An approximation to the invariant measure of the limiting diffusion of G/Ph/n+GI queues in the Halfin-Whitt regime and related asymptotics

TL;DR

This paper introduces a stochastic algorithm using Euler--Maruyama to approximate the invariant measure of the limiting diffusion in complex queueing systems, providing error bounds and asymptotic results.

Contribution

It develops a non-asymptotic error bound for the approximation of the invariant measure of the limiting diffusion in G/Ph/n+GI queues using Stein's method and Euler--Maruyama scheme.

Findings

Established a non-asymptotic error bound for the approximation.

Proved the CLT and MDP for occupation measures of the diffusion and its approximation.

Determined variances using Stein's equation and Malliavin calculus.

Abstract

In this paper, we develop a stochastic algorithm based on the Euler--Maruyama scheme to approximate the invariant measure of the limiting multidimensional diffusion of $G/Ph/n+GI$ queues in the Halfin-Whitt regime. Specifically, we prove a non-asymptotic error bound between the invariant measures of the approximate model from the algorithm and the limiting diffusion. To establish the error bound, we employ the recently developed Stein's method for multi-dimensional diffusions, in which the regularity of Stein's equation developed by Gurvich (2014, 2022) plays a crucial role. We further prove the central limit theorem (CLT) and the moderate deviation principle (MDP) for the occupation measures of the limiting diffusion of $G/Ph/n+GI$ queues and its Euler-Maruyama scheme. In particular, the variances in the CLT and MDP associated with the limiting diffusion are determined by Stein's equation and Malliavin calculus, in which properties of a mollified diffusion and an associated weighted occupation time play a crucial role.