The Waiting Time Distribution in Poisson-Driven Deterministic Systems

We consider systems with Poisson arrivals and a deterministic delay structure given by some increasing sequence. We characterize the distribution of waiting times in the transient regime, and describe effective ways to evaluate it. We also compute the stationary waiting time distribution in the case where the deterministic structure becomes ultimately periodic. This analysis is strongly connected to the theory of linear $(\max,+)$ systems and has the same range of applications: we give examples from stochastic Petri Net theory and stochastic task graph theory. It also applies in general to single server queues with Poisson arrivals and known service times, in particular to the $M/D/1$ queue with periodic service durations and the $E/D/1$ queue.