Comparative analysis of three user equilibrium models under stochastic demand

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 α‐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.     

Day-to-day traffic dynamics considering social interaction: From individual route choice behavior to a network flow model

Social interaction is increasingly recognized as an important factor that influences travelers’ behaviors. It remains challenging to incorporate its effect into travel choice behaviors, although there has been some research into this area. Considering random interaction among travelers, we model travelers’ day-to-day route choice under the uncertain traffic condition. We further explore the evolution of network flow based on the individual-level route choice model, though that travelers are heterogeneous in decision-making under the random-interaction scheme. We analyze and prove the existence of equilibrium and the stability of equilibrium. We also analyzed and described the specific properties of the network flow evolution and travelers’ behaviors. Two interesting phenomena are found in this study. First, the number of travelers that an individual interacts with can affect his route choice strategy. However, the interaction count exerts no influence on the evolution of network flow at the aggregate-level. Second, when the network flow reaches equilibrium, the route choice strategy at the individual-level is not necessarily invariable. Finally, two networks are used as numerical examples to show model properties and to demonstrate the two study phenomena. This study improves the understanding of travelers’ route choice dynamics and informs how the network flow evolves under the influence of social interaction.     

A stochastic multimodal reliable network design problem under adverse weather conditions

This paper formulates a network design problem (NDP) for finding the optimal public transport service frequencies and link capacity expansions in a multimodal network with consideration of impacts from adverse weather conditions. The proposed NDP aims to minimize the sum of expected total travel time, operational cost of transit services, and construction cost of link capacity expansions under an acceptable level of variance of total travel time. Auto, transit, bus, and walking modes are considered in the multimodal network model for finding the equilibrium flows and travel times. In the proposed network model, demands are assumed to follow Poisson distribution, and weather‐dependent link travel time functions are adopted. A probit‐based stochastic user equilibrium, which is based on the perceived expected travel disutility, is used to determine the multimodal route of the travelers. This model also considers the strategic behavior of the public transport travelers in choosing their routes, that is, common‐line network. Based on the stochastic multimodal model, the mean and variance of total travel time are analytical estimated for setting up the NDP. A sensitivity‐based solution algorithm is proposed for solving the NDP, and two numerical examples are adopted to demonstrate the characteristics of the proposed model. Copyright © 2014 John Wiley & Sons, Ltd.     

Modeling the evolutions of day‐to‐day route choice and year‐to‐year ATIS adoption with stochastic user equilibrium

Suppose that in an urban transportation network there is a specific advanced traveler information system (ATIS) which acts for reducing the drivers' travel time uncertainty through provision of pre‐trip route information. Because of the imperfect information provided, some travelers are not in compliance with the ATIS advice although equipped with the device. We thus divide all travelers into three groups, one group unequipped with ATIS, another group equipped and in compliance with ATIS advice and the third group equipped but without compliance with the advice. Each traveler makes route choice in a logit‐based manner and a stochastic user equilibrium with multiple user classes is reached for every day. In this paper, we propose a model to investigate the evolutions of daily path travel time, daily ATIS compliance rate and yearly ATIS adoption, in which the equilibrium for every day's route choice is kept. The stability of the evolution model is initially analyzed. Numerical results obtained from a test network are presented for demonstrating the model's ability in depicting the day‐to‐day and year‐to‐year evolutions.     

Effect of providing traffic information estimated by a stochastic network equilibrium model with stochastic demand

To estimate travel times through road networks, in this study, we assume a stochastic demand and formulate a stochastic network equilibrium model whose travel times, flows, and demands are stochastic. This model enables us to examine network reliability under stochastic circumstances and to evaluate the effect of providing traffic information on travel times. For traffic information, we focus on travel time information and propose methods to evaluate the effect of providing that information. To examine the feasibility and validity of the proposed model and methods, we apply them to a simple network and the real road network of Kanazawa, Japan. The results indicate that providing ambulance drivers in Kanazawa with travel time information leads to an average reduction in travel time of approximately three minutes.     

Preferences for travel time under risk and ambiguity: Implications in path selection and network equilibrium

In this paper, we study the preferences for uncertain travel times in which probability distributions may not be fully characterized. In evaluating an uncertain travel time, we explicitly distinguish between risk, where the probability distribution is precisely known, and ambiguity, where it is not. In particular, we propose a new criterion called ambiguity-aware CARA travel time (ACT) for evaluating uncertain travel times under various attitudes of risk and ambiguity, which is a preference based on blending the Hurwicz criterion and Constant Absolute Risk Aversion (CARA). More importantly, we show that when the uncertain link travel times are independently distributed, finding the path that minimizes travel time under the ACT criterion is essentially a shortest path problem. We also study the implications on Network Equilibrium (NE) model where travelers on the traffic network are characterized by their knowledge of the network uncertainty as well as their risk and ambiguity attitudes under the ACT. We derive and analyze the existence and uniqueness of solutions under NE. Finally, we obtain the Price of Anarchy that characterizes the inefficiency of this new equilibrium. The computational study suggests that as uncertainty increases, the influence of selfishness on inefficiency diminishes.     

Speed limits,speed selection and network equilibrium

This paper investigates the local and global impact of speed limits by considering road users’ non-obedient behavior in speed selection. Given a link-specific speed limit scheme, road users will take into account the subjective travel time cost, the perceived crash risk and the perceived ticket risk as determinant factors for their actual speed choice on each link. Homogeneous travelers’ perceived crash risk is positively related to their driving speed. When travelers are heterogeneous, the perceived crash risk is class-specific: different user classes interact with each other and choose their own optimal speed, resulting in a Nash equilibrium speed pattern. With the speed choices on particular roads, travelers make route choices, resulting in user equilibrium in a general network. An algorithm is proposed to solve the user equilibrium problem with heterogeneous users under link-specific speed limits. The models and algorithms are illustrated with numerical examples.     

Travel mental budgeting under road toll: An investigation based on user equilibrium

With the approach of introducing the conceptions of mental account and mental budgeting into the process of travelers’ route choice, we try to identify why the usages of tolled roads are often overestimated. Assuming that every traveler sets a mental account for his/her travel to keep track of their expense and keep out-of-pocket spending under control, it addresses these questions such that “How much money can I spend on the travel?” and “What if I spend too much?”. Route tolls that exceed the budget are much more unacceptable compared to those within budget due to the non-fungibility of money between different accounts. A simple network with two nodes and two routes is analyzed firstly, the analytical solutions are obtained and the optimal road tolls supporting the user equilibrium as a system optimum are also derived. The proposed model is then extended to a generalized network. The multiclass user equilibrium conditions with travel mental budgeting are formulated into an equivalent variational inequality (VI) problem and an equivalent minimization problem. Through analyses with numerical examples, it is found that the main reason that the usages of high tolled roads are often overestimated is due to the fact that travelers with low and moderate out-of-pocket travel budget perceive a much higher travel cost than their actual cost on the high tolled roads.     

Managing network mobility with tradable credits

A system of tradable travel credits is explored in a general network with homogeneous travelers. A social planner is assumed to initially distribute a certain number of travel credits to all eligible travelers, and then there are link-specific charges to travelers using that link. Free trading of credits among travelers is assumed. For a given credit distribution and credit charging scheme, the existence of a unique equilibrium link flow pattern is demonstrated with either fixed or elastic demand. It can be obtained by solving a standard traffic equilibrium model subject to a total credit consumption constraint. The credit price at equilibrium in the trading market is also conditionally unique. The appropriate distribution of credits among travelers and correct selection of link-specific rates is shown to lead to the most desirable network flow patterns in a revenue-neutral manner. Social optimum, Pareto-improving and revenue-neutral, and side-constrained traffic flow patterns are investigated.     

Evaluation of exclusive bus lanes in a bi-modal degradable road network

In this paper, we proposed an evaluation method of exclusive bus lanes (EBLs) in a bi-modal degradable road network with car and bus transit modes. Link travel time with and without EBLs for two modes is analyzed with link stochastic degradation. Furthermore, route general travel costs are formulated with the uncertainty of link travel time for both modes and the uncertainty of waiting time at a bus stop and in-vehicle congestion costs for the bus mode. The uncertainty of bus waiting time is considered to be relevant to the degradation of the front links of the bus line. A bi-modal user equilibrium model incorporating travelers’ risk adverse behavior is proposed for evaluating EBLs. Finally, two numerical examples are used to illustrate how the road degradation level, travelers’ risk aversion level and the front link’s correlation level with the uncertainty of the bus waiting time affect the results of the user equilibrium model with and without EBLs and how the road degradation level affects the optimal EBLs setting scheme. A paradox of EBLs setting is also illustrated where adding one exclusive bus lane may decrease share of bus.     

The Impact of Travel Time Information on Travelers’ Learning under Uncertainty

In this work, laboratory experiment was conducted in order to evaluate the effect of feedback on decision-making under uncertainty, with and without provided information about travel times. We discuss the prediction of travelers’ response to uncertainty in two route–choice situations. In the first situation travelers are faced with a route–choice problem in which travel times are uncertain but some external information about routes’ travel times is provided. The second situation takes place in a more uncertain environment in which external information about travel times is not provided, and the travelers’ only source of information is their own experience. Experimental results are in conflict with the paradigm about traveler information systems: As a consequence of information, the propensity of travelers to minimize expected travel time is not necessarily increased. Providing travelers with static information about expected travel times reveals an increase in the heterogeneity of travelers’ choices and reduces the maximization rate.     

Simultaneous determination of the equilibrium market penetration and compliance rate of advanced traveler information systems

We consider a specific advanced traveler information systems (ATIS) whose objective is to reduce drivers’ travel time uncertainty with recurrent network congestion through provision of traffic information. Since the provided information is still partial or imperfect, drivers equipped with an ATIS cannot always find the shortest travel time route and thus may not always comply with the advice provided by ATIS. Thus, there are three classes of drivers on a specific day: drivers without ATIS, drivers with ATIS but without compliance with ATIS advice, drivers with ATIS and in compliance with ATIS advice. All three classes of drivers make route choice in a stochastic manner, but with different degree of uncertainty of travel time on the network. In this paper we investigate the interactions among the three classes of drivers in an ATIS environment using a multiple behavior stochastic user equilibrium model. By assuming that the market penetration of ATIS is an increasing function of the actual private gain (time saving minus the cost associated with system use) derived from ATIS service, and the ATIS compliance rate of equipped drivers is given as the probability of the actual travel time of complied drivers being less than that of non-complied drivers, we determine the equilibrium market penetration and compliance rate of ATIS and the resulting equilibrium network flow pattern using an iterative solution procedure.     

Assessing the effects of stochastic perception error under travel time variability

Perceived mean-excess travel time is a new risk-averse route choice criterion recently proposed to simultaneously consider both stochastic perception error and travel time variability when making route choice decisions under uncertainty. The stochastic perception error is conditionally dependent on the actual travel time distribution, which is different from the deterministic perception error used in the traditional logit model. In this paper, we investigate the effects of stochastic perception error at three levels: (1) individual perceived travel time distribution and its connection to the classification by types of travelers and trip purposes, (2) route choice decisions (in terms of equilibrium flows and perceived mean-excess travel times), and (3) network performance measure (in terms of the total travel time distribution and its statistics). In all three levels, a curve fitting method is adopted to estimate the whole distribution of interest. Numerical examples are also provided to illustrate and visualize the above analyses. The graphical illustrations allow for intuitive interpretation of the effects of stochastic perception error at different levels. The analysis results could enhance the understanding of route choice behaviors under both (subjective) stochastic perception error and (objective) travel time uncertainty. Some suggestions are also provided for behavior data collection and behavioral modeling.     

A discrete dynamical system of formulating traffic assignment: Revisiting Smith’s model

Through relaxing the behavior assumption adopted in Smith’s model (Smith, 1984), we propose a discrete dynamical system to formulate the day-to-day evolution process of traffic flows from a non-equilibrium state to an equilibrium state. Depending on certain preconditions, the equilibrium state can be equivalent to a Wardrop user equilibrium (UE), Logit-based stochastic user equilibrium (SUE), or boundedly rational user equilibrium (BRUE). These equivalence properties indicate that, to make day-to-day flows evolve to equilibrium flows, it is not necessary for travelers to choose their routes based on actual travel costs of the previous day. Day-to-day flows can still evolve to equilibrium flows provided that travelers choose their routes based on estimated travel costs which satisfy these preconditions. We also show that, under a more general assumption than the monotonicity of route cost function, the trajectory of the dynamical system converges to a set of equilibrium flows by reasonably setting these parameters in the dynamical system. Finally, numerical examples are presented to demonstrate the application and properties of the dynamical system. The study is helpful for understanding various processes of forming traffic jam and designing an algorithm for calculating equilibrium flows.     

A bi-level programming for bus lane network design

This paper proposes a bi-level programming model to solve the design problem for bus lane distribution in multi-modal transport networks. The upper level model aims at minimizing the average travel time of travelers, as well as minimizing the difference of passengers’ comfort among all the bus lines by optimizing bus frequencies. The lower level model is a multi-modal transport network equilibrium model for the joint modal split/traffic assignment problem. The column generation algorithm, the branch-and-bound algorithm and the method of successive averages are comprehensively applied in this paper for the solution of the bi-level model. A simple numerical test and an empirical test based on Dalian economic zone are employed to validate the proposed model. The results show that the bi-level model performs well with regard to the objective of reducing travel time costs for all travelers and balancing transit service level among all bus lines.     

Driver response to variable message signs: a stated preference investigation

This paper uses a Stated Preference approach to undertake a detailed assessment of the effect on drivers’ route choice of information provided by variable message signs (VMS). Although drivers’ response to VMS information will vary according to the availability of alternative routes and the extent to which they are close substitutes, our findings show that route choice can be strongly influenced by the provision of information about traffic conditions ahead. This has important implications for the use of VMS systems as part of comprehensive traffic management and control systems. The principal findings are that the impact of VMS information depends on: the content of the message, such as the cause of delay and its extent; local circumstances, such as relative journey times in normal conditions; and drivers’ characteristics, such as their age, sex and previous network knowledge. The impact of qualitative indicators, visible queues and delays were examined. It was found that not only is delay time more highly valued than normal travel time, which is to be expected, but that drivers become more sensitive to delay time as delay times increased across the range presented.     

Capturing value of reliability through road pricing in congested traffic under uncertainty

Empirical studies showed that travel time reliability, usually measured by travel time variance, is strongly correlated with travel time itself. Travel time is highly volatile when the demand approaches or exceeds the capacity. Travel time variability is associated with the level of congestion, and could represent additional costs for travelers who prefer punctual arrivals. Although many studies propose to use road pricing as a tool to capture the value of travel time (VOT) savings and to induce better road usage patterns, the role of the value of reliability (VOR) in designing road pricing schemes has rarely been studied. By using road pricing as a tool to spread out the peak demand, traffic management agencies could improve the utility of travelers who prefer punctual arrivals under traffic congestion and stochastic network conditions. Therefore, we could capture the value of travel time reliability using road pricing, which is rarely discussed in the literature. To quantify the value of travel time reliability (or reliability improvement), we need to integrate trip scheduling, endogenous traffic congestion, travel time uncertainty, and pricing strategies in one modeling framework. This paper developed such a model to capture the impact of pricing on various costs components that affect travel choices, and the role of travel time reliability in shaping departure patterns, queuing process, and the choice of optimal pricing. The model also shows the benefits of improving travel time reliability in various ways. Findings from this paper could help to expand the scope of road pricing, and to develop more comprehensive travel demand management schemes.     

Traffic dynamics in a bi-modal transportation network with information provision and adaptive transit services

This paper has two major components. The first one is the day-to-day evolution of travelers’ mode and route choices in a bi-modal transportation system where traffic information (predicted travel cost) is available to travelers. The second one is a public transit operator adjusting or adapting its service over time (from period to period) based on observed system conditions. Particularly, we consider that on each day both travelers’ past travel experiences and the predicted travel cost (based on information provision) can affect travelers’ perceptions of different modes and routes, and thus affect their mode choice and/or route choice accordingly. This evolution process from day to day is formulated by a discrete dynamical model. The properties of such a dynamical model are then analyzed, including the existence, uniqueness and stability of the fixed point. Most importantly, we show that the predicted travel cost based on information provision may help stabilize the dynamical system even if it is not fully accurate. Given the day-to-day traffic evolution, we then model an adaptive transit operator who can adjust frequency and fare for public transit from period to period (each period contains a certain number of days). The adaptive frequency and fare in one period are determined from the realized transit demands and transit profits of the previous periods, which is to achieve a (locally) maximum transit profit. The day-to-day and period-to-period models and their properties are also illustrated by numerical experiments.     

Value of travel time reliability: A review of current evidence

Travel time reliability is a fundamental factor in travel behavior. It represents the temporal uncertainty experienced by travelers in their movement between any two nodes in a network. The importance of the time reliability depends on the penalties incurred by the travelers. In road networks, travelers consider the existence of a trip travel time uncertainty in different choice situations (departure time, route, mode, and others). In this paper, a systematic review of the current state of research in travel time reliability, and more explicitly in the value of travel time reliability is presented. Moreover, a meta-analysis is performed in order to determine the reasons behind the discrepancy among the reliability estimates.