Convergence in a continuous dynamic queueing model for traffic networks

The paper considers a dynamic traffic assignment model with deterministic queueing and inelastic demand for each origin-destination (OD) pair in the network. Two types of time-varying behaviour are modelled. First, within-day time is regarded as a continuous variable. During each day, flows propagating through routes connecting OD pairs are represented by non-negative, essentially bounded and measurable functions. Also, day-to-day time is (slightly surprisingly) modelled as if it were continuous. The day-to-day dynamical system that is adopted is derived naturally from the usual user equilibrium condition. The route cost is shown to be a Lipschitz continuous function of route flow in the single bottleneck per route case. Global convergence to equilibrium is shown to be guaranteed when the route cost vector is a non-decreasing (monotone) function of the route flow vector. In the single bottleneck per route case, the route cost function is shown to be a monotone function of the route flow if the bottleneck capacities are all non-decreasing as functions of within-day time. Monotonicity of the route cost function is also shown to hold when each bottleneck has at most one route passing through it. (c) 2005 Elsevier Ltd. All rights reserved.