Stein’s Method for Steady-state Diffusion Approximations: An Introduction through the Erlang-A and Erlang-C Models

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.