Stability and Control of Piecewise-Deterministic Queueing Systems

We consider a piecewise-deterministic queueing (PDQ) model to study traffic queues due to stochastic capacity fluctuations in transportation facilities. The saturation rate (capacity) of the PDQ model switches between a finite set of values (modes) according to a Markov chain. The inflow to the PDQ is controlled by a state-feedback policy. The main results of this article are stability conditions of PDQs, i.e. conditions under which the distribution of the queue length converges to a unique invariant probability measure. On one hand, a necessary condition for stability is that the average inflow does not exceed the average saturation rate. On the other hand, based on the Foster-Lyapunov criteria, we derive a sufficient condition that requires a bilinear matrix inequality to admit positive solutions and the invariant probability measure to be unique. We also study the rate of convergence for stable PDQs. Furthermore, for PDQs with two modes, a necessary and sufficient condition for stability is available. In addition, we present examples for the stability analysis of feedback control policies for single PDQs as well as a network of two PDQ links in parallel.