Multidimensional reflected Brownian motions, also called regulated Brownian motions or simply RBM's, arise as approximate models of queueing networks. Thus the stationary distributions of these diffusion processes are of interest for steady-state analysis of the corresponding queueing systems. This paper considers two-dimensional semimartingale RBM's with rectangular state space, which include the RBM's that serve as approximate models of finite queues in tandem. The stationary distribution of such an RBM is uniquely characterized by a certain basic adjoint relationship, and an algorithm is proposed for numerical solution of that relationship. We cannot offer a general proof of convergence, but the algorithm has been coded and applied to special cases where the stationary distribution can be determined by other means; the computed solutions agree closely with previously known results and convergence is reasonably fast. Our current computer code is specific to two-dimensional rectangles, but the basic logic of the algorithm applies equally well to any semimartingale RBM with bounded polyhedral state space, regardless of dimension. To demonstrate the role of the algorithm in practical performance analysis, we use it to derive numerical performance estimates for a particular example of finite queues in tandem; our numerical estimates of both the throughput loss rate and the average queue lengths are found to agree with simulated values to within about five percent.
"Steady-State Analysis of RBM in a Rectangle: Numerical Methods and A Queueing Application." Ann. Appl. Probab. 1 (1) 16 - 35, February, 1991. https://doi.org/10.1214/aoap/1177005979