| Abstract | Unreliable link capacities cause a significant amount of delay in transportation networks. In this paper, we propose a novel approach to studying the traffic queues due to capacity-reducing events under a class of control policies. First, we propose a Piecewise-Deterministic Queueing (PDQ) model in which the link saturation rates switch between a finite set of values (modes) according to a Markov chain, which captures the occurrence and clearance of capacity-reducing events. Second, we derive results on the stability of PDQ networks, i.e. when the joint distribution of the link queue sizes converges to a unique invariant probability measure. On one hand, a necessary condition for stability is that the average inflow to each link is less than the link's effective capacity. On the other hand, a sufficient condition is that a set of bilinear matrix inequalities involving model parameters and the control policy has a feasible solution. Third, we provide an analytical characterization of the steady-state distribution of bimodal PDQ systems, which enables us to obtain the optimal static/mode-dependent routing policy for bimodal PDQ networks by solving a convex min-cost problem. |
Stability and Control of Piecewise-Deterministic Queueing Systems