Stability of a Queue Fed by Scheduled Traffic at Critical Loading

TL;DR

This paper investigates the stability of a critically loaded single-server queue with scheduled arrivals perturbed by randomness, revealing conditions under which the queue remains stable despite critical loading and showing asymmetry in stability under time reversal.

Contribution

It provides a necessary and sufficient stability condition for queues with scheduled traffic and finite mean perturbations, highlighting non-reversibility of stability.

Findings

Queue can be stable at critical load with scheduled traffic.

Stability depends on a specific non-reversible criterion.

Stability differs when traffic process is reversed in time.

Abstract

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 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 identify a necessary and sufficient condition for stability when the perturbations have finite mean. Perhaps surprisingly, the 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.