On the Distribution of Multidimensional Reflected Brownian Motion

Let Z = \ Z(t),t > 0\ be a reflected Brownian motion on the K-dimensional nonnegative orthant, with the direction of reflection constant over each boundary surface. Such processes arise in heavy traffic theory for K-station networks of queues. This paper continues our study of the diffusion process Z,with attention restricted to the case $k = 2$ for simplicity. (Most of the results extend directly to higher dimensions; our notation and style of argument are designed to suggest appropriate generalizations for arbitrary K wherever possible.) The backward equation for the transition density function (with boundary and initial conditions), the corresponding forward equation and the equation for the steady-state distribution are all derived informally. Also presented are various calculations relating to steady-state distributions, including a moment formula and the derivation of a condition (involving the drifts and directions of reflection), that we conjecture to be necessary and sufficient for existence of a steady-state. Finally, the relevance of reflected Brownian motion for approximate analysis of queuing networks is reviewed, and some further references on the associated differential equations are given.