Link-based day-to-day network traffic dynamics and equilibria

A general dynamical system model with link-based variables is formulated to characterize the processes of achieving equilibria from a non-equilibrium state in traffic networks. Several desirable properties of the dynamical system model are established, including the equivalence between its stationary state and user equilibrium, the invariance of its evolutionary trajectories, and the uniqueness and stability of its stationary points. Moreover, it is shown that not only a link-based version of two existing day-to-day traffic dynamics models but also two existing link-based dynamical system models of traffic flow are the special cases of the proposed model. The stabilities of stationary states of these special cases are also analyzed and discussed. In addition, an extension is made to the case with elastic demand. The study is helpful for better understanding the day-to-day adjustment mechanism of traffic flows in networks.     

Modeling the day-to-day traffic evolution process after an unexpected network disruption

Although various approaches have been proposed for modeling day-to-day traffic flow evolution, none of them, to the best of our knowledge, have been validated for disrupted networks due to the lack of empirical observations. By carefully studying the driving behavioral changes after the collapse of I-35W Mississippi River Bridge in Minneapolis, Minnesota, we found that most of the existing day-to-day traffic assignment models would not be suitable for modeling the traffic evolution under network disruption, because they assume that drivers’ travel cost perception depends solely on their experiences from previous days. When a significant network change occurs unexpectedly, travelers’ past experience on a traffic network may not be entirely useful because the unexpected network change could disturb the traffic greatly. To remedy this, in this paper, we propose a prediction-correction model to describe the traffic equilibration process. A “predicted” flow pattern is constructed inside the model to accommodate the imperfect perception of congestion that is gradually corrected by actual travel experiences. We also prove rigorously that, under mild assumptions, the proposed prediction-correction process has the user equilibrium flow as a globally attractive point. The proposed model is calibrated and validated with the field data collected after the collapse of I-35W Bridge. This study bridges the gap between theoretical modeling and practical applications of day-to-day traffic equilibration approaches and furthers the understanding of traffic equilibration process after network disruption.     

Learning marginal-cost pricing via a trial-and-error procedure with day-to-day flow dynamics

This paper investigates the convergence of the trial-and-error procedure to achieve the system optimum by incorporating the day-to-day evolution of traffic flows. The path flows are assumed to follow an ‘excess travel cost dynamics’ and evolve from disequilibrium states to the equilibrium day by day. This implies that the observed link flow pattern during the trial-and-error procedure is in disequilibrium. By making certain assumptions on the flow evolution dynamics, we prove that the trial-and-error procedure is capable of learning the system optimum link tolls without requiring explicit knowledge of the demand functions and flow evolution mechanism. A methodology is developed for updating the toll charges and choosing the inter-trial periods to ensure convergence of the iterative approach towards the system optimum. Numerical examples are given in support of the theoretical findings.     

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.     

A day-to-day dynamical model for the evolution of path flows under disequilibrium of traffic networks with fixed demand

Transportation networks are often subjected to perturbed conditions leading to traffic disequilibrium. Under such conditions, the traffic evolution is typically modeled as a dynamical system that captures the aggregated effect of paths-shifts by drivers over time. This paper proposes a day-to-day (DTD) dynamical model that bridges two important gaps in the literature. First, existing DTD models generally consider current path flows and costs, but do not factor the sensitivity of path costs to flow. The proposed DTD model simultaneously captures all three factors in modeling the flow shift by drivers. As a driver can potentially perceive the sensitivity of path costs with the congestion level based on past experience, incorporating this factor can enhance real-world consistency. In addition, it smoothens the time trajectory of path flows, a desirable property for practice where the iterative solution procedure is typically terminated at an arbitrary point due to computational time constraints. Second, the study provides a criterion to classify paths for an origin–destination pair into two subsets under traffic disequilibrium: expensive paths and attractive paths. This facilitates flow shifts from the set of expensive paths to the set of attractive paths, enabling a higher degree of freedom in modeling flow shift compared to that of shifting flows only to the shortest path, which is behaviorally restrictive. In addition, consistent with the real-world driver behavior, it also helps to preclude flow shifts among expensive paths. Improved behavioral consistency can lead to more meaningful path/link time-dependent flow profiles for developing effective dynamic traffic management strategies for practice. The proposed DTD model is formulated as the dynamical system by drawing insights from micro-economic theory. The stability of the model and existence of its stationary point are theoretically proven. Results from computational experiments validate its modeling properties and illustrate its benefits relative to existing DTD dynamical models.     

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.     

Bounded rationality and irreversible network change

A network change is said to be irreversible if the initial network equilibrium cannot be restored by revoking the change. The phenomenon of irreversible network change has been observed in reality. To model this phenomenon, we develop a day-to-day dynamic model whose fixed point is a boundedly rational user equilibrium (BRUE) flow. Our BRUE based approach to modeling irreversible network change has two advantages over other methods based on Wardrop user equilibrium (UE) or stochastic user equilibrium (SUE). First, the existence of multiple network equilibria is necessary for modeling irreversible network change. Unlike UE or SUE, the BRUE multiple equilibria do not rely on non-separable link cost functions, which makes our model applicable to real-world large-scale networks, where well-calibrated non-separable link cost functions are generally not available. Second, travelers’ boundedly rational behavior in route choice is explicitly considered in our model. The proposed model is applied to the Twin Cities network to model the flow evolution during the collapse and reopening of the I-35 W Bridge. The results show that our model can to a reasonable level reproduce the observed phenomenon of irreversible network change.     

Exploration of day-to-day route choice models by a virtual experiment

This paper examines existing day-to-day models based on a virtual day-to-day route choice experiment using the latest mobile Internet technologies. With the realized day-to-day path flows and path travel times in the experiment, we calibrate several well-designed path-based day-to-day models that take the Wardrop’s user equilibrium as (part of) their stationary states. The nonlinear effects of path flows and path time differences on path switching are then investigated. Participants’ path preferences, time-varying sensitivity, and learning behavior in the day-to-day process are also examined. The prediction power of various models with various settings (nonlinear effects, time-varying sensitivity, and learning) is compared. The assumption of “rational behavior adjustment process” in Yang and Zhang (2009) is further verified. Finally, evolutions of different Lyapunov functions used in the literature are plotted, and no obvious diversity is observed.     

A traffic assignment model for a ridesharing transportation market

A nascent ridesharing industry is being enabled by new communication technologies and motivated by the many possible benefits, such as reduction in travel cost, pollution, and congestion. Understanding the complex relations between ridesharing and traffic congestion is a critical step in the evaluation of a ridesharing enterprise or of the convenience of regulatory policies or incentives to promote ridesharing. In this work, we propose a new traffic assignment model that explicitly represents ridesharing as a mode of transportation. The objective is to analyze how ridesharing impacts traffic congestion, how people can be motivated to participate in ridesharing, and, conversely, how congestion influences ridesharing, including ridesharing prices and the number of drivers and passengers. This model is built by combining a ridesharing market model with a classic elastic demand Wardrop traffic equilibrium model. Our computational results show that (i) the ridesharing base price influences the congestion level, (ii) within a certain price range, an increase in price may reduce the traffic congestion, and (iii) the utilization of ridesharing increases as the congestion increases. Copyright © 2014 John Wiley & Sons, Ltd.     

Entropy weighted average method for the determination of a single representative path flow solution for the static user equilibrium traffic assignment problem

The formulation of the static user equilibrium traffic assignment problem (UETAP) under some simplifying assumptions has a unique solution in terms of link flows but not in terms of path flows. Large variations are possible in the path flows obtained using different UETAP solution algorithms. Many transportation planning and management applications entail the need for path flows. This raises the issue of generating a meaningful path flow solution in practice. Past studies have sought to determine a single path flow solution using the maximum entropy concept. This study proposes an alternate approach to determine a single path flow solution that represents the entropy weighted average of the UETAP path flow solution space. It has the minimum expected Euclidean distance from all other path flow solution vectors of the UETAP. The mathematical model of the proposed entropy weighted average method is derived and its solution stability is proved. The model is easy to interpret and generalizes the proportionality condition of Bar-Gera and Boyce (1999). Results of numerical experiments using networks of different sizes suggest that the path flow solutions for the UETAP using the proposed method are about identical to those obtained using the maximum entropy approach. The entropy weighted average method requires low computational effort and is easier to implement, and can therefore serve as a potential alternative to the maximum entropy approach in practice.     

On seat capacity in traffic assignment to a transit network

Seating or standing make distinct on‐board states to a transit rider, yielding distinct discomfort costs, with potential influence on the passenger route choice onto the transit network. The paper provides a transit assignment model that captures the seating capacity and its occupancy along any transit route. The main assumptions pertain to: the seat capacity by service route, selfish user behaviour, a seat allocation process with priority rules among the riders, according to their prior state either on‐board or at boarding. To each transit leg from access to egress station is associated a set of ‘service modes’, among which the riders are assigned in a probabilistic way, conditionally on their priority status and the ratio between the available capacity and the flow of them. Thus the leg cost is a random variable, with mean value to be included in the trip disutility. Computationally efficient algorithms are provided for, respectively, loading the leg flows and evaluating the leg costs along a transit line. At the network level, a hyperpath formulation is provided for supply‐demand equilibrium, together with a property of existence and an method of successive averages equilibration algorithm. It is shown that multiple equilibria may arise. Copyright © 2010 John Wiley & Sons, Ltd.     

A new schedule-based transit assignment model with travel strategies and supply uncertainties

This paper proposes a new scheduled-based transit assignment model. Unlike other schedule-based models in the literature, we consider supply uncertainties and assume that users adopt strategies to travel from their origins to their destinations. We present an analytical formulation to ensure that on-board passengers continuing to the next stop have priority and waiting passengers are loaded on a first-come-first-serve basis. We propose an analytical model that captures the stochastic nature of the transit schedules and in-vehicle travel times due to road conditions, incidents, or adverse weather. We adopt a mean variance approach that can consider the covariance of travel time between links in a space–time graph but still lead to a robust transit network loading procedure when optimal strategies are adopted. The proposed model is formulated as a user equilibrium problem and solved by an MSA-type algorithm. Numerical results are reported to show the effects of supply uncertainties on the travel strategies and departure times of passengers.     

Hierarchical network-based equilibrium model and algorithm for a mixed-traffic urban transport system

Abstract

Many equilibrium models and algorithms based on homogeneous motorized traffic have been devised to model urban transport systems in developed countries, but they are inadequate when it comes to represent mixed-traffic urban transport systems, including automobiles, transit, bicycles, and pedestrians, in developing countries such as China or India. In these cases, traffic flow on a road segment is an aggregated result of travellers' combined mode/route choices and corresponding interactions. Therefore, a special assignment model and algorithm are needed for modeling these distinct behaviors. In this article, the structure of a mixed-traffic urban transport system is analyzed and then expanded and represented using a hierarchical network model based on graph theory. Based on the analysis of travelers' combined mode/route choices, generalized travel cost functions and link impedance functions for different modes are formulated, where the interferences between different modes on the same road segments are taken into account. Due to the ‘asymmetric’ nature of these functions, a variational inequality model is proposed to represent the equilibrium assignment problem in a mixed-traffic urban transport system. The corresponding solution algorithm is also presented. Finally, a numerical example is provided to illustrate the practicality of the proposed model and algorithm.     

Quasi-dynamic traffic assignment with residual point queues incorporating a first order node model

Static traffic assignment models are still widely applied for strategic transport planning purposes in spite of the fact that such models produce implausible traffic flows that exceed link capacities and predict incorrect congestion locations. There have been numerous attempts to constrain link flows to capacity. Capacity constrained models with residual queues are often referred to as quasi-dynamic traffic assignment models. After reviewing the literature, we come to the conclusion that an important piece of the puzzle has been missing so far, namely the inclusion of a first order node model. In this paper we propose a novel path-based static traffic assignment model for finding a stochastic user equilibrium in general transportation networks. This model includes a first order (steady-state) node model that yields more realistic turn capacities, which are then used to determine consistent capacity constrained traffic flows, residual point (vertical) queues (upstream bottleneck links), and path travel times consistent with queuing theory. The route choice part of the model is specified as a variational inequality problem, while the network loading part is formulated as a fixed point problem. Both problems are solved using existing techniques to find a solution. We illustrate the model using hypothetical examples, and also demonstrate feasibility on large-scale networks.     

Schedule-based transit assignment model with vehicle capacity and seat availability

In this paper, we propose a new schedule-based equilibrium transit assignment model that differentiates the discomfort level experienced by sitting and standing passengers. The notion of seat allocation has not been considered explicitly and analytically in previous schedule-based frameworks. The model assumes that passengers use strategies when traveling from their origin to their destination. When loading a vehicle, standing on-board passengers continuing to the next station have priority to get available seats and waiting passengers are loaded on a First-Come-First-Serve (FCFS) principle. The stimulus of a standing passenger to sit increases with his/her remaining journey length and time already spent on-board. When a vehicle is full, passengers unable to board must wait for the next vehicle to arrive. The equilibrium conditions can be stated as a variational inequality involving a vector-valued function of expected strategy costs. To find a solution, we adopt the method of successive averages (MSA) that generates strategies during each iteration by solving a dynamic program. Numerical results are also reported to show the effects of our model on the travel strategies and departure time choices of passengers.     

On the degradation of performance for traffic networks with oblivious users

We consider the problem of characterizing user equilibria and optimal solutions for routing in a given network. We extend the known models by considering users oblivious to congestion in the following sense: While in the typical user equilibrium setting the users follow a strategy that minimizes their individual cost by taking into account the (dynamic) congestion due to the current routing pattern, an oblivious user ignores congestion altogether; instead, he or she decides his routing on the basis of cheapest routes on a network without any flow whatsoever. These cheapest routes can be, for example, the shortest paths in the network without any flow. This model tries to capture the fact that a certain percentage of travelers base their route simply on the distances they observe on a map, without thinking (or knowing, or caring) about the delays experienced on this route due to their fellow travelers. In this work we study the effect of such users using as the measure of network performance its price of anarchy, i.e., the ratio of the total latency experienced by the users (oblivious or not) at equilibrium over the social optimum.     

Development and calibration of a long-distance passenger traffic assignment model

This paper studies the assignment of long-distance passenger traffic on a highway corridor network. First, we propose a traditional model for the long-distance traffic assignment considering interactions with local commuter traffic. It addresses the effect of local networks on highway corridors. An iterative algorithm is developed to solve for the exact solution. Then, to address the potential computational issues that arise therein, a decomposition method is proposed by introducing a new concept of corridor elasticity. An assignment procedure for long-distance passenger traffic is developed accordingly. Numerical tests show that the proposed decomposition method makes significant improvements in computational performance at a small loss of optimality. This decomposition method well approximates the exact assignment from the traditional formulation, especially when the highway corridors are near-saturation. The proposed decomposition method appears practical for application.     

A dynamic traffic assignment model for a continuum transportation system

A predictive continuum dynamic user-optimal (PDUO-C) model is formulated in this study to investigate the dynamic characteristics of traffic flow and the corresponding route-choice behavior of travelers within a region with a dense urban road network. The modeled region is arbitrary in shape with a single central business district (CBD) and travelers continuously distributed over the region. Within this region, the road network is represented as a continuum and travelers patronize a two-dimensional continuum transportation system to travel to the CBD. The PDUO-C model is solved by a promising solution algorithm that includes elements of the finite volume method (FVM), the finite element method (FEM), and the explicit total variation diminishing Runge-Kutta (TVD-RK) time-stepping method. A numerical example is given to demonstrate the utility of the proposed model and the effectiveness of the solution algorithm in solving this PDUO-C problem.     

Formulating the within-day dynamic stochastic traffic assignment problem from a Bayesian perspective

This study proposes a formulation of the within-day dynamic stochastic traffic assignment problem. Considering the stochastic nature of route choice behavior, we treat the solution to the assignment problem as the conditional joint distribution of route traffic, given that the network is in dynamic stochastic user equilibrium. We acquire the conditional joint probability distribution using Bayes’ theorem. A Metropolis–Hastings sampling scheme is developed to estimate the characteristics (e.g., mean and variance) of the route traffic. The proposed formulation has no special requirements for the traffic flow models and user behavior models, and so is easily implemented.     

Modeling stochastic perception error in the mean-excess traffic equilibrium model

In this paper, we extend the α-reliable mean-excess traffic equilibrium (METE) model of Chen and Zhou (Transportation Research Part B 44(4), 2010, 493-513) by explicitly modeling the stochastic perception errors within the travelers’ route choice decision processes. In the METE model, each traveler not only considers a travel time budget for ensuring on-time arrival at a confidence level α, but also accounts for the impact of encountering worse travel times in the (1 − α) quantile of the distribution tail. Furthermore, due to the imperfect knowledge of the travel time variability particularly in congested networks without advanced traveler information systems, the travelers’ route choice decisions are based on the perceived travel time distribution rather than the actual travel time distribution. In order to compute the perceived mean-excess travel time, an approximation method based on moment analysis is developed. It involves using the conditional moment generation function to derive the perceived link travel time, the Cornish-Fisher Asymptotic Expansion to estimate the perceived travel time budget, and the Acerbi and Tasche Approximation to estimate the perceived mean-excess travel time. The proposed stochastic mean-excess traffic equilibrium (SMETE) model is formulated as a variational inequality (VI) problem, and solved by a route-based solution algorithm with the use of the modified alternating direction method. Numerical examples are also provided to illustrate the application of the proposed SMETE model and solution method.