Diffusion approximation error for queueing systems with general primitives

TL;DR

This paper analyzes the steady-state diffusion approximation error in queueing systems with general primitives, decomposing it into interior and boundary parts, and extends Stein's method to handle the complexities of PDMPs.

Contribution

It introduces a novel generator comparison approach for PDMPs and provides a detailed error decomposition for queueing systems with general primitives.

Findings

Interior error bounds depend on low-order moments.

Boundary error bounds require model-specific insights.

The method extends Stein's approach to discontinuous processes.

Abstract

We investigate the steady-state diffusion-approximation error for continuous-time queueing systems with generally distributed primitives. Across four canonical systems -- the $G/G/1$ and $G/M/\infty$ queues, the join-the-shortest-queue system, and a two-station tandem queue -- a common picture emerges: the error decomposes into interior and boundary terms. The former are simpler to handle and can be bounded using only low-order moments of the system's primitives -- when the approximation error is measured using the Wasserstein distance, three moments suffice. The boundary terms are inherently more delicate: while crude bounds are easy to obtain, sharper (e.g., order optimal) bounds require deeper, model specific, insights. Methodologically, we extend the generator comparison approach of Stein's method to piecewise-deterministic Markov processes (PDMPs). The discontinuous nature of the PDMP at jump times necessitates using the basic adjoint relationship (BAR), instead of the infinitesimal generator, to characterize the stationary distribution. A second-order Taylor expansion of the BAR jump terms, coupled with a Palm-inversion step that converts event-averaged quantities into time averages, yields the candidate diffusion generator and a transparent interior/boundary error decomposition. In parallel, we show how the prelimit generator approach -- working with the Poisson equation of the queueing system instead of the diffusion process -- offers a promising avenue for bounding the challenging boundary terms.