More paradoxes in the equilibrium assignment problem

The sensitivity of travel costs to changes in input flows in the Wardrop equilibrium problem is studied. Examples are given showing that both origin to destination and global travel costs may decrease as a result of an increase in input flows. Other examples show that, in the two-mode equilibrium assignment problem transit origin to destination travel costs may decrease as a result of an increase in automobile input flows.     

Private road competition and equilibrium with traffic equilibrium constraints

Toll road competition is one of the important issues under a build-operate-transfer (BOT) scheme, which is being encountered nowadays in many cities. When there are two or more competing firms and each firm operates a competitive toll road, their profits are interrelated due to the competitors' choices and demand inter-dependence in the network. In this paper we develop game-theoretic approaches to the study of the road network, on which multiple toll roads are operated by competitive private firms. The strategic interactions and market equilibria among the private firms are analyzed both in determining their supply (road capacity) and price (toll level) over the network. The toll road competition problems in general traffic equilibrium networks are formulated as an equilibrium program with equilibrium constraints or bi-level variational inequalities. Heuristic solution methods are proposed and their convergences are demonstrated with simple network examples. It is shown that private pricing and competition can be both profitable and welfare-improving.     

Some developments in equilibrium traffic assignment

A network optimization problem is formulated which yields a probabilistic equilibrated traffic assignment incorporating congestion effects and which as a special case, reduces to a user optimized equilibrium solution. In the resulting model, path choice is determined by a logit formula in which path costs are functions of the assigned flows. The article also demonstrates the similarity between some fixed demand incremental methods of traffic assignment and the minimization problem associated with computing the user equilibrium assignment.     

Two alternative definitions of traffic equilibrium

The paper shows that, under reasonable conditions, if there are junction or modal interactions then a Wardrop equilibrium exists but a user—optimised flow may well not.     

Comparative tests of multimodal traffic equilibrium methods

In this study we proceeded to test the relative efficiencies of the two main computational techniques now available for calculating the traffic equilibrium in multimodal networks, namely, the relaxation and projection methods. Since both of these methods solve at each step a mathematical programming problem, we first set out to test the Frank-Wolfe algorithm and the Dafermos-Sparrow algorithm and found the latter to be the more efficient algorithm. As expected, this was also the case when these algorithms were used to solve the mathematical programming problem at each step of the relaxation method. We then investigated how different versions of the projection method enhance its performance. Subsequently, we proceeded to our main goal to compare the best projection method with the relaxation method. We tested multimodal networks with three different classes of monotone travel cost functions and found that the form of the travel cost functions affects the performance of the two basic techniques available for computing the multimodal equilibrium.     

Lane‐based optimization of traffic equilibrium settings for area traffic control

Area traffic control is an important element in Intelligent Transportation System (ITS). This paper extends the lane‐based optimization method to a traffic equilibrium network, which improves the operational performance of signal‐controlled network. We formulate a decomposition approach to simultaneously optimize the lane markings and signal settings for a signal‐controlled network that comprises two levels of optimization. At the junction level, the lane markings, control sequence, and other aspects of the signal settings are optimized for individual junctions, whereas at the network level, the group‐based signal settings are optimized to take into account the re‐routing characteristics of travelers and signal coordination effects that are based on a TRANSYT traffic model, which is a well‐known procedure for evaluating the performance of signal‐controlled networks. We use a numerical example to demonstrate the effectiveness of the proposed methodology.     

An iterative group-based signal optimization scheme for traffic equilibrium networks

This paper presents an iterative scheme for a combined signal optimization and assignment problem, using a traffic model from the well-known procedure TRANSYT. The signal settings are optimized by means of a group-based technique, in which the signal timings are specified by the common cycle time, the start time and duration of the period of right of way for each signal group in the network. The optimization problem was formulated as an integer program and solved by a set of heuristics. Given the optimized signal settings determined from the group-based technique, a path-based assignment algorithm is employed to obtain the equilibrium traffic pattern using the sensitivity information for TRANSYT model and a Frank-Wolf method. Based on the equilibrium flow pattern, the group-based optimization algorithm is then used to determine a better set of signal timings. The procedure is repeated until certain convergence criteria are satisfied. A numerical example is employed to demonstrate the benefits obtained from this iterative scheme. Encouraging results are obtained.     

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.     

General stochastic user equilibrium traffic assignment problem with link capacity constraints

This paper addresses a general stochastic user equilibrium (SUE) traffic assignment problem with link capacity constraints. It first proposes a novel linearly constrained minimization model in terms of path flows and then shows that any of its local minimums satisfies the generalized SUE conditions. As the objective function of the proposed model involves path‐specific delay functions without explicit mathematical expressions, its Lagrangian dual formulation is analyzed. On the basis of the Lagrangian dual model, a convergent Lagrangian dual method with a predetermined step size sequence is developed. This solution method merely invokes a subroutine at each iteration to perform a conventional SUE traffic assignment excluding link capacity constraints. Finally, two numerical examples are used to illustrate the proposed model and solution method.     

An equilibrium algorithm for the spatial aggregation problem of traffic assignment

Present traffic assignment methods require that all possible origins and destinations of trips taking place within a study area be represented as if they were taking place to and from a small set of points or centroids. Each centroid is supposed to represent the location of all trip-ends within a given zone, and this necessarily misrepresents points located at the edges of the zone.In order to alleviate this problem (which we refer to as the spatial aggregation problem) one could use smaller zones and more centroids, but existing traffic assignment algorithms cannot efficiently handle many centroids.This paper introduces an algorithm procedure which is designed to handle a substantially larger number of centroids. In the paper that follows, the technique is further developed to take into account a continuous distribution of population.     

A kinematic wave theory of lane-changing traffic flow

Frequent lane-changes in highway merging, diverging, and weaving areas could disrupt traffic flow and, even worse, lead to accidents. In this paper, we propose a simple model for studying bottleneck effects of lane-changing traffic and aggregate traffic dynamics of a roadway with lane-changing areas. Based on the observation that, when changing its lane, a vehicle affects traffic on both its current and target lanes, we propose to capture such lateral interactions by introducing a new lane-changing intensity variable. With a modified fundamental diagram, we are able to study the impacts of lane-changing traffic on overall traffic flow. In addition, the corresponding traffic dynamics can be described with a simple kinematic wave model. For a location-dependent lane-changing intensity variable, we discuss kinematic wave solutions of the Riemann problem of the new model and introduce a supply–demand method for its numerical solutions. With both theoretical and empirical analysis, we demonstrate that lane-changes could have significant bottleneck effects on overall traffic flow. In the future, we will be interested in studying lane-changing intensities for different road geometries, locations, on-ramp/off-ramp flows, as well as traffic conditions. The new modeling framework could be helpful for developing ramp-metering and other lane management strategies to mitigate the bottleneck effects of lane-changes.     

Properties of a traffic control policy which ensure the existence of a traffic equilibrium consistent with the policy

We consider a network with interactions and capacity constraints at each junction. We give conditions on the interactions and constraints which, if satisfied at each separate junction, ensure that any feasible assignment problem has an equilibrium solution. Two illustrative examples are provided; the first arises naturally and does not satisfy our conditions, while the second does satisfy our conditions but is somewhat unnatural.     

Dynamic prediction of traffic volume through Kalman filtering theory

Two models employing Kalman filtering theory are proposed for predicting short-term traffic volume. Prediction parameters are improved using the most recent prediction error and better volume prediction on a link is achieved by taking into account data from a number of links. Based on data collected from a street network in Nagoya City, average prediction error is found to be less than 9% and maximum error less than 30%. The new models perform substantially (up to 80%) better than UTCS-2.     

The impact of speed limits on traffic equilibrium and system performance in networks

Speed limits are usually imposed on roads in an attempt to enhance safety and sometimes serve the purpose of reducing fuel consumption and vehicular emissions as well. Most previous studies up to date focus on investigation of the effects of speed limits from a local perspective, while network-wide traffic reallocation effects are overlooked. This paper makes the first attempt to investigate how a link-specific speed limit law reallocates traffic flow in an equilibrium manner at a macroscopic network level. We find that, although the link travel time–flow relationship is altered after a speed limit is imposed, the standard traffic assignment method still applies. With the commonly adopted assumptions, the uniqueness of link travel times at user equilibrium (UE) remains valid, and the UE flows on links with non-binding speed limits are still unique. The UE flows on other links with binding speed limits may not be unique but can be explicitly characterized by a polyhedron or a linear system of equalities and inequalities. Furthermore, taking into account the traffic reallocation effects of speed limits, we compare the capability of speed limits and road pricing for decentralizing desirable network flow patterns. Although from a different perspective for regulating traffic flows with a different mechanism, a speed limit law may play the same role as a toll charge scheme and perform better than some negative (rebate) toll schemes under certain conditions for network flow management.     

An application of shock wave theory to traffic signal control

A real time control policy minimizing total intersection delays subject to queue length constraints at an isolated signalized intersection is developed in this paper. The policy is derived from a new traffic model which describes the simultaneous evolution of queue lengths of two conflicting traffic streams, controlled by a traffic light, in both time and space. The model is based on the examination of shock waves generated upstream of the stop lines by the intermittent service of traffic at the signal. The proposed policy was tested against the existing pre-timed control policy at a high volume intersection and it was found superior, especially when demands increase well above the saturation level.     

Joint optimization of freight facility location and pavement infrastructure rehabilitation under network traffic equilibrium

Establishment of industry facilities often induces heavy vehicle traffic that exacerbates congestion and pavement deterioration in the neighboring highway network. While planning facility locations and land use developments, it is important to take into account the routing of freight vehicles, the impact on public traffic, as well as the planning of pavement rehabilitation. This paper presents an integrated facility location model that simultaneously considers traffic routing under congestion and pavement rehabilitation under deterioration. The objective is to minimize the total cost due to facility investment, transportation cost including traffic delay, and pavement life-cycle costs. Building upon analytical results on optimal pavement rehabilitation, the problem is formulated into a bi-level mixed-integer non-linear program (MINLP), with facility location, freight shipment routing and pavement rehabilitation decisions in the upper level and traffic equilibrium in the lower level. This problem is then reformulated into an equivalent single-level MINLP based on Karush–Kuhn–Tucker (KKT) conditions and approximation by piece-wise linear functions. Numerical experiments on hypothetical and empirical network examples are conducted to show performance of the proposed algorithm and to draw managerial insights.     

On a link-based day-to-day traffic assignment model

In this paper, we perform a rigorous analysis on a link-based day-to-day traffic assignment model recently proposed in He et al. (2010). Several properties, including the invariance set and the constrained stability, of this dynamical process are established. An extension of the model to the asymmetric case is investigated and the stability result is also established under slightly more restrictive assumptions. Numerical experiments are conducted to demonstrate the findings.     

On the macroscopic stability of freeway traffic

A simple model of traffic flow is used to analyze the spatio-temporal distribution of flow and density on closed-loop homogeneous freeways with many ramps, which produce inflows and allow outflows. As we would expect, if the on-ramp demand is space-independent then this distribution tends toward uniformity in space if the freeway is either: (i) uncongested; or (ii) congested with queues on its on-ramps and enough inflow to cause the average freeway density to increase with time. In all other cases, however, including any recovery phase of a rush hour where the freeway’s average density declines, the distribution of flow and density quickly becomes uneven. This happens even under conditions of perfect symmetry, where the percentage of vehicles exiting at every off ramp is the same. The flow-density deviations from the average are shown to grow exponentially in time and propagate backwards in space with a fixed wave speed. A consequence of this type of instability is that, during recovery, gaps of uncongested traffic will quickly appear in the unevenly congested stream, reducing average flow. This extends the duration of recovery and invariably creates clockwise hysteresis loops on scatter-plots of average system flow vs. density during any rush hour that oversaturates the freeway. All these effects are quantified with formulas and verified with simulations. Some have been observed in real networks. In a more practical vein, it is also shown that the negative effects of instability diminish (i.e., freeway flows increase) if (a) some drivers choose to exit the freeway prematurely when it is too congested and/or (b) freeway access is regulated in a certain traffic-responsive way. These two findings could be used to improve the algorithms behind VMS displays for driver guidance (finding a), and on-ramp metering rates (finding b).