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 $d$-dimensional semimartingale reflecting Brownian motion (SRBM) that lives in the nonnegative orthant. The $d$-dimensional SRBM data can be equivalently specified by $d+1$ geometric objects: an ellipse and $d$ 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 $d$-dimensional problem to $\frac{1}{2}d(d-1)$ two-dimensional SRBMs, each of which is determined by an ellipse and two rays. This characterization contrasts with the algebraic condition of [14]. A $d$-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.