Many-Sources Large Deviations for Max-Weight Scheduling

TL;DR

This paper establishes a large deviations principle for the transient and stationary workloads in multi-queue systems under max-weight scheduling, enabling asymptotic analysis of buffer overflow probabilities.

Contribution

It introduces a novel large deviations framework for multi-queue max-weight systems, including the first LDP for stationary workloads under certain conditions.

Findings

LDP for transient workload established using Garcia's contraction principle.

LDP for stationary workload derived for simplex rate-region with additional assumptions.

Explicit rate function for stationary workload related to finite-horizon workload rate functions.

Abstract

In this paper, a many-sources large deviations principle (LDP) for the transient workload of a multi-queue single-server system is established where the service rates are chosen from a compact, convex and coordinate-convex rate region and where the service discipline is the max-weight policy. Under the assumption that the arrival processes satisfy a many-sources LDP, this is accomplished by employing Garcia's extended contraction principle that is applicable to quasi-continuous mappings. For the simplex rate-region, an LDP for the stationary workload is also established under the additional requirements that the scheduling policy be work-conserving and that the arrival processes satisfy certain mixing conditions. The LDP results can be used to calculate asymptotic buffer overflow probabilities accounting for the multiplexing gain, when the arrival process is an average of \emph{i.i.d.} processes. The rate function for the stationary workload is expressed in term of the rate functions of the finite-horizon workloads when the arrival processes have \emph{i.i.d.} increments.