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 $$d$$d-dimensional semimartingale reflecting Brownian motion (SRBM) that lives in the nonnegative orthant. The $$d$$d-dimensional SRBM data can be equivalently specified by $$d+1$$d+1 geometric objects: an ellipse and $$d$$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 $$d$$d-dimensional problem to $$\frac{1}{2}d(d-1)$$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 (Ann Probab 15:115–137, 1987b). A $$d$$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.

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