Comparative analysis of three user equilibrium models under stochastic demand

Journal of Advanced Transportation, Fall, 2008 by Zhong Zhou, Anthony Chen

Recent empirical studies on the value of time and reliability reveal that travel time variability plays an important role on travelers' route choice decision process. It can be considered as a risk to travelers making a trip. Therefore, travelers are not only interested in saving their travel time but also in reducing their risk. Typically, risk can be represented by two different aspects: acceptable risk and unacceptable risk. Acceptable risk refers to the reliability aspect of acceptable travel time, which is defined as the average travel time plus the acceptable additional time (or buffer time) needed to ensure more frequent on-time arrivals, while unacceptable risk refers to the unreliability aspect of unacceptable late arrivals (though infrequent) that have a travel time excessively higher than the acceptable travel time. Most research in the network equilibrium based approach to modeling travel time variability ignores the unreliability aspect of unacceptable late arrivals. This paper examines the effects of both reliability and unreliability aspects in a network equilibrium framework. Specifically, the traditional user equilibrium model, the demand driven travel time reliability-based user equilibrium model, and the [alpha]-reliable mean-excess travel time user equilibrium model are considered in the investigation under an uncertain environment due to stochastic travel demand. Numerical results are presented to examine how these models handle risk under travel time variability.

Keywords: Travel time reliability, Travel time budget, Mean-excess travel time, User equilibrium, Variational inequality

1. Introduction

In real life, travel time is uncertain. The sources contributing to travel time variability could be exogenous and/or endogenous, which often lead to non-recurrent congestion (Chen et al., 2002, Al-Deek and Emam, 2006) and recurrent congestion (Heydecker et al., 2007), respectively. Recent empirical studies (Brownstone et al., 2003; Liu et al., 2004) revealed that travelers actually consider travel time variability as a risk in their route choice decisions. They are interested in not only travel time saving but also risk reduction. However, the traditional user equilibrium (UE) model neglects travel time variability in the route choice decision process. It uses only the expected travel time as the sole criterion for making route choices, which implicitly assumes all travelers to be risk-neutral.

To model the route choice decision process under travel time variability, various models have been proposed. Mirchandani and Soroush (1987) were the first to propose the generalized traffic equilibrium model that incorporates both probabilistic travel times and variable perceptions in the route choice decision process. Under the assumptions that travelers are highly pessimistic about the travel time variability and behave in a very risk-averse way, Bell and Cassir (2002) provided a risk-averse traffic equilibrium model formulated as a noncooperative, mixed-strategy game. Based on the concept of schedule delay, Watling (2006) proposed a late arrival penalized user equilibrium (LAPUE) model by incorporating a schedule delay term to the disutility function to penalize late arrival under fixed departure times. Lo et al. (2006) proposed a probabilistic user equilibrium model to account for the effects of within budget time reliability due to link degradation. By considering daily travel demand variation, Shao et al. (2006) proposed a demand driven travel time reliability-based user equilibrium (DRUE) model. The key concept adopted in these models (Lo et al., 2006; Shao et al., 2006) is the travel time budget (TTB), which is defined as the average travel time plus an extra time as an acceptable travel time, such that the probability of completing the trip within the TTB is no less than a predefined reliability threshold (or a confidence level et). The concept of TTB is analogous to the Value-at-Risk (VaR), which is by far the most widely applied risk measure in the finance area. However, it has been determined that VaR is not even a weakly coherent measure of risk (Artzner, 1999). Models using VaR are unable to deal with the possibility that the losses associated with the worst scenarios are excessively higher than the VaR, and reduction of VaR may lead to stretch of tail exceeding VaR (Yamai and Yoshiba, 2001). In the same spirit, TTB may also be an inadequate risk measure, which could introduce overwhelmingly high trip times to travelers 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 reliability aspect may not be adequate to describe travelers' risk preferences. On the one hand, the FHWA report (2006) documented that travelers, especially commuters, do add a 'buffer time' to their expected travel time to ensure more frequent 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 also examined in the literature (Noland et al., 1999) and appreciated as the 'value' of unreliability (Watling, 2006). It represents travelers' concern of the unreliability aspect of travel time variability in their route choice decision process, where trip time longer than they expected would be considered as 'unreliable' or 'unacceptable' (Cambridge Systematics, 2003).