An analytical model for traffic delays and the dynamic user equilibrium problem

In urban transportation planning, it has become critical (1) to determine the travel time of a traveler and how it is affected by congestion, and (2) to understand how traffic distributes in a transportation network. In the first part of this paper, we derive an analytical function of travel time, based on the theory of kinematic waves. This travel-time function integrates the traffic dynamics as well as the effects of shocks. Numerical examples demonstrate the quality of the analytical function, in comparison with simulated travel times. In the second part of this paper, we incorporate the travel-time model within a dynamic user equilibrium setting. We prove that the travel-time function is continuous and strictly monotone if the flow varies smoothly. We illustrate how the model applies to solve a large network assignment problem through a numerical example.