From Local to Global Stability in Stochastic Processing Networks through Quadratic Lyapunov Functions

TL;DR

This paper introduces a simple, randomized scheduling policy for stochastic processing networks that guarantees global stability using quadratic Lyapunov functions, applicable across various network types.

Contribution

It presents a novel framework linking local quadratic Lyapunov functions to global stability, enabling broad applicability to different network models.

Findings

Network stability is achieved under the proposed policy.

The framework applies to queueing, switch, and wireless networks.

Construction of global Lyapunov functions from local ones is demonstrated.

Abstract

We construct a generic, simple, and efficient scheduling policy for stochastic processing networks, and provide a general framework to establish its stability. Our policy is randomized and prioritized: with high probability it prioritizes jobs which have been least routed through the network. We show that the network is globally stable under this policy if there exists an appropriate quadratic local Lyapunov function that provides a negative drift with respect to nominal loads at servers. Applying this generic framework, we obtain stability results for our policy in many important examples of stochastic processing networks: open multiclass queueing networks, parallel server networks, networks of input-queued switches, and a variety of wireless network models with interference constraints. Our main novelty is the construction of an appropriate global Lyapunov function from quadratic local Lyapunov functions, which we believe to be of broader interest.