On the stability of a class of non-monotonic systems of parallel queues

TL;DR

This paper analyzes the stability of a class of non-monotonic parallel queue systems with a novel customer assignment policy, providing conditions for the existence of stationary workloads and revealing a system splitting in heavy traffic.

Contribution

It introduces a new stability analysis for non-monotonic queue systems with a unique customer routing policy, extending classical models.

Findings

Conditions for the existence of stationary workloads are established.

The system can split into a loss system and an FCFS system in heavy traffic.

An original sufficient condition for stationary workload existence in loss systems is provided.

Abstract

We investigate, under general stationary ergodic assumptions, the stability of systems of $S$ parallel queues in which any incoming customer joins the queue of the server having the $p+1$-th shortest workload ($p < S$), or a free server if any. This change in the allocation policy makes the analysis much more challenging with respect to the classical FCFS model with $S$ servers, as it leads to the non-monotonicity of the underlying stochastic recursion. We provide sufficient conditions of existence of a stationary workload, which indicate a "splitting" of the system in heavy traffic, into a loss system of $p$ servers plus a FCFS system of $S-p$ servers. To prove this result, we show {\em en route} an original sufficient condition for existence and uniqueness of a stationary workload for a multiple-server loss system.