Decomposable stationary distribution of a multidimensional SRBM

TL;DR

This paper characterizes when the stationary distribution of a multidimensional SRBM can be decomposed into lower-dimensional distributions, providing conditions and methods to compute these distributions for specific classes.

Contribution

It offers a novel characterization of decomposability for SRBM stationary distributions and identifies necessary and sufficient conditions for certain classes of SRBMs.

Findings

Decomposable stationary distributions are products of lower-dimensional SRBM distributions.

Conditions for decomposability are established for SRBMs with feed-forward queueing network structures.

Stationary distributions of lower-dimensional SRBMs can be used to derive the original distribution under decomposability.

Abstract

We call a multidimensional distribution to be decomposable with respect to a partition of two sets of coordinates if the original distribution is the product of the marginal distributions associated with these two sets. We focus on the stationary distribution of a multidimensional semimartingale reflecting Brownian motion (SRBM) on a nonnegative orthant. An SRBM is uniquely determined (in distribution) by its data that consists of a covariance matrix, a drift vector, and a reflection matrix. Assume that the stationary distribution of an SRBM exists. We first characterize two marginal distributions under the decomposability assumption. We prove that they are the stationary distributions of some lower dimensional SRBMs. We also identify the data for these lower dimensional SRBMs. Thus, under the decomposability assumption, we can obtain the stationary distribution of the original SRBM by computing those of the lower dimensional ones. However, this characterization of the marginal distributions is not sufficient for the decomposability. So, we next consider necessary and sufficient conditions for the decomposability. We obtain those conditions for several classes of SRBMs. These classes include SRBMs arising from Brownian models of queueing networks that have two sets of stations with feed-forward routing between these two sets. This work is motivated by applications of SRBMs and geometric interpretations of the product form stationary distributions.