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.