The finite element method for computing the stationary distribution of a semimartingale reflecting Brownian motion in a hypercube with applications to finite buffer queuing networks

This paper proposes an algorithm, referred to as BNAfm (Brownian network analyzer with finite element method), for computing the stationary distribution of a semimartingale reflecting Brownian motion (SRBM) in a hypercube. The SRBM serves as an approximate model of queueing networks with finite buffers. Our BNAfm algorithm is based on the finite element method and an extension of a generic algorithm developed by Dai and Harrison. It uses piecewise polynomials to form an approximate subspace of an infinite-dimensional functional space. The BNAfm algorithm is shown to produce good estimates for stationary probabilities, in addition to stationary moments. This is in contrast to the BNAsm algorithm (Brownian network analyzer with spectral method) of Dai and Harrison, which uses global polynomials to form the approximate subspace and which sometimes fails to produce meaningful estimates of these stationary probabilities. Extensive computational experiences from our implementation are reported, which may be useful for future numerical research on SRBMs. A three-station tandem network with finite buffers is presented to illustrate the effectiveness of the Brownian approximation model and our BNAfm algorithm.