Coupling in the queue with impatience: case of several servers

TL;DR

This paper develops a mathematical framework for analyzing a multi-server queue with impatient customers, establishing conditions for the existence and uniqueness of a stationary state using stochastic comparison and Renovation theory.

Contribution

It introduces explicit constructions and sufficient conditions for the stability and stationary distribution of multi-server queues with impatience, extending existing methods.

Findings

Established a sufficient condition for the existence of a unique stationary state.

Proved the existence of stationary states under relaxed conditions using extension techniques.

Provided explicit stochastic comparison methods for workload sequences.

Abstract

We present the explicit construction of a stable queue with several servers and impatient customers, under stationary ergodic assumptions. Using a stochastic comparison of the (multivariate) workload sequence with two monotonic stochastic recursions, we propose a sufficient condition of existence of a unique stationary state of the system using Renovation theory. Whenever this condition is relaxed we use extension techniques to prove the existence of a stationary state in some cases.