Construction of a stationary FIFO queue with impatient customers

TL;DR

This paper investigates the stability and stationarity conditions of FIFO queues with impatient customers, providing new criteria for regenerative behavior and stationary workload existence under general assumptions.

Contribution

It introduces novel conditions for queue regeneration and stationary workload existence in queues with impatience, using stochastic recursions and Palm calculus.

Findings

Conditions for queue regeneration under ergodic assumptions

Existence of stationary workload in FIFO queues with impatience

Use of stochastic comparison and Palm calculus techniques

Abstract

In this paper, we study the stability of queues with impatient customers. Under general stationary ergodic assumptions, we first provide some conditions for such a queue to be regenerative (i.e. to empty a.s. an infinite number of times). In the particular case of a single server operating in First in, First out, we prove the existence (in some cases, on an enlarged probability space) of a stationary workload. This is done by studying stochastic recursions under the Palm settings, and by stochastic comparison of stochastic recursions.