Asymptotically Tight Steady-State Queue Length Bounds Implied By Drift Conditions

TL;DR

This paper develops a method using Lyapunov drift conditions to derive tight steady-state queue length bounds in heavy traffic, capturing resource pooling effects and demonstrating near-optimality in wireless networks.

Contribution

It introduces a novel approach linking Lyapunov drift conditions with state-space collapse to obtain asymptotically tight queue bounds.

Findings

Bounds are tight in the heavy-traffic limit.

Method applies to wireless networks under MaxWeight scheduling.

Demonstrates heavy-traffic optimality of the bounds.

Abstract

The Foster-Lyapunov theorem and its variants serve as the primary tools for studying the stability of queueing systems. In addition, it is well known that setting the drift of the Lyapunov function equal to zero in steady-state provides bounds on the expected queue lengths. However, such bounds are often very loose due to the fact that they fail to capture resource pooling effects. The main contribution of this paper is to show that the approach of "setting the drift of a Lyapunov function equal to zero" can be used to obtain bounds on the steady-state queue lengths which are tight in the heavy-traffic limit. The key is to establish an appropriate notion of state-space collapse in terms of steady-state moments of weighted queue length differences, and use this state-space collapse result when setting the Lyapunov drift equal to zero. As an application of the methodology, we prove the steady-state equivalent of the heavy-traffic optimality result of Stolyar for wireless networks operating under the MaxWeight scheduling policy.