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.     

Route switching behavior on freeways with the provision of different types of real-time traffic information

This study investigates route switching behavior on freeways in reaction to the provision of different types of real-time traffic information. The experimental design of the stated preference survey is based on four types of real-time information provided to travelers who were randomly selected at rest areas. The four types of real-time information defined in this paper are qualitative, quantitative, qualitative guidance, and quantitative guidance. The bounded rationality framework, also known as indifference band approach, is applied to model the freeway route switching behavior. Two important variables, travel time and travel cost, are included in the indifference band. In this study, the best route switching rule, travelers’ current routes as compared to the best route, is investigated to further provide valuable insights into freeways travelers’ route switching behavior with the provision of different types of real-time traffic information.     

Traveler route choice: Travel time implications of improved information and adaptive decisions

The Wardrop user equilibrium model states that travelers choose the fastest available route and always choose the same route on repeated trips. However, travelers are not always capable of choosing the fastest route, and if travel time is uncertain, they may acquire information on the day of travel that helps to select a better route. Thus, travelers can reduce their travel time over the Wardrop “optimum” by selecting routes adaptively. The focus of this paper is to find the most promising approach for improving actual transit route choice through providing better traveler information. Actual and ideal travel time were estimated for each of six information scenarios, ranging from one where travelers use transit maps, to one where travelers use adaptive route choice, and to the hypothetical situation referred to as perfect information. Travelers using maps and travelers using maps and schedules took significantly longer than ideally possible on an experimental trip (24% longer with maps, 42% longer with maps and schedules). Ideal travel time under perfect information was 49% less than actual travel time with no information, and 6% less than that of the best non-adaptive decision rule. Time adaptive route choice resulted in no travel time reduction. The potential travel time improvement from giving travelers more information was not as great as that from making information more understandable. Adaptive route choice did not offer great potential on the studied trip. To be effective there must be several nearly equal route options, and trips must involve transfers, which excludes most travel on transit today.     

A day-to-day route flow evolution process towards the mixed equilibria

This study investigates a travelers’ day-to-day route flow evolution process under a predefined market penetration of advanced traveler information system (ATIS). It is assumed that some travelers equipped with ATIS will follow the deterministic user equilibrium route choice behavior due to the complete traffic information provided by ATIS, while the other travelers unequipped with ATIS will follow the stochastic user equilibrium route choice behavior. The interaction between these two groups of travelers will result in a mixed equilibrium state. We first propose a discrete day-to-day route flow adjustment process for this mixed equilibrium behavior by specifying the travelers’ route adjustment principle and adjustment ratio. The convergence of the proposed day-to-day flow dynamic model to the mixed equilibrium state is then rigorously demonstrated under certain assumptions upon route adjustment principle and adjustment ratio. In addition, without affecting the convergence of the proposed day-to-day flow dynamic model, the assumption concerning the adjustment ratio is further relaxed, thus making the proposed model more appealing in practice. Finally, numerical experiments are conducted to illustrate and evaluate the performance of the proposed day-to-day flow dynamic model.     

Physics of day-to-day network flow dynamics

This paper offers a new look at the network flow dynamics from the viewpoint of physics by demonstrating that the traffic system, in terms of the aggregate effects of human behaviors, may exhibit like a physical system. Specifically, we look into the day-to-day evolution of network flows that arises from travelers’ route choices and their learning behavior on perceived travel costs. We show that the flow dynamics is analogous to a damped oscillatory system. The concepts of energies are introduced, including the potential energy and the kinetic energy. The potential energy, stored in each link, increases with the traffic flow on that link; the kinetic energy, generated by travelers’ day-to-day route swapping, is proportional to the square of the path flow changing speed. The potential and kinetic energies are converted to each other throughout the whole flow evolution, and the total system energy keeps decreasing owing to travelers’ tendency to stay on their current routes, which is analogous to the damping of a physical system. Finally, the system will approach the equilibrium state with minimum total potential energy and zero kinetic energy. We prove the stability of the day-to-day dynamics and provide numerical experiments to elucidate the interesting findings.     

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 study of traveler behavior under traffic information-provided conditions in the Beijing area

ABSTRACT

The quality of traffic information has become one of the most important factors that can affect the distribution of urban and highway traffic flow by changing the travel route, transportation mode, and travel time of travelers and trips. Past research has revealed traveler behavior when traffic information is provided. This paper summarizes the related study achievements from a survey conducted in the Beijing area with a specially designed questionnaire considering traffic conditions and the provision of traffic information services. With the survey data, a Logit model is estimated, and the results indicate that travel time can be considered the most significant factor that affects highway travel mode choice between private vehicles and public transit, whereas trip purpose is the least significant factor for private vehicle usage for both urban and highway travel.     

Optimal transit fare in a bimodal network under demand uncertainty and bounded rationality

This paper investigates the optimal transit fare in a simple bimodal transportation system that comprises public transport and private car. We consider two new factors: demand uncertainty and bounded rationality. With demand uncertainty, travelers are assumed to consider both the mean travel cost and travel cost variability in their mode choice decision. Under bounded rationality, travelers do not necessarily choose the travel mode of which perceived travel cost is absolutely lower than the one of the other mode. To determine the optimal transit fare, a bi‐level programming is proposed. The upper‐level objective function is to minimize the mean of total travel cost, whereas the lower‐level programming adopts the logit‐based model to describe users' mode choice behaviors. Then a heuristic algorithm based on a sensitivity analysis approach is designed to solve the bi‐level programming. Numerical examples are presented to illustrate the effect of demand uncertainty and bounded rationality on the modal share, optimal transit fare and system performance. Copyright © 2013 John Wiley & Sons, Ltd.     

Interactive travel choices and traffic forecast in a doubly dynamical system with user inertia and information provision

This study models the joint evolution (over calendar time) of travelers’ departure time and mode choices, and the resulting traffic dynamics in a bi-modal transportation system. Specifically, we consider that, when adjusting their departure time and mode choices, travelers can learn from their past travel experiences as well as the traffic forecasts offered by the smart transport information provider/agency. At the same time, the transport agency can learn from historical data in updating traffic forecast from day to day. In other words, this study explicitly models and analyzes the dynamic interactions between transport users and traffic information provider. Besides, the impact of user inertia is taken into account in modeling the traffic dynamics. When exploring the convergence of the proposed model to the dynamic bi-modal commuting equilibrium, we find that appropriate traffic forecast can help the system converge to the user equilibrium. It is also found that user inertia might slow down the convergence speed of the day-to-day evolution model. Extensive sensitivity analysis is conducted to account for the impacts of inaccurate parameters adopted by the transport agency.     

Transportation serviceability analysis for metropolitan commuting corridors based on modal choice modeling

Major commuting corridors in metropolitan areas generally comprise multiple transportation modes for commuters, such as transit (subways or buses), private vehicles, or park-and-ride combinations. During the morning peak hour, the commuters would choose one of the available transportation modes to travel through the corridors from rural/suburban living areas to urban working areas. This paper introduces a concept of transportation serviceability to evaluate a transportation mode’s service status in a specific link, route, road, or network during a certain period. The serviceability can be measured by the possibility that travelers choose a specific type of transportation service at a certain travel cost. The commuters’ modal-choice possibilities are calculated using a stochastic equilibrium model based on general travel cost. The modeling results illustrate how transportation serviceability is influenced by background traffic flow in a corridor, value of comfort for railway mode, and parking fee distribution.     

Passenger route guidance system for multi‐modal transit networks

In many public transport oriented cities in the world, especially Asian cities, the public transport system has been developed extensively, to the extent that it has become increasingly difficult to navigate. Although inter‐modal transfers are common and often necessary, a complete set of the routes across transport modes is generally not presented in a form that is accessible to travelers, as each operator would only publish its own routes. Moreover, the common nonlinear fare tables together with inter‐modal fare discounts pose challenges to travelers in deciding their best routes. This study develops a multi‐modal passenger route guidance system, called eFinder, to aid travelers with their combined mode‐route choices. We discuss the architecture and features of this system in this study. This system forms a platform for disseminating public transit information and should complement further development and use of the public transport network by enabling travelers to make informed choices.     

The Downs–Thomson Paradox with responsive transit service

Downs (1962) and Thomson (1977) suggested that highway capacity expansion may produce counterproductive effects on the two-mode (auto and transit) transport system (Downs–Thomson Paradox). This paper investigates the occurrence of this paradox when transit authority can have different economic objectives (profit-maximizing or breakeven) and operating schemes (frequency, fare, or both frequency and fare). For various combinations of economic objectives and operating schemes, the interaction between highway expansion and transit service is explored, as well as its impact on travelers’ mode choices and travel utilities. Further, for each combination, the conditions for occurrence of the Downs–Thomson Paradox are established. We show that the paradox never occurs when transit authority is profit-maximizing, but it is inevitable when the transit authority is running to maximize travelers’ utility while maintaining breakeven. This is because the former transit authority tends to enhance transit service (e.g., raise frequency or reduce fare) when facing an expanded highway; and on the contrary, the latter tends to compromise transit service (e.g., reduce frequency or raise fare). Both analytical and numerical examples are provided to verify the theoretical results.     

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.     

Transit competitiveness in polycentric metropolitan regions

This paper analyzes the potential to, and impacts of, increasing transit modal split in a polycentric metropolitan area – the Philadelphia, Pennsylvania region. Potential transit riders are preselected as those travelers whose trips begin and end in areas with transit-supportive land uses, defined as “activity centers,” areas of high-density employment and trip attraction. A multimodal traffic assignment model is developed and solved to quantify the generalized cost of travel by transit services and private automobile under (user) equilibrium conditions. The model predicts transit modal split by identifying the origin–destination pairs for which transit offers lower generalized cost. For those origin–destination pairs for which transit does not offer the lowest generalized cost, I compute a transit competitiveness measure, the ratio of transit generalized cost to auto generalized cost. The model is first formulated and solved for existing transit service and regional pricing schemes. Next, various transit incentives (travel time or fare reductions, increased service) and auto disincentives (higher out of pocket expenses) are proposed and their impacts on individual travel choices and system performance are quantified. The results suggest that a coordinated policy of improved transit service and some auto disincentives is necessary to achieve greater modal split and improved system efficiency in the region. Further, the research finds that two levels of coordinated transit service, between and within activity centers, are necessary to realize the greatest improvements in system performance.     

Tracking a system of shared autonomous vehicles across the Austin,Texas network using agent-based simulation

This study provides a large-scale micro-simulation of transportation patterns in a metropolitan area when relying on a system of shared autonomous vehicles (SAVs). The six-county region of Austin, Texas is used for its land development patterns, demographics, networks, and trip tables. The agent-based MATSim toolkit allows modelers to track individual travelers and individual vehicles, with great temporal and spatial detail. MATSim’s algorithms help improve individual travel plans (by changing tour and trip start times, destinations, modes, and routes). Here, the SAV mode requests were simulated through a stochastic process for four possible fare levels: $0.50, $0.75, $1, and $1.25 per trip-mile. These fares resulted in mode splits of 50.9, 12.9, 10.5, and 9.2% of the region’s person-trips, respectively. Mode choice results show longer-distance travelers preferring SAVs to private, human-driven vehicles (HVs)—thanks to the reduced burden of SAV travel (since one does not have to drive the vehicle). For travelers whose households do not own an HV, SAVs (rather than transit, walking and biking) appear preferable for trips under 10 miles, which is the majority of those travelers’ trip-making. It may be difficult for traditional transit services and operators to survive once SAVs become available in regions like Austin, where dedicated rail lines and bus lanes are few. Simulation of SAV fleet operations suggest that higher fare rates allow for greater vehicle replacement (ranging from 5.6 to 7.7 HVs per SAV, assuming that the average SAV serves 17–20 person-trips per day); when fares rise, travel demands shift away from longer trip distances. Empty vehicle miles traveled by the fleet of SAVs ranged from 7.8 to 14.2%, across the scenarios in this study. Implications of mobility and sustainability benefits of SAVs are also discussed in the paper.     

Making time count: Traveler activity engagement on urban transit

In practice, travel time is assigned a cost and treated as a disutility to be minimized. There is a growing body of research supporting the hypothesis that travel time has some value of its own, and the proliferation of information and communication technology (ICT) may be contributing to that value. Travelers’ attitudes are confounded with their mode choice, and as telecommunications mediate travel behavior, analysts must recognize the interaction between time use and customer satisfaction for appropriate travel demand management. To that end, this paper presents results from jointly estimated models of travelers’ latent satisfaction and on-board activity engagement using Chicago transit rider data gathered in April 2010. The simple questionnaire and small sample corroborate the findings of past research indicating travel attitudes and activity engagement have potential to influence travelers’ value of time, and many transit riders consider transit a better use of time and/or money than driving. The findings affirm the need for a more holistic understanding of value of time for travel demand management and infrastructure valuation. As time use has an influence on users’ valuation of the transit mode, offering opportunities to conduct certain leisure activities could improve the perceived value of travel time.     

Modeling drivers' diversion from normal routes under ATIS using generalized estimating equations and binomial probit link function

Advanced Traveler Information Systems (ATIS) provide travelers with real time traffic information to optimize their travel choices. The objective of this paper is to model drivers' diversion from their normal routes in the provision of ATIS. Five different scenarios of traffic information are used. Generalized Estimating Equations (GEE) framework with repeated observations and binomial probit link function is introduced and implemented. GEE with four different correlation structures including the independent case are developed and compared with each other and with regular Maximum Likelihood Estimation (MLE). A travel simulator was used. Sixty-five subjects have traveled 10 simulated trial days each on a 40-link realistic network with real historical congestion levels. The results showed that providing traffic information increases the probability of drivers' diversion from their normal routes. Adding advice to the pre-trip and/or en-route information encourages drivers to divert. Providing en-route in addition to the pre-trip information with or without advice increases the diversion probability. High travel time on the normal route and less travel time on the diverted route increase the probability of diversion. High-educated drivers are less likely to divert. Expressway users are more likely to divert from their normal routes under ATIS. Drivers' familiarity with the device that provides the information and high number of traffic signals on the normal route increase the diversion probability.     

Policy analysis of third party electronic coupons for public transit fares

Mobile technologies are generating new business models for urban transport systems, as is evident from recent startups cropping up from the private sector. Public transport systems can make more use of mobile technologies than just for measuring system performance, improving boarding times, or for analyzing travel patterns. A new transaction model is proposed for public transport systems where travelers are allowed to pre-book their fares and trade that demand information to private firms. In this public-private partnership model, fare revenue management is outsourced to third party private firms such as big box retail or large planned events (such as sports stadiums and theme parks), who can issue electronic coupons to travelers to subsidize their fares. This e-coupon pricing model is analyzed using marginal cost theory for the transit service and shown to be quite effective for monopolistic coupon rights, particularly for demand responsive transit systems that feature high cost fares, non-commute travel purposes, and a closed access system with existing pre-booking requirements. However, oligopolistic scenarios analyzed using game theory and network economics suggest that public transport agencies need to take extreme care in planning and implementing such a policy. Otherwise, they risk pushing an equivalent tax on private firms or disrupting the urban economy and real estate values while increasing ridership.     

A path-based flow formulation for the traffic assignment problem

In order to improve cooperation between traffic management and travelers, traffic assignment is the key component to achieve the objectives of both traffic management and route choice decisions for travelers. Traffic assignment can be classified into two models based on the behavioral assumptions governing route choices: User Equilibrium (UE) and System Optimum (SO) traffic assignment. According to UE and SO traffic assignment, travelers usually compete to choose the least cost routes to minimize their own travel costs, while SO traffic assignment requires travelers to work cooperatively to minimize overall cost in the road network. Thus, the paradox of benefits between UE and SO indicates that both are not practical. Thus, a solution technique needs to be proposed to balance UE and SO models, which can compromise both sides and give more feasible traffic assignments. In this paper, Stackelberg game theory is introduced to the traffic assignment problem, which can achieve the trade-off process between traffic management and travelers. Since traditional traffic assignments have low convergence rates, the gradient projection algorithm is proposed to improve efficiency.     

Dynamic congestion pricing with day-to-day flow evolution and user heterogeneity

This paper investigates evolutionary implementation of congestion pricing schemes to minimize the system cost and time, measured in monetary and time units, respectively, with the travelers’ day-to-day route adjustment behavior and their heterogeneity. The travelers’ heterogeneity is captured by their value-of-times. First, the multi-class flow dynamical system is proposed to model the travelers’ route adjustment behavior in a tolled transportation network with multiple user classes. Then, the stability condition and properties of equilibrium is examined. We further investigate the trajectory control problem via dynamic congestion pricing scheme to derive the system cost, time optimum, and generally, Pareto optimum in the sense of simultaneous minimization of system cost and time. The trajectory control problem is modeled by a differential–algebraic system with the differential sub-system capturing the flow dynamics and the algebraic one capturing the pricing constraint. The explicit Runge–Kutta method is proposed to calculate the dynamic flow trajectories and anonymous link tolls. The method allows the link tolls to be updated with any predetermined periods and forces the system cost and/or time to approach the optimum levels. Both analytical and numerical examples are adopted to examine the efficiency of the method.