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

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 universal, valid in any load condition from lightly loaded to heavily loaded.

Keywords

• Stein’s method
• steady-state
• diffusion approximation
• convergence rates
• Erlang-A
• Erlang-C