David Watling

User equilibrium traffic network assignment with stochastic travel times and late arrival penalty

Abstract The classical Wardrop User Equilibrium (UE) assignment model assumes traveller choices are based on fixed, known travel times, yet these times are known to be rather variable between trips, both within and between days; typically, then, only mean travel times are represented. Classical Stochastic User Equilibrium (SUE) methods allow the mean travel times to be differentially perceived across the population, yet in a conventional application neither the UE or SUE approach recognises the travel times to be inherently variable. That is to say, there is no recognition that drivers risk arriving late at their destinations, and that this risk may vary across different paths of the network and according to the arrival time flexibility of the traveller. Recent work on incorporating risky elements into the choice process is seen either to neglect the link to the arrival constraints of the traveller, or to apply only to restricted problems with parallel alternatives and inflexible travel time distributions. In the paper, an alternative approach is described based on the ‘schedule delay’ paradigm, penalising late arrival under fixed departure times. The approach allows flexible travel time densities, which can be fitted to actual surveillance data, to be incorporated. A generalised formulation of UE is proposed, termed a Late Arrival Penalised UE (LAPUE). Conditions for the existence and uniqueness of LAPUE solutions are considered, as well as methods for their computation. Two specific travel time models are then considered, one based on multivariate Normal arc travel times, and an extended model to represent arc incidents, based on mixture distributions of multivariate Normals. Several illustrative examples are used to examine the sensitivity of LAPUE solutions to various input parameters, and in particular its comparison with UE predictions. Finally, paths for further research are discussed, including the extension of the model to include elements such as distributed arrival time constraints and penalties.

A Genetic Algorithm Approach for Optimizing Traffic Control Signals Considering Routing

It is well known that coordinated, area-wide traffic signal control offers great potential for improvements in delays, safety, and environmental measures. However, an aspect of this problem commonly neglected in practice is the potentially confounding effect of drivers rerouting in response to changes in travel times on competing routes, brought about by changes in signal timings. This paper considers the problem of optimizing signal green and cycle timings over an urban network, in such a way that optimization anticipates the impact on traffic routing patterns. This is achieved by including a network equilibrium model as a constraint to the optimization. A genetic algorithm (GA) is devised for solving the resulting problem, using total travel time across the network as an illustrative fitness function, and with a widely used traffic simulation-assignment model providing the equilibrium flows. The procedure is applied to a case study of the city of Chester, UK, and the performance of the algorithms is analyzed with respect to parameters of the GA method. Results show a better performance of signal timing as optimized by the GA method as compared to a method that does not consider rerouting. This improvement is found to be more significant with more congested networks, whereas under a relatively mild congestion situation the improvement is not very clear.

Reliable Network Design Problem: Case with Uncertain Demand and Total Travel Time Reliability

In the reliable network design problem (RNDP) the main sources of uncertainty are variable demand and route choice. The objective is to maximize network total travel time reliability (TTR), which is defined as the probability that the network total travel time will be less than a threshold. A framework is presented for a stochastic network model with Poisson-distributed demand and uncertain route choice. The travelers are assumed to choose their routes to minimize their perceived expected travel cost following the probit stochastic user equilibrium condition. An analytical method is presented for approximation of the first and second moments of the total travel time. These moments are then fitted with a log-normal distribution. Then the design problem is tackled in which the analytical derivative of the TTR is derived with the sensitivity analysis of the equilibrated path choice probability. This derivative is then supplied to a gradient-based optimization algorithm to solve the RNDP. The algorithm is tes...

Sensitivity analysis of the probit-based stochastic user equilibrium assignment model

Abstract The probit-based stochastic user equilibrium (SUE) model has the advantage of being able to represent perceptual differences in utility across the driver population, while taking proper account of the natural correlations in these utilities between overlapping routes within the network (which the simpler logit SUE is unable to do). Its main drawback is the potentially heavy computational demands, and this has previously been thought to preclude a consideration of the sensitivity analysis of probit-based SUE, whereby an approximation to changes in the equilibrium solution is deduced as its input parameters (specifically origin/destination (O–D) flows and link cost-flow function parameters) are perturbed. In the present paper, an efficient computational method for performing such an analysis for general networks is described. This approach uses information on SUE path flows, but is not specific to any particular equilibrium solution algorithm. Problems inherent in the consideration of general network topologies are identified, and methods proposed for overcoming them. The paper concludes with an application of the method to a realistic network, and compares the approximate solutions with those obtained by direct estimation methods.

Asymmetric problems and stochastic process models of traffic assignment

There is a spectrum of asymmetric assignment problems to which existing results on uniqueness of equilibrium do not apply. Moreover, multiple equilibria may be seen to exist in a number of simple examples of real-life phenomena, including interactions at priority junctions, responsive traffic signals, multiple user classes, and multi-modal choices. In contrast, recent asymptotic results on the stochastic process approach to traffic assignment establish the existence of a unique, stationary, joint probability distribution of flows under mild conditions, that include problems with multiple equilibria. In studying the simple examples mentioned above, this approach is seen to be a powerful tool in suggesting the relative, asymptotic attractiveness of alternative equilibrium solutions. It is seen that the stationary distribution may have multiple peaks, approximated by the stable equilibria, or a unimodal shape in cases where one of the equilibria dominates. It is seen, however, that the convergence to stationarity may be extremely slow. In Monte Carlo simulations of the process, this gives rise to different types of pseudo-stable behaviour (flows varying in an apparently stable manner, with a mean close to one of the equilibria) for a given problem, and this may prevail for long periods. The starting conditions and random number seed are seen to affect the type of pseudo-stable behaviour over long, but finite, time horizons. The frequency of transitions between these types of behaviour (equivalently, the average sojourn in a locally attractive, pseudo-stable set of states) is seen to be affected by behavioural parameters of the model. Recommendations are given for the application of stochastic process models, in the light of these issues.

The modelling of dynamic route guidance systems

Abstract In attempting to simulate the operation of a dynamic route guidance system, the modelling task is concerned both with the operation of the control system and with the implications this has for modelling driver behaviour (whether or not the driver is receiving information from the controller) and network conditions. The aim of this paper is to provide an overview of the modelling issues that need to be considered when addressing such a problem, and that have been identified by various authors in reports on experimental/survey work and in discussion papers. We achieve this by presenting a structured survey of recent research into dynamic route guidance and highlighting issues that are critical to its effectiveness. It is our belief that the development of a model that adequately represents the performance of a dynamic route guidance system is of the utmost importance to the success of the system. It will not only provide a means for evaluating the potential benefits, but should also provide an essential insight into the most appropriate means for its implementation and improve our understanding of transportation networks.

Modelling sources of variation in transportation systems: theoretical foundations of day-to-day dynamic models

The last 20 years has seen a growing interest in models of transportation networks which explicitly represent the epoch-to-epoch adaptive behaviour of travellers, such as the day-to-day dynamics of drivers’ route choices. These models may represent the system as either a stochastic or deterministic process (DP). A body of theoretical literature now exists on this topic, and the purpose of the present paper is to both synthesise and advance this theory. To provide a focus to the work we analyse such models in terms of their ability to capture various contributory sources of variance in transportation systems. Dealing separately with the cases of uncongested and congested networks, we examine how moment-based deterministic dynamical systems may be exactly or approximately derived from some underlying stochastic process (SP). This opens up such problems to the tools of both deterministic dynamical systems (e.g. stability analysis) and SPs (e.g. Monte Carlo methods, statistical inference). In analysing these sources of variation, we also make several new advances to the existing body of theory, in terms of: extending the model assumptions (e.g. randomly varying choice probabilities and stochastic demand); deriving exact, explicit connections between stochastic and DPs in uncongested networks; applying stability analysis in novel ways to moment characterisations; and last, but not least, providing new limit theorems for asymptotic (large demand) analysis of the dynamics of SP models in congested networks.

Urban traffic network models and dynamic driver information systems

The potential for using advanced technology to influence traffic movement and travel behaviour has generated a massive interest throughout the developed world. In‐vehicle, dynamic driver information and dynamic route guidance systems have probably attracted the most attention, with many research activities and field trials planned or currently in progress. Network simulation models have a key role to play, both in assessing the potential benefits (in order to justify the initial investment) and in determining the best means of implementation. There are many who believe that existing models are deficient in a number of ways for evaluating such strategies, and a number have sought to develop new approaches. However, it takes a great many years for a new network model to be fully developed, tested, validated and accepted in the practical world. This review article has three main objectives. First, a review of the state‐of‐the‐art in existing network models is given, with respect to their ability to meet the ...

A statistical procedure for estimating a mean origin-destination matrix from a partial registration plate survey

This paper addresses the problem of matching partial registration plate data collected at a number of origins and destinations in a traffic system, the aim being to reconstruct the pattern of vehicle movements. The matching algorithm proposed here embraces two of the most appealing features of the methods used in most commercially available packages -- the use of passage times to improve the accuracy of the matching process and the ability to handle data at any number of origins and destinations -- but does so within a statistical framework. The new method is seen to have a number of significant advantages over existing approaches. A model is firstly proposed for the two observation point case and the algorithm derived, in which, alternately, the most probable complete matching consistent with the observed data and the maximum is described. The approach is then generalised to handle data collected at any number of origins and destinations. Finally, the performance of the proposed method is studied in applications to simulated data sets, where the solution yielded by the algorithm is compared with the (known) true matching of the data.

PROBIT-BASED SENSITIVITY ANALYSIS FOR GENERAL TRAFFIC NETWORKS

The probit-based stochastic user-equilibrium (SUE) model is widely recognized as one of the most intuitively robust traffic network assignment techniques. Its advantages include explicit consideration of random perceptual differences in utility across the driver population and the ability to take proper account of the correlations in these utilities between overlapping routes, which the simpler logit SUE is unable to do. Its main drawback is the potentially heavy computational demands of the method, but various efficient approximation methods have been observed to provide reasonable solutions. This computational complexity was previously, however, thought to preclude a consideration of the sensitivity analysis of probit-based SUE, whereby an approximation to changes in the equilibrium solution is deduced as its input parameters are perturbed. An efficient computational method for performing such an analysis in general networks is described. A simple network example is described in detail to illustrate the method, and it is followed by a larger network example that further demonstrates the practical nature of the method.