Heavy traffic approximation for the stationary distribution of a generalized Jackson network: the BAR approach

TL;DR

This paper introduces a novel BAR-based approach to justify the heavy traffic steady-state approximation of generalized Jackson networks, offering a more natural alternative to the traditional limit interchange method.

Contribution

It proposes using the basic adjoint relationship (BAR) directly for steady-state analysis, expanding the toolkit for studying stochastic processing networks.

Findings

BAR approach effectively justifies heavy traffic steady-state approximation

Method potentially applicable to other complex queueing networks

Provides a more natural framework than limit interchange method

Abstract

In the seminal paper of Gamarnik and Zeevi (2006), the authors justify the steady-state diffusion approximation of a generalized Jackson network (GJN) in heavy traffic. Their approach involves the so-called limit interchange argument, which has since become a popular tool employed by many others who study diffusion approximations. In this paper we illustrate a novel approach by using it to justify the steady-state approximation of a GJN in heavy traffic. Our approach involves working directly with the basic adjoint relationship (BAR), an integral equation that characterizes the stationary distribution of a Markov process. As we will show, the BAR approach is a more natural choice than the limit interchange approach for justifying steady-state approximations, and can potentially be applied to the study of other stochastic processing networks such as multiclass queueing networks.