Steady-state analysis of reflected Brownian motions: Characterization, numerical methods and queueing applications

This dissertation is concerned with multidimensional diffusion processes that arise as approximate models of queueing networks. To be specific, we consider two classes of semimartingale reflected Brownian motions (SRBM's), each with polyhedral state space. For one class the state space is a two-dimensional rectangle, and for the other class it is the general d-dimensional nonnegative orthant ;Motivated by the applications in queueing theory, our focus is on steady-state analysis of SRBM, which involves three tasks: (a) determining when a stationary distribution exists; (b) developing an analytical characterization of the stationary distribution; and (c) computing the stationary distribution from that characterization.;To make practical use of SRBM's as approximate models of queueing networks, one needs practical methods for determining stationary distributions, and it is very unlikely that general analytical solutions will ever be found. We describe an approach to computation of stationary distributions that seems to be widely applicable. That approach gives rise to a family of algorithms, and we investigate one version of the algorithm. Under one mild assumption, we are able to provide a full proof of the algorithm's convergence. We compare the numerical results from our algorithm with known analytical results for SRBM, and also use the algorithm to estimate the performance measures of several illustrative open queueing networks. All the numerical comparisons show that our method gives reasonably accurate estimates and the convergence is relatively fast.;Our ultimate goal is to implement this approach in a general routine for computing the stationary distribution of SRBM in an arbitrary polyhedral state space. The plan is to combine that routine with appropriate "front end" and "back end" program modules to form a software package, tentatively called QNET, for analysis of complex queueing networks. (Abstract shortened with permission of author.).;SRBM in a rectangle has been identified as an approximate model of a two-station queueing network with finite storage space at each station. We show how SRBM in a rectangle can be constructed by means of localization. The process is shown to be unique in law, and therefore to be a Feller continuous strong Markov process.