Tight matrices and heavy traffic steady state convergence in queueing networks

TL;DR

This paper investigates the conditions under which the reflection matrix of a semimartingale reflecting Brownian motion (SRBM) is tight, ensuring the convergence of queueing network stationary distributions to SRBM in heavy traffic, with specific results for 2D and higher dimensions.

Contribution

It provides necessary and sufficient conditions for the tightness of the reflection matrix in 2D SRBMs and sufficient conditions in higher dimensions, applied to queueing network diffusion approximations.

Findings

Tightness of R is always achieved in 2D for completely-S SRBMs.

R is tight for reentrant lines with LBFS discipline in higher dimensions.

R is not always tight for reentrant lines with FBFS discipline.

Abstract

We are interested to prove that the stationary distribution of a multiclass queueing network converges to the stationary distribution of a semimartingale reflecting Brownian motion (SRBM) in heavy traffic. A key condition for this convergence is that the sequence of the pre-limit stationary distributions under appropriate scaling is tight. In Braverman et al.(2025), a sufficient condition for this tightness is introduced in the term of the reflection matrix $R$ of the SRBM, which is coined for $R$ to be ``tight''. In this paper, we study how we can verify this tightness of $R$ of an SRBM. For a $2$-dimensional SRBM, we give necessary and sufficient conditions for $R$ to be tight, while, for a general dimension, we only give sufficient conditions. We then apply these results to the SRBMs arising from the diffusion approximations of multiclass queueing networks with static buffer priority service disciplines that are studied in Braverman et al.(2025). It is shown that $R$ is always tight for this network with two stations if $R$ is completely-$\sr{S}$. For the case of more than two stations, it is shown that $R$ is tight for reentrant lines with last-buffer-first-service (LBFS) discipline, but it is not always tight for reentrant line with first-buffer-first-service (FBFS) discipline.