 
Summary: On Large Deviations
for Polyhedral Regulated Brownian Motion
Florin Avram
Heriot Watt University
Edinburgh, Scotland
June 24, 1998
Abstract We show below that the two dimensional RBM variational problem has a
local structure which reduces its solution to a finite choice problem. This is illustrated by
obtaining a complete description of possible tail behaviors for the case of the two dimensional
quadrant.
Keywords Large Deviations, Reflected Brownian Motion, Calculus of Variations
AMS subject classifications: Primary 60K25; Secondary 93E20, 90B15
1 Introduction
The ''regulated Brownian motion'' approximation for queuing networks promoted by J.M.
Harrison has become one of the most useful tools for obtaining insights in the behavior and
optimization of networks. Some of the pioneering papers in this field are [28],[16],[21],[29],
[20], [32], [33], [18], [19]. Unfortunately the stationary distribution of regulated Brownian
motion may be not be obtained explicitly, except for a few particular symmetry cases (see
for example [16]) or cases when it has product form (see [19] or Example 1 below).
There are however important applications in which just the asymptotic behavior of the
