Computation of equilibrium distributions of Markov traffic-assignment models

Markov traffic-assignment models explicitly represent the day-to-day evolving interaction between traffic congestion and drivers' information acquisition and choice processes. Such models can, in principle, be used to investigate traffic flows in stochastic equilibrium, yielding estimates of the equilibrium mean and covariance matrix of link or route traffic flows. However, in general these equilibrium moments cannot be written down in closed form. While Monte Carlo simulations of the assignment process may be used to produce “empirical” estimates, this approach can be extremely computationally expensive if reliable results (relatively free of Monte Carlo error) are to be obtained. In this paper an alternative method of computing the equilibrium distribution is proposed, applicable to the class of Markov models with linear exponential learning filters. Based on asymptotic results, this equilibrium distribution may be approximated by a Gaussian process, meaning that the problem reduces to determining the first two multivariate moments in equilibrium. The first of these moments, the mean flow vector, may be estimated by a conventional traffic-assignment model. The second, the flow covariance matrix, is estimated through various linear approximations, yielding an explicit expression. The proposed approximations are seen to operate well in a number of illustrative examples. The robustness of the approximations (in terms of network input data) is discussed, and shown to be connected with the “volatility” of the traffic assignment process.