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

Abstract

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 non-negative orthant R+. SRBM in a rectangle has been identified as an approximate model of a two-station queueing network with finite storage space at each station. Until now, however, the foundational characteristics of the process have not been rigorously established. Building on previous work by Varadhan and Williams [53] and by Taylor and Williams [50], 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. Taylor and Williams [50] have proved the analogous foundational result for SRBM in the non-negative orthant R+, which arises as an approximate model of a d-station open queueing network with infinite storage space at every station. 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. With regard to (a), we give a sufficient condition for the existence of a stationary distribution in terms of Liapunov functions. With regard to (b), for a special class of SRBM’s in an orthant, Harrison and Williams [26] showed that the stationary distribution must satisfy a weak form of an adjoint linear elliptic partial differential equation with oblique derivative boundary conditions, which they called the basic adjoint relationship (BAR). They further conjectured that (BAR) characterizes the stationary distribution. We give two proofs of their conjecture. For an SRBM in a rectangle, using Echeverria’s Theorem [10], we give a direct proof of their

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