Explicit Characterization of Stability Region for Stationary Multi-Queue Multi-Server Systems

TL;DR

This paper explicitly characterizes the stability region of multi-queue multi-server systems with stationary channels and arrivals, providing finite linear inequalities, conditions for stability, and bounds on delay, advancing understanding of system capacity and performance.

Contribution

It introduces a finite set of linear inequalities to describe the stability region of MQMS systems and derives necessary and sufficient stability conditions for stationary arrivals.

Findings

Stability region is a polytope defined by finite linear inequalities.

Derived delay bounds for the Maximum Weight server policy.

Stability region can sometimes be characterized by nonlinear inequalities.

Abstract

In this paper, we characterize the network stability region (capacity region) of multi-queue multi-server (MQMS) queueing systems with stationary channel distribution and stationary arrival processes. The stability region is specified by a finite set of linear inequalities. We first show that the stability region is a polytope characterized by the finite set of its facet defining hyperplanes. We explicitly determine the coefficients of the linear inequalities describing the facet defining hyperplanes of the stability region polytope. We further derive the necessary and sufficient conditions for the stability of the system for general arrival processes with finite first and second moments. For the case of stationary arrival processes, the derived conditions characterize the system stability region. Furthermore, we obtain an upper bound for the average queueing delay of Maximum Weight (MW) server allocation policy which has been shown in the literature to be a throughput optimal policy for MQMS systems. Using a similar approach, we can characterize the stability region for a fluid model MQMS system. However, the stability region of the fluid model system is described by an infinite number of linear inequalities since in this case the stability region is a convex surface. We present an example where we show that in some cases depending on the channel distribution, the stability region can be characterized by a finite set of non-linear inequalities instead of an infinite number of linear inequalities.