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 \(\frac{1}{2}d(d-1)\) two-dimensional SRBMs, each of which is determined by an ellipse and two rays. This characterization contrasts with the algebraic condition of Harrison and Williams (Ann Probab 15:115–137, 1987b). 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 Avram et al. (Queueing Syst 37:259–289, 2001), Dai and Miyazawa (Queueing Syst 74:181–217, 2013), we discuss potential optimal paths for a variational problem associated with the three-station tandem queue.