Modelling network travel time reliability under stochastic demand

A technique is proposed for estimating the probability distribution of total network travel time, in the light of normal day-to-day variations in the travel demand matrix over a road traffic network. A solution method is proposed, based on a single run of a standard traffic assignment model, which operates in two stages. In stage one, moments of the total travel time distribution are computed by an analytic method, based on the multivariate moments of the link flow vector. In stage two, a flexible family of density functions is fitted to these moments. It is discussed how the resulting distribution may in practice be used to characterise unreliability. Illustrative numerical tests are reported on a simple network, where the method is seen to provide a means for identifying sensitive or vulnerable links, and for examining the impact on network reliability of changes to link capacities. Computational considerations for large networks, and directions for further research, are discussed

A Link-Based Day-to-Day Traffic Assignment Model

Existing day-to-day traffic assignment models are all built upon path flow variables. This paper demonstrates two essential shortcomings of these path-based models. One is that their application requires a given initial path flow pattern, which is typically unidentifiable, i.e., mathematically nonunique and practically unobservable. In particular, we show that, for the path-based models, different initial path flow patterns constituting the same link flow pattern generally gives different day-to-day link flow evolutions. The other shortcoming of the path-based models is the path-overlapping problem. That is, the path-based models ignore the interdependence among paths and thus can give very unreasonable results for networks with paths overlapping with each other. These two path-based problems exist for most (if not all) deterministic day-to-day dynamics whose fixed points are the classic Wardrop user equilibrium. To avoid the two path-based problems, we propose a day-to-day traffic assignment model that directly deals with link flow variables. Our link-based model captures travelers\u27 cost-minimization behavior in their path finding as well as their inertia. The fixed point of our link-based dynamical system is the classic Wardrop user equilibrium

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\u27 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-35W Bridge. The results show that our model can to a reasonable level reproduce the observed phenomenon of irreversible network change

A General Stochastic Process for Day-to-Day Dynamic Traffic Assignment: Formulation, Asymptotic Behaviour, and Stability Analysis

This paper presents a general modelling approach to day-to-day dynamic assignment to a congested network through discrete-time stochastic and deterministic process models including an explicit modelling of users’ habit as a part of route choice behaviour, through an exponential smoothing filter, and of their memory of network conditions on past days, through a moving average or an exponentially smoothing filter. An asymptotic analysis of the mean process is carried out to provide a better insight. Results of such analyses are also used for deriving conditions, about values of the system parameters, assuring that the mean process is dissipative and/or converges to some kind of attractor. Numerical small examples are also provided in order to illustrate the theoretical results obtained

Stochastic network equilibrium under stochastic demand

A generalisation of the conventional stochastic user equilibrium (SUE) model is developed in order to represent day-to-day variability in traffic flows due to stochastic variation in both a) the inter-zonal trip demand matrix, and b) the route choice proportions conditional on the demands. The equilibrated variables in this new problem are the link flow means and covariance matrix. A heuristic solution algorithm is proposed, based on the solution of a sequence of SUE subproblems. Numerical results are reported from the application of this technique to a realistic network, under the assumption of probit-based choice probabilities. In these tests, as the level of demand variability is increased (but the mean demand held fixed), the link flow variances predicted by the proposed model are seen to increase, but the effect on mean flows is relatively small. The increased variation in flows is, however, seen to have an inflationary effect on one of the prime indicators of network congestion, mean total travel cost

A Second Order Stochastic Network Equilibrium Model, II: Solution Method and Numerical Experiments

Real traffic networks typically exhibit considerable day-to-day variations in traffic flows and travel times, yet these variations are commonly neglected in network performance models. Recently, two alternative theoretical approaches were proposed for incorporating stochastic flow variation in the equilibration of route choices: the stochastic process (SP) approach (Cantarella and Cascetta 1995) and the second order generalized stochastic user equilibrium (GSUE(2)) model (Watling 2002). The theoretical basis of the two approaches is contrasted, and the paper goes on to present a heuristic solution method for the GSUE(2) model, and two alternative simulation methods for the SP model, each applicable to the realistic case of probit-based choice probabilities. These solution methods are then applied to two realistic networks. Factors affecting convergence and reproducibility are first identified, followed by comparisons of the GSUE(2) and SP predictions. It is seen that a quasi-periodic behaviour commonly arises in the SP model, with the predictions radically different from the GSUE(2) model. However, by modifying the link performance functions in the overcapacity regime, the nature of the SP solution changes, and for a memory filter based on a large number of days' experience, its moments are seen to be approximated by those of the GSUE(2) model. Implications for the application of these models are discussed

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

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