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

作者：Dai, J. G.; Miyazawa, Masakiyo; Wu, Jian^*

来源：Queueing Systems, 2014, 78(4): 313-335.

DOI：10.1007/s11134-014-9411-0

• Semimartingale reflecting Brownian motion
• Variational problem
• Skew symmetry condition
• Queueing networks
• Diffusion approximations

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 Harrison and Williams (Ann Probab 15:115-137, 1987b). 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 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.

• 出版日期2014-12