A variational inequality formulation of the dynamic traffic assignment problem

This paper deals with the modelling of the Dynamic Traffic Assignment Problem (DTAP) for analyzing the day time varying flows of urban transportation networks. During the past two decades, many models have been proposed in the literature, but some of them are based on heuristic concepts, and most of them incorporate important limitations. The paper proposes a Dynamic Traffic Assignment Model which is mainly based on the following assumption: the time spent by a vehicle on a link may be decomposed into a fixed travel time and a waiting time. The fixed travel time is the free flow travel time, after which vehicles are put in the link exit queue, until there is room for them to proceed their trip. The paper shows that this model leads to a network structure (a temporal expansion of the base network, including the queues), and then formulate the DTAP as a network equilibrium problem over the expanded network. The mathematical formulation is achieved through a variational inequality where the variables are the path (or link) flows over the space-time expanded network. Numerical results show that the model may handle large networks, so it may be used in practice to analyze the traffic congestion moves in real cities, as well in space (physical links) as in day time.