Technical Note–Stability of a Queue Fed by Scheduled Traffic at Critical Loading

Consider the workload process for a single server queue with deterministic service times in which customers arrive according to a scheduled traffic process. A scheduled arrival sequence is one in which customers are scheduled to arrive at constant interarrival times, but each customer’s actual arrival time is perturbed from her scheduled arrival time by a random perturbation. In this paper, we consider a critically loaded queue in which the service rate equals the arrival rate. Unlike a queue fed by renewal traffic, this queue can be stable even in the presence of critical loading. We show that for finite mean perturbations, a necessary and sufficient condition for stability is when the positive part of the perturbation has bounded support, with no requirement on the negative part of the perturbation. Perhaps surprisingly, this criterion is not reversible, in the sense that such a queue can be stable for a scheduled traffic process in forward time, but unstable for the time-reversal of the same traffic process.