A new class of doubly stochastic day-to-day dynamic traffic assignment models

Real-life systems are known to exhibit considerable day-to-day variability. A better understanding of such variability has increasing policy-relevance in the context of network reliability assessment and the design of intelligent transport systems. Conventional equilibrium models are ill-suited, because deterministic models such as these do not account for any kind of variability. At best, these types of models are restricted to finding a steady state of the mean flow patterns, they cannot capture the variance in flows as well. A more suitable alternative are stochastic day-to-day dynamic models studied by Cascetta in Trans Res 23:1–17, (1989). These types of traffic assignment models represent the traffic flows via a Markov process, where the current route flows are modelled as a function of previous traffic conditions. Day-to-day dynamic models differ from equilibrium models in that day-to-day changes in the system are modelled dependent on the time and thus allow for a far wider representation of traveller behaviour. However, to some degree they still suffer from some of the limitations of equilibrium analyses, in that while they permit variation they are still wedded to the concept of ‘stationarity’. In this paper, we show how these Markovian day-to-day dynamic traffic assignment models can be extended by replacing a subset of the fixed parameters in the Markov model with random processes. The resulting models are analogous to Cox process models. They are conditionally non-stationary given any realization of the parameter processes. We present numerical examples that demonstrate that this new class of doubly stochastic day-to-day traffic assignment models can indeed reproduce features such as the heteroscedasticity of traffic flows observed in real-life settings.