A multi-dimensional SRBM: Geometric views of its product form stationary distribution
Abstract
We present a geometric interpretation of a product form stationary distribution for a -dimensional semimartingale reflecting Brownian motion (SRBM) that lives in the nonnegative orthant. The -dimensional SRBM data can be equivalently specified by geometric objects: an ellipse and rays. Using these geometric objects, we establish necessary and sufficient conditions for characterizing product form stationary distribution. The key idea in the characterization is that we decompose the -dimensional problem to two-dimensional SRBMs, each of which is determined by an ellipse and two rays. This characterization contrasts with the algebraic condition of [14]. A -station tandem queue example is presented to illustrate how the product form can be obtained using our characterization. Drawing the two-dimensional results in [1,7], we discuss potential optimal paths for a variational problem associated with the three-station tandem queue. Except Appendix D, the rest of this paper is almost identical to the QUESTA paper with the same title.
Keywords
Cite
@article{arxiv.1312.1758,
title = {A multi-dimensional SRBM: Geometric views of its product form stationary distribution},
author = {J. G. Dai and Masakiyo Miyazawa and Jian Wu},
journal= {arXiv preprint arXiv:1312.1758},
year = {2015}
}