The join-the-shortest-queue system in the Halfin-Whitt regime: rates of convergence to the diffusion limit

TL;DR

This paper proves that the steady-state distribution of the join-the-shortest-queue system converges to its diffusion limit at a rate of at least 1/√n in the Halfin-Whitt regime, using Stein's method.

Contribution

It introduces a novel application of Stein's method with generator comparison to analyze convergence rates in a complex queueing system.

Findings

Convergence rate of at least 1/√n established

Stein's method effectively applied to high-dimensional queueing models

Provides a framework for analyzing similar stochastic systems

Abstract

We show that the steady-state distribution of the join-the-shortest-queue (JSQ) system converges, in the Halfin-Whitt regime, to its diffusion limit at a rate of at least $1/\sqrt{n}$, where $n$ is the number of servers. Our proof uses Stein's method and, specifically, the recently proposed prelimit generator comparison approach. The JSQ system is non-trivial, high-dimensional, and has a state-space collapse component, and our analysis may serve as a helpful example to readers wishing to apply the approach to their own setting.