A Numerical Approach to Stability of Multiclass Queueing Networks

TL;DR

This paper introduces a simulation-based numerical method to determine the stability region of multi-class queueing networks, addressing the lack of analytical conditions for stability in complex multi-class scenarios.

Contribution

The paper develops a minimal set of conditions under which a numerical method can reliably identify the stability region of multi-class queueing networks, extending monotonicity-based approaches.

Findings

Method accurately identifies stability regions in various network configurations.

Numerical experiments demonstrate effectiveness for both single and multi-class networks.

Provides practical tools for stability analysis where analytical solutions are unavailable.

Abstract

The Multi-class Queueing Network (McQN) arises as a natural multi-class extension of the traditional (single-class) Jackson network. In a single-class network subcriticality (i.e. subunitary nominal workload at every station) entails stability, but this is no longer sufficient when jobs/customers of different classes (i.e. with different service requirements and/or routing scheme) visit the same server; therefore, analytical conditions for stability of McQNs are lacking, in general. In this note we design a numerical (simulation-based) method for determining the stability region of a McQN, in terms of arrival rate(s). Our method exploits certain (stochastic) monotonicity properties enjoyed by the associated Markovian queue-configuration process. Stochastic monotonicity is a quite common feature of queueing models and can be easily established in the single-class framework (Jackson networks); recently, also for a wide class of McQNs, including first-come-first-serve (FCFS) networks, monotonicity properties have been established. Here, we provide a minimal set of conditions under which the method performs correctly. Eventually, we illustrate the use of our numerical method by presenting a set of numerical experiments, covering both single and multi-class networks.