A Continuous Macroscopic Model

In this paper the Dynamic User Equilibrium is formulated and solved as a fixed point problem in terms of time-continuous real valued temporal profiles of arc inflows and arc performances, where travel demand is specified both through a deterministic and a stochastic implicit path choice model. By extending to the dynamic case the concept of Network Loading Map, yielding arc flows for given demand flows consistently with certain arc performances, the equilibrium is formulated without introducing the Dynamic Network Loading as a subproblem. The proposed assignment algorithm reflects this approach as the temporal consistency of the flow pattern through the arc performance model is attained only jointly with the equilibrium. In addiction, a new dynamic shortest path algorithm is presented to evaluate the Network Loading Map in the deterministic case, which can also be utilized within a Montecarlo simulation to evaluate it in the stochastic case. The main feature of this algorithm is its capability to handle a continuous travel time pattern by approximating the temporal profiles with piece-wise linear functions of time defined on long time intervals.