Stein's method for steady-state diffusion approximations: an introduction through the Erlang-A and Erlang-C models

TL;DR

This paper introduces the Stein method for steady-state diffusion approximations, demonstrating universal error bounds in Wasserstein and Kolmogorov distances for Erlang models across all load conditions.

Contribution

It applies the Stein method to Erlang-A and Erlang-C models, providing universal error bounds for diffusion approximations in steady-state distributions.

Findings

Wasserstein and Kolmogorov distances decrease at rate 1/√R.

Error bounds are valid across all load conditions.

Stein method effectively quantifies steady-state diffusion approximation errors.

Abstract

This paper provides an introduction to the Stein method framework in the context of steady-state diffusion approximations. The framework consists of three components: the Poisson equation and gradient bounds, generator coupling, and moment bounds. Working in the setting of the Erlang-A and Erlang-C models, we prove that both Wasserstein and Kolmogorov distances between the stationary distribution of a normalized customer count process, and that of an appropriately defined diffusion process decrease at a rate of $1/\sqrt{R}$, where $R$ is the offered load. Futhermore, these error bounds are \emph{universal}, valid in any load condition from lightly loaded to heavily loaded.