The emergence of stochastic user equilibria in day-to-day traffic models

Stochastic user equilibrium (SUE) is a commonly used deterministic model of traffic assignment. It is well known that SUE provides an approximation to the stationary mean of a wide class of Markov day-to-day traffic assignment models. However, as is shown in this article, it is generally impossible to find any time homogeneous random process models for discrete traffic in which SUE precisely matches the stationary mean. In response, we develop a time-inhomogeneous model with this property. That model relies on a rather unnatural description of route choice utility. We therefore propose a second time-inhomogeneous model with a more standard type of utility function, where the stationary mean equals a generalized version of SUE.& nbsp; & nbsp; & nbsp;While the time-inhomogeneous models are largely of theoretical interest, they also provide highly serviceable approximations to standard types of day-to-day traffic models in which travellers update their route choice utilities relatively slowly in response to recent events. For such models, the generalized form of SUE provides a closer approximation to the stationary mean than classical SUE, particularly when traffic demands are small.