Higher-Order Approximations of Sojourn Times in M/G/1 Queues via Stein's Method

TL;DR

This paper develops higher-order approximations for the stationary sojourn time in M/G/1 queues under heavy traffic using Stein's method, improving the classical exponential limit by matching more moments.

Contribution

It introduces a Stein's method-based approach for higher-order expansions of sojourn time distributions, providing explicit error bounds and systematic improvements.

Findings

Error bounds decay as a high-order power of the slack parameter

Approximation accuracy improves with more moment matching

Results apply to bounds in Wasserstein and Zolotarev metrics

Abstract

We study the stationary sojourn time distribution in an M/G/1 queue operating under heavy traffic. It is known that the sojourn time converges to an exponential distribution in the limit. Our focus is on obtaining pre-asymptotic, higher-order approximations that go beyond the classical exponential limit. Using Stein's method, we develop an approach based on higher-order expansions of the generator of the underlying Markov process. The key technical step is to represent higher-order derivatives in terms of lower-order ones and control the resulting error via derivative bounds of the Stein equation. Under suitable moment-matching conditions on the service distribution, we show that the approximation error decays as a high-order power of the slack parameter $\varepsilon=1-\rho$. Error bounds are established in the Zolotarev metric, which further imply bounds on the Wasserstein distance as well as the moments. Our results demonstrate that the accuracy of the exponential approximation can be systematically improved by matching progressively more moments of the service distribution.