Modelling sources of variation in transportation systems: Theoretical foundations of day-to-day dynamic models.

The last 20 years has seen a growing interest in models of transportation networks which explicitly represent the epoch-to-epoch adaptive behaviour of travellers, such as the day-to-day dynamics of drivers'route choices. These models may represent the system as either a stochastic or deterministic process (DP).A body of theoretical literature now exists on this topic, and the purpose of the present paper is to both synthesise and advance this theory. To provide a focus to thework we analyse such models in terms of their ability to capture various contributory sources of variance in transportation systems. Dealing separately with the cases of uncongested and congested networks, we examine how moment-based deterministic dynamical systems may be exactly or approximately derived from some underlying stochastic process (SP). This opens up such problems to the tools of both deterministic dynamical systems (e.g. stability analysis) and SPs (e.g. Monte Carlo methods, statistical inference). In analysing these sources of variation, we also make several new advances to the existing body of theory, in terms of: extending the model assumptions (e.g. randomly varying choice probabilities and stochastic demand); deriving exact, explicit connections between stochastic and DPs in uncongested networks; applying stability analysis in novel ways to moment characterisations; and last, but not least, providing new limit theorems for asymptotic (large demand) analysis of the dynamics of SP models in congested networks