst-degraded capacity to be a fraction qa of the designcapacity), Lo et al. (2006) derived the mean and variance of Ta as below:

EðTaÞ ¼ t0a þ bt0

avna

ð1� q1�na Þ

�cnað1� qaÞð1� nÞ ; ð29Þ

VarðTaÞ ¼ b2ðt0aÞ

2v2na

1� q1�2na

�c2na ð1� qaÞð1� 2nÞ �

1� q1�na

�cnað1� qaÞð1� nÞ

� �2( )

: ð30Þ

To relax the assumption of uniform distribution, they suggested adopting the Mellin Transform technique as discussed inLo and Tung (2003).

2.5.3. Demand variationAnother main source of travel time variability is the stochastic travel demand. In view of the day-to-day travel demand fluc-

tuation, the traffic demand between each OD pair is assumed to be a random variable with a given probability distribution:

Q rs ¼ qrs þ ers; 8r 2 R; s 2 S; ð31Þ

where qrs ¼ EðQrsÞ is the mean demand, and ers is the random term with EðersÞ ¼ 0. Consequently, the route flow Frsp and link

flow Va are also random variables that contribute to travel time variability. By assuming that the route flow follows the sametype of probability distribution as the OD demand, the route flow’s coefficient of variation (CV) is equal to that of the ODdemand, and the route flows are mutually independent, Shao et al. (2006b) analytically derived the mean and variance of


Ta based on the assumption that the OD demands are normally distributed (Asakura and Kashiwadani, 1991; Clark and Wa-tling, 2005) as follows:

E½Ta� ¼ t0a þ

bt0a

ðcaÞnXn

i¼0;i¼even

ni

� �rv

a

iðvaÞn�iði� 1Þ!!; 8a 2 A; ð32Þ

Var½Ta� ¼bt0

a

ðcaÞn� �2 X2n

i¼0;i¼even

2n

i

� �ðrv

a ÞiðvaÞ2n�iði� 1Þ!!�

Xn

i¼0;i¼even

n

i

� �ðrv

a ÞiðvaÞn�iði� 1Þ!!

!20@

1A; 8a 2 A; ð33Þ

where ði� 1Þ!! is the double factorial of i� 1, i.e., ði� 1Þ!! ¼ ði� 1Þði� 3Þ . . . 2 (i is even), ni

� �is binomial coefficient, i.e.,

ni

� �¼ n!ðn�iÞ!i!, and rv

a is the standard deviation of the random link flow.

Under similar assumptions, except that the route flow’s variance-to-mean ratio (VMR) is assumed to be equal to that ofthe OD demand to maintain the flow conservation constraints, Zhou and Chen (2008) analytically derived the mean and var-iance of Ta based on the log-normal distribution as follows:

E½Ta� ¼ t0a þ

bt0a

Cna

enlvaþ

n2 ðrva Þ

2

2

� �; 8a 2 A; ð34Þ

Var½Ta� ¼bt0

a

ðCaÞn� �2

en2ðrva Þ

2 � 1h i

� e2nlvaþn2ðrv

a Þ2; 8a 2 A: ð35Þ

where lva and rv

a are the parameters of the link flow distribution Lognðlva ;rv

a Þ.Recently, some researches (Lam et al., 2008; Shao, 2007; Shao et al., 2008; Siu and Lo, 2008) were conducted to model

travelers’ route choice behavior under travel time variability due to both demand fluctuation and link capacity degradation.Lam et al. (2008) considered the rain effects on road network, where the link free-flow travel time is represented by a non-decreasing function of rainfall intensity, the link capacity is represented by a non-increasing function of rainfall intensity,and travel demand is stochastic. Shao et al. (2008) extended the model by Lam et al. (2008) to consider multiple user classes.Siu and Lo (2008) considered the travel demand is composed of two parts: commuters and non-commuters, where the ran-domness of travel demand is assumed to be introduced by the volume of non-commuters, and the link capacity is subject tostochastic degradation.

2.5.4. Derivation of METTTo facilitate the presentation of the essential ideas, we assume that the link travel times are independent. This assump-

tion is also adopted by Lo et al. (2006), Shao et al. (2006b), Siu and Lo (2006), and Watling (2006). Hence, the mean and var-iance of Trs

p can be written as:

lrsp ¼ E½Trs

p � ¼Xa2A

E½Ta�drspa; 8p 2 Prs; r 2 R; s 2 S; ð36Þ

ðrrsp Þ

2 ¼ Var½Trsp � ¼

Xa2A

Var½Ta�drspa; 8p 2 Prs; r 2 R; s 2 S: ð37Þ

According to the Central Limit Theorem, for routes consisting of many links, the random route travel times tend to benormally distributed regardless of what the underlying link travel time distribution is (Lo and Tung, 2003; Shao et al.,2006b):

Trsp � N lrs

p ; ðrrsp Þ

2� �

: ð38Þ

Then, according to the definition of METT in Eq. (1) and assuming the travelers’ confidence level is a, the route METT canbe represented as the following minimization problem:

min1

ð1� aÞ

Z 1

nrsp

Trsp �


rrsp

exp �ðTrs

p � lrsp Þ

2

2ðrrsp Þ

2

!dðTrs

p Þ; ð39Þ

s:tZ nrs

p

�1


rrsp

exp �ðTrs

p � lrsp Þ

2

2ðrrsp Þ

2

!dðTrs

p ÞP a; ð40Þ

where Eq. (40) represents the travel time reliability chance constraints.Following some calculus manipulations, the METT of route p can be represented as

grsp ¼ lrs

p þrrs

pffiffiffiffiffiffiffi2pp

ð1� aÞexp �ðU

�1ðaÞÞ2

2

!: ð41Þ


Note that, in reality, the link travel times may not be totally independent due to the network topology or the sources ofvariations. Therefore, the effects of the covariance of link travel times should be considered in future research.

Other forms of travel time distributions were also proposed in the literature. For example, exponential and uniform traveltime distributions were adopted in Noland and Small (1995) for studying the morning commute problem. A family of dis-tributions known as the ‘‘Johnson curves” was studied by Clark and Watling (2005) to model the total network travel timeunder random demand. Gamma type distributions were tested by Fan and Nie (2006) in the stochastic optimal routing prob-lem. To allow for a more flexible control over the right-hand tail and better fit the data, a mixture of normal distribution wassuggested in Watling (2006). However, no matter what kind of distribution is assumed, the concept of METT and the METEmodel are still valid. METT is a simple, convenient representation of risk, which fits quite well to the way that travelers’ as-sess the reliability and unreliability aspects of travel time variability and make their route choice decisions to tradeoff thebuffer time measure (i.e., the reliability aspect measured by TTB) and the tardy time measure (i.e., the unreliability aspectthat measures the worst travel times beyond TTB) accordingly.

3. Solution algorithm

A basic assumption of the traditional traffic equilibrium models is additivity, i.e. the route cost is simply the sum of thecosts on the links that constitute that route. This additive assumption enables the application of a number of well-knownalgorithms (e.g., the Frank–Wolfe algorithm), and the traffic equilibrium problem can be solved without the need to storeroutes. This is a significant benefit when one needs to solve large-scale network problems (Boyce et al., 2004). However,these types of algorithms are not applicable to the METE model, since the METT is nonadditive in general. Therefore, to solvethe proposed model, a route-based solution algorithm is needed (Bernstein and Gabriel, 1997; Chen et al., 2001; Gabriel andBernstein, 1997; Lo and Chen, 2000).

By exploring the special structure of the METE model, a modified alternating direction (MAD) algorithm (Han, 2002),which is a kind of projection-based algorithm, is adopted in this study for solving the VI problem (Eq. (21)). By attachinga Lagrangian multiplier vector y to the demand conservation constraints (Eq. (17)), we can reformulate the METE modelas an equivalent VI problem, denoted as VIðF;KÞ shown below:

Find u� 2 K , such that

Fðu�ÞTðu� u�ÞP 0; 8u 2 K; ð42Þ

where

u ¼fy

� �; FðuÞ ¼ gðf Þ �KT y

Kf � q

!; K ¼ Rn

þ � Rk; ð43Þ

where K denotes the OD-route incidence matrix, q denotes the demand vector ð. . . ; qrs; . . . ÞT , n is the total number of routes,and k is the total number of OD pairs. Based on this transformation, we can see that a projection on the new feasible region Kis much easier than on the original set X. Therefore, in each iteration, the MAD algorithm can make a simple projection onthe set K to update the solution vector u. This simple projection makes the MAD algorithm very attractive. Furthermore, aself-adaptive stepsize updating scheme is embedded in the MAD algorithm, where the stepsize is automatically updatedaccording to the information of the previous iterations (i.e., route flows and route METTs). These features make the MADalgorithm efficient and robust. The global convergence of the MAD algorithm can be rigorously proven under mild conditions,which only require the underlying mapping gðf Þ to be continuous and monotone (Han, 2002; Zhou et al., 2007). However, itshould be noted that convergence may not always hold since the monotonicity of the route METT is hard to guarantee.

The remaining complication is how to generate the route set in real applications. Lo and Chen (2000) and Chen et al.(2001) proposed two alternative approaches: the first approach works with a set of predefined routes, which could be de-rived from personal interviews and hence constitutes a set of likely used routes; the second approach is to use a heuristiccolumn generation procedure. Chen and Ji (2005) proposed a genetic algorithm for finding a-reliable routes. This approachmay also be extended to find routes with the minimum METT. To facilitate the presentation of the essential ideas, we assumea set of working routes is available in advance in our implementation when solving the METE model. Behaviorally, using aworking route set (i.e., generated from a choice set generation scheme) has the advantage of identifying routes that wouldlikely to be used (Bekhor et al., 2008; Cascetta et al., 1997). Future work should include developing more efficient algorithmfor finding routes with the minimum METT and combining it as a column generation procedure in the proposed solutionprocedure.

4. Numerical examples

To demonstrate the proposed METE model and solution algorithm, two networks are adopted in the numerical experi-ments. First, a small network is used to illustrate the features of the proposed model, its differences compared to other trafficequilibrium models, and the correctness of the solution algorithm. Then, a medium-sized network is employed to demon-strate the applicability of the solution algorithm to larger networks.


4.1. Small network

To illustrate the proposed METE model, a simple network consists of four nodes, five links and three routes as shown inFig. 7 is adopted, where the route–link relationship is provided in Table 1. There is one OD pair (1, 4) with 1000 units of de-mand. The free-flow travel time for each individual link is assumed to follow a normal distribution. The reason for choosing anormal distribution in this numerical experiment is because it enables deriving analytical expressions of the METT for eachroute. Therefore, we can solve the problem analytically and use the results as a benchmark to compare the METE model withthe traditional traffic equilibrium model and the travel time budget model. The variances of the link free-flow travel timesare exogenously defined in Table 2 and the mean link travel times are calculated from the Bureau of Public Road (BPR) func-tion as below:

Table 2Networ

Link

12345

Table 1Route–l

Rout

123

�ta ¼ E½Ta� ¼ t0a 1þ 0:15

va

ca

� �2 !

; 8a 2 A; ð44Þ

where �ta, t0a , va, and ca are respectively the mean travel time, free-flow travel time, flow, and capacity of link a. The corre-

sponding network characteristics are also shown in Table 2.Without loss of generality, in the following tests, we assume the confidence level of all travelers is 90%. Note that, to facil-

itate the presentation in this study, we assume that the link travel times are independent from each other. Under theseassumptions, the route travel time distribution can be acquired from Eq. (41). However, the effects of link correlations couldalso be incorporated in the proposed model scheme, as long as the joint probability distribution function of the route traveltime can be derived.

The equilibrium route flow pattern is shown in Table 3, as well as the corresponding mean travel time, TTB and METT. Tocheck the validity of the results, we examine two conditions: travel demand conservation constraint and METE conditions. Asexpected, both conditions are satisfied. The route flows sum up to the OD travel demand and the METT of all used routes areequal and minimal. Furthermore, to analyze the impact of travel time variability on travelers’ route choice decisions, the per-centages of the three travel time components (mean travel time, buffer time, and ‘expected excess delay’) that compose theMETT for each route are depicted in Fig. 8. As we discussed before, the summation of the mean travel time and the buffertime gives the TTB, which represents the travel time reliability requirement of the travelers. Specifically, in this experiment,the travelers are assumed to have a 90% confidence level of on-time arrival. At the same time, the ‘‘expected excess delay”describes the unreliability aspect of travel time variability, which evaluates the risk associated with the unacceptable late

Fig. 7. Small network.

k characteristics.

# Free-flow travel time (t0a) Capacity (Ca) Variance of the free-flow travel time

5 600 212 400 6

7 400 110 400 5

8 600 2

ink relationship of the small network.

e # Link sequence

1–21–3–54–5

Table 3Equilibrium results of the METE model.

Route # Route flow Mean travel time Travel time budget Mean-excess travel time

1 499.68 20.43 24.06 25.402 47.82 21.47 24.34 25.403 452.50 20.75 24.15 25.40

14%

80%

5%

85%

11%4%

82%

13%5%

MTT

Route 1

MTT

Route 2

MTT

Route 3

BufferTime

BufferTime

BufferTime




Fig. 8. Analysis of the reliability and unreliability aspects of travel time variability on travelers’ route choice decision.

Table 4Equilibrium route flows of different user equilibrium models.

Route # UE RUE METE

1 532.40 517.77 499.682 0.00 13.23 47.823 467.60 469.00 452.50


arrivals (though infrequent) that have a travel time higher than the TTB. From the figure, we can see that the reliability andunreliability aspects of travel time variability make up about 15–20% of METT on travelers’ route choice decisions. Thoughthe percentage of METT associated with the unreliability aspect is not high (around 5%) in this simple example, which is dueto the normally distributed link free-flow travel time, it is expected to play a more significant role in the travelers’ routechoice decisions under highly skewed link travel time distributions.

In the following, we further compare the METE model with the conventional UE model and the reliability-based userequilibrium (RUE) model. Recall that the route choice criterion in the conventional UE model adopts the mean route traveltime, while the RUE model uses the concept of TTB. Conceptually, the WBTR model (Lo et al., 2006) and DRUE model (Shaoet al., 2006a) are all RUE model, since they all share the common route choice criterion despite that the original sources oftravel time variation are different. The equilibrium results of the three models are demonstrated in Table 4 and Figs. 9–11,where the x-axis represents different equilibrium models, the y-axis represents the mean travel time, TTB and METT of eachroute under the equilibrium state of each individual model, respectively.

From the table and figures, it is clear that the three user equilibrium models give quite different equilibrium flow pat-terns. In the following, we further investigate the differences among these three models. First, we examine the results underthe conventional UE model (see Fig. 9). In the equilibrium state, two used routes (route 1 and route 3) with equal and min-imum mean travel times have positive flows, while the unused route (route 2) with a higher mean travel time carries noflow. However, as we discussed above, the route choice model under the mean travel time cannot account for the risk asso-ciated with the travel time variation. Therefore, it shows that, under the equilibrium state of the conventional UE model,though route 2 has the highest mean travel time, it actually has the second lowest TTB and the lowest mean-excess traveltime. This is because the travel time variance of route 2 is the lowest among the all three routes. Therefore, under the con-sideration of the reliability aspects of travel time variability, travelers on route 1 are willing to switch to route 2 or route 3.When more travelers shift to route 2 and route 3, according to the relationship of travel time and flow, the mean travel time,the TTB on these two routes will increase accordingly. A new equilibrium will reach when the TTB for all three routes becomeequal.

Similarly, in the equilibrium state of the RUE model (see Fig. 10), the travelers on each route can acquire the same traveltime reliability under the same TTB, i.e., all of them can ensure the same confidence level of on-time arrival (90% in thisexample). However, the RUE model is unable to account for the unreliable impacts beyond the TTB, thus travelers do nothave an assessment of the possible risk involved in the 1 � a (i.e., 10% in this case) unreliability of the distribution tail. There-fore, by taking both the reliability and unreliability aspects into consideration, travelers on route 1 and route 3 are willing to

UE RUE METE24

24.1

24.2

24.3

24.4

24.5

R o

u t

e T

r a

v e

l T

i m e

B u

d g

e t

R o u t e 1 R o u t e 2R o u t e 3

Fig. 10. Travel time budgets of different user equilibrium models.

UE RUE METE25.2

25.4

25.6

25.8

R o

u t

e M

e a

n E

x c

e s

s T

r a

v e

l T

i m e

R o u t e 1 R o u t e 2 R o u t e 3

Fig. 11. Mean-excess travel times of different user equilibrium models.

UE RUE METE20

20.4

20.8

21.2

21.6

R o

u t

e M

e a

n T

r a

v e

l T

i m

e

R o u t e 1 R o u t e 2R o u t e 3

Fig. 9. Mean travel times of different user equilibrium models.


switch to route 2 in order to avoid the higher risk in their original route while still ensure the same confidence level of punc-tual arrival. Finally, travelers will reach equilibrium under the METE model, where the mean-excess travel times in all threeroutes are equal (see Fig. 11). From these three figures, the trend of the changing of flow, mean travel time, TTB and METT canbe easily discovered.


4.2. Medium-size network

In this section, the proposed model and solution procedure are demonstrated using the Sioux Falls network (Leblanc,1973), which is a medium-sized network with 24 nodes, 76 links, and 550 OD pairs (see Fig. 12). The working route setof the network are from Bekhor et al. (2008), where the routes are generated by using a combination of the link eliminationmethod (Azevedo et al., 1993) and the penalty method (De La Barra et al., 1993). The total number of routes is 3441, themaximum number of routes generated for any OD pair is 13, and the average number of routes is 6.3 per OD pair.

In this experiment, we assume the travel time variability comes from the day-to-day travel demand fluctuations. Here,the stochastic travel demands are assumed to follow a log-normal distribution, which is a nonnegative, asymmetrical distri-bution. It has been adopted in the literature as a more realistic approximation of the stochastic travel demand to examine theuncertainty of the four-step travel demand forecasting model (Zhao and Kockelman, 2002). The PDF of the log-normal dis-tribution is

f ðxjl;rÞ ¼ 1xr

ffiffiffiffiffiffiffi2pp e

�ðln x�lÞ2

2r2 ; 8x > 0; ð45Þ

where x is the random variable, l and r are the distribution parameters, and the mean and variance are m ¼ elþr2=2 andv ¼ e2lþr2 ðer2 � 1Þ, respectively. Under some commonly adopted assumptions, the distribution of the random route traveltime can be analytically derived (see Zhou and Chen (2008) for details). For simplicity, the variance-to-mean ratio (VMR)of route flows are assumed to be 0.3, the root mean squared error (RMSE) of the route flows between two consecutive iter-ations is used as the stopping criterion, tolerance for termination (e) is set to be 5e�3, and the initial route flow for a givenOD pair is set to be the OD demand divided by the number of routes connecting this OD. Then, the route-based traffic assign-ment algorithm is tested on a personal computer with 2.4 G Pentium-IV processor and 768M RAM.

The MAD algorithm terminates after 70 iterations and the CPU time is 24.52 s. The convergence of the MAD algorithm interms of the RMSE is shown in Fig. 13.

To further demonstrate the convergence of the proposed solution procedure, without loss of generality, we examine tworoutes connecting OD pair (1, 10). The link sequence of route 1 and route 2 are 2–6–9–13–25 and 2–6–10–32, respectively.The evolution of the route flow and the route METT during the iteration process are depicted in Figs. 14 and 15.

From these figures, we can see that the algorithm quickly converges to the required solution precision. The METE solutionis achieved as the RMSE approaches zero after 70 iterations. At the same time, the METTs of used routes for a given OD pair

1

8

4 5 63

2

15 19

17

18

7

12 11 10 16

9

20

23 22

14

13 24 21

31

6

8

9

11

5

15

12231321

16 1917

20

55

5048

295149 52

58

24

27

32

33

36

4034

41

44

57

45

72

70

46 67

69 65

25

28 43

53

59 61

56

66 62

6863

7673

30

7142

6475

74

38

26

4 14

22 47

10 31

39

37

60

2

7 35

Fig. 12. Sioux Falls network.

0 10 20 30 40 50 60 70

0.015

0.025

0.035

N u m b e r o f I t e r a t i o n s

R M

S E

Fig. 13. Convergence curve of the MAD algorithm.

0 10 20 30 40 50 60 700

0.2

0.4

0.6

0.8

1

1.2

1.4


R o

u t

e F

l o

w R o u t e 1R o u t e 2

Fig. 14. Evolution of the route flows for OD pair (1, 10) during the iteration process.

0 10 20 30 40 50 60 700

1

2

3

4

5

6

7

8

9


R o

u t

e M

E T

T

R o u t e 1R o u t e 2

Fig. 15. Evolution of the route METTs for OD pair (1, 10) during the iteration process.


are getting closer to each other during the iteration process. Furthermore, it can be observed that the fluctuation of the RMSEis more frequent and larger at the early iterations, and getting smaller along with the iteration process. It demonstrates thatthe proposed algorithm has the ability to reach a stable equilibrium solution.


5. Conclusions and future research

In this study, we proposed a mean-excess traffic equilibrium (METE) model under stochastic travel times. The new modelexplicitly considers both reliability and unreliability aspects of travel time variability in the travelers’ route choice decisionprocess. The METE model is formulated as a variational inequality (VI) problem. Qualitative properties, such as equivalenceand existence of the solution, were also rigorously proved. A route-based traffic assignment algorithm based on the modifiedalternating direction method was adopted to solve the proposed model. Numerical examples were also provided to highlightthe essential ideas of the model and to demonstrate the proposed algorithm.

Many further works are worthy of exploring based on the proposed METE model. From the computational point of view,more efficient algorithms for finding the nonadditive METT routes and solving the proposed model need to be developed andtested on large networks. From the behavioral aspect, empirical studies need to be performed to obtain a better understand-ing of the travelers’ risk preference and attitudes to the reliability and unreliability aspects of travel time variability (seeFranklin and Karlstrom (2009) for details). It would be interesting to extend the proposed model to consider a ‘generalizedcost’ structure with individual preference parameter for each term (mean travel time, buffer time and expected excess delay)in the METT route cost and construct a generalized disutility function that better describes travelers’ route choice criterionunder travel time variability. Furthermore, in this study, travelers were assumed to have perfect knowledge about the traveltime distribution. This assumption could be relaxed by incorporating a perception error in the METT of each route. Thus, dur-ing the equilibrium process, travelers try to minimize their perceived METT, which corresponds to the stochastic user equi-librium counterpart of the METE model (Chen and Zhou, 2009). Note that, in reality, different user classes may have differentattitudes towards risk or perception differences when making their route choice decisions. Therefore, further extensionscould be to incorporate multiple user classes corresponding to traveler’s different risk preferences or perception errors intothe model.

Finally, by incorporating the specific sources of travel time variation, such as link capacity degradation and traffic controldevices, into the consideration of the proposed modeling approach, the METE model can be regarded as a fundamental com-ponent of high level transportation system risk assessment framework. That is, the METE model can be extended to the net-work design problem, where the functions of the METT (e.g., total mean travel time of the whole network or mean traveltime between specific OD pairs) as system risk measures are to be minimized or act as constraints by optimally determiningthe design variables subject to a budgetary constraint. To design a more reliable/robust network by developing appropriaterisk assessment measure and efficient solution algorithm remains an important and active research topic in the future.

Acknowledgements

The work described in this paper was supported by a CAREER Grant from the National Science Foundation of the UnitedStates (CMS-0134161).

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Further reading

Small, K.A., 1982. The scheduling of consumer activities: work trips. The American Economic Review 73 (3), 467–479.

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The α-reliable mean-excess traffic equilibrium model with stochastic travel times