Diffusion approximation for stationary analysis of queues and their networks: A review

TL;DR

This paper reviews diffusion approximations in queueing networks, focusing on their theoretical foundations, limitations in higher dimensions, and the challenges in extracting useful information from the limiting processes.

Contribution

It provides a comprehensive overview of diffusion approximations, including technical background and a new perspective on their stationary distributions and applicability.

Findings

Diffusion approximations are well-developed for low-dimensional cases.

Higher-dimensional cases remain intractable and less understood.

The paper proposes a new approach focusing on stationary distributions.

Abstract

Diffusion processes have been widely used for approximations in the queueing theory. There are different types of diffusion approximations. Among them, we are interested in those obtained through limits of a sequence of models which describe queueing networks. Such a limit is typically obtained by the weak convergence of either stochastic processes or stationary distributions. We already have nice reviews and text books for them. However, this area is still actively studied, and it seems getting hard to have a comprehensive overview because mathematical results are highly technical. We try to fill this gap presenting technical background. Although those diffusion approximations have been well developed, there remains a big problem, which is difficulty to get useful information from the limiting diffusion processes. Their state spaces are multidimensional, whose dimension corresponds to the number of nodes for a single-class case and the number of customer types for a multi-class case. We now have a better view for the two dimensional case, but still know very little about the higher dimensional case. This intractability is somehow against a spirit of diffusion approximation. This motivates us to reconsider diffusion approximation from scratch. For this, we highlight the stationary distributions, and make clear a mechanism to produce diffusion approximations.