Asymptotic approximations of transient behaviour for day-to-day traffic models

We consider a wide class of stochastic process traffic assignment models that capture the day-to-day evolving interaction between traffic congestion and drivers’ information acquisition and choice processes. Such models provide a description of not only transient change and ‘steady’ behaviour, but also represent additional variability that occurs through probabilistic descriptions. They are therefore highly suited to modelling both the disturbance and subsequent ‘drift’ of networks that are subject to some systematic change, be that a road closure or capacity reduction, new policy measure or general change in demand patterns. In this paper we derive analytic results to probabilistically capture the nature of the transient effects following such a systematic change. This can be thought of as understanding what happens as a system moves from varying about one equilibrium state to varying about a new equilibrium state. The results capture analytically the changes over time in descriptors of the system, in terms of link flow means, variances and covariances. Formally, the analytic results hold asymptotically as approximations, as we imagine demand increasing in tandem with capacities; however, our interest is in general cases where such tandem increases do not occur, and so we provide conditions under which our approximations are likely to work well. Numerical results of applying the methods are reported on several examples. The quality of the approximations is assessed through comparisons with Monte Carlo simulations from the true underlying process.