A note on the stability of multiclass Markovian queueing networks

TL;DR

This paper proves that in multiclass Markovian queueing networks with unit rate servers, ensuring the average load at each server is below one guarantees stability regardless of scheduling policy or routing, using advanced Markov chain criteria.

Contribution

It establishes a sufficient stability condition for multiclass Markovian networks that holds universally across scheduling policies and routing schemes, extending previous understanding.

Findings

Average load less than one guarantees stability

Stability holds for any work conserving policy

Applicable to class-independent routing

Abstract

In this paper we show that in a multiclass Markovian network with unit rate servers, the condition that the average load $\rho$ at every server is less than unity is indeed sufficient for the stability or positive recurrence for \emph{any} work conserving scheduling policy and \emph{class-independent} routing. We use a variation of the positive recurrence criterion for multidimensional discrete-time Markov chains over countable state spaces due to Rosberg (JAP, Vol.~17, No.~3, 1980) and a monotonicity argument to establish this assertion.