Convergence to Equilibrium in Dynamic Traffic Networks when Route Cost Is Decay Monotone

This paper addresses the issue of convergence to equilibrium in dynamic traffic assignment. Within-day time is considered to be a continuous variable, so that traffic flows and costs are functions of within-day time. Flow propagates along routes connecting origin-destination (OD) pairs with the demand for travel between each OD pair considered to be rigid (fixed from day to day although it can vary within day). Day-to-day time is also modelled as continuous with the day-to-day dynamical system derived naturally from the usual dynamical user equilibrium (DUE) condition. This paper focuses on the bottleneck model, which has deterministic vertical queueing at bottleneck link exits when flow exceeds capacity. A new property called decay monotonicity is introduced. The link delay (and hence link cost) function is shown to be a decay monotone function of link flow provided that the link capacity is continuously differentiable and positive. In a restricted version of the single bottleneck per route case, it is shown that link cost decay monotonicity implies route cost decay monotonicity. Decay monotonicity of the route cost function is shown to be sufficient for convergence to equilibrium of the dynamical system.