Construction of a stationary queue with impatient customers

TL;DR

This paper investigates the stability and stationary behavior of queues with impatient customers, providing conditions for regeneration and stationary workload existence under general assumptions.

Contribution

It introduces new conditions for queue regeneration and proves stationary workload existence in single-server FIFO queues with impatient customers.

Findings

Conditions for queue regeneration under ergodic assumptions

Existence of stationary workload in FIFO queues with impatience

Use of stochastic recursions and Palm calculus for analysis

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 a non-monotonic stochastic recursion under the Palm settings, and by stochastic comparison of stochastic recursions.