A pathwise comparison result for parallel queues

TL;DR

This paper develops a framework for pathwise comparison of parallel queue systems, demonstrating the benefits of additional servers under certain policies and exploring the limitations of these comparisons in more complex settings.

Contribution

It introduces a new pathwise comparison framework for parallel queues under stationary ergodic assumptions, extending existing semi-orderings and analyzing their applicability.

Findings

More servers improve system performance under FCFS and JSW policies.

The comparison results hold for certain policies but not for semi-cyclic allocation.

The framework is based on recursive representations and generalized semi-orderings.

Abstract

We introduce the appropriate framework for pathwise comparison of multiple server queues under general stationary ergodic assumptions. We show in what sense it is better to have more servers for a system under FCFS ('First Come, First Served') or equivalently, more queues in a system of parallel queues under the JSW ('Join the Shortest Workload') allocation policy. This comparison result is based on the recursive representation of Kiefer and Wolfowitz, and on a non-mass conservative generalization of the Schur-Convex semi-ordering. We also show that the latter result does not hold true in general, for the larger class of systems applying the semi-cyclic allocation policy introduced by Scheller-Wolf in \cite{SW03}.