A multi-dimensional SRBM: geometric views of its product form stationary distribution

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 12d(d − 1) two-dimensional SRBMs, each of which is determined by an ellipse and two rays. This characterization contrasts with the algebraic condition of Harrison and Williams [12]. A d-station tandem queue example is presented to illustrate how the product form can be geometrically understood. Drawing the two-dimensional results in [1, 7], we discuss potential optimal paths for a variational problem associated with the tandem queue.