A Lyapunov view on positive Harris recurrence of multiclass queueing networks

TL;DR

This paper introduces a Lyapunov-based method to determine positive Harris recurrence of Markov processes in multiclass queueing networks by linking fluid model stability with Lyapunov functions.

Contribution

It provides a new approach to establish positive Harris recurrence using explicit Lyapunov functions derived from fluid models.

Findings

Fluid model stability implies positive Harris recurrence.

Explicit Lyapunov functions can be constructed from fluid models.

The method offers a practical criterion for queueing network stability.

Abstract

This paper addresses the question when the underlying Markov process of a multiclass queueing network is positive Harris recurrent. It is well-known that stability of the fluid limit model is a sufficient condition for this. Hence, stability of fluid (limit) models is of vital interest. Recently, it has been shown that if the fluid model satisfies certain properties, it is stable if and only if there exists a Lyapunov function. In this paper a new method is provided to conclude that the underlying Markov process is positive Harris recurrent if the fluid model is stable by using explicitly the Lyapunov function of the fluid model.