The Finite Element Method for Computing the Stationary Distribution of an SRBM in a Hypercube with Applications to Finite Buffer Queueing Networks

This paper proposes an algorithm, referred to as BNA/FM (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 BNA/FM algorithm is based on finite element method and an extension of a generic algorithm developed by Dai and Harrison (1991). It uses piecewise polynomials to form an approximate subspace of an infinite dimensional functional space. The BNA/FM algorithm is shown to produce good estimates for stationary probabilities, in addition to stationary moments. This is in contrast to BNA/SM (Brownian network analyzer with spectral method) of Dai and Harrison (1991), where global polynomials are used to form the approximate subspace and it sometime fails to produce meaningful estimates of these stationary probabilities. Extensive computational experiences from our implementation are reported that may be useful for future numerical research on SRBMs. A three-station tandem network with finite buffers are presented to illustrate the effectiveness of the Brownian approximation model and our BNA/FM algorithm.