Understand the behavior of queueing networks in heavy tra c is very important
due to its importance in evaluating the network performance in related applications.
However, in many cases, the stationary distributions of such networks are
intractable. Based on di usion limits of queueing networks, we can use Re
ected
Brownian Motion (RBM) processes as reasonable approximations. As such, we are
interested in obtaining the stationary distribution of RBM. Unfortunately, these distributions
are also in most cases intractable. However, the tail behavior (large deviations)
of RBM may give insight into the stationary distribution. Assuming that
a large deviations principle holds, we need only solve the corresponding variational
problem to obtain the rate function. Our research is mainly focused on how to solve
variational problems in the case of rotationally symmetric (RS) data.
The contribution of this dissertation primarily consists of three parts. In the rst
part we give out the speci c stability condition for the RBM in the octant in the RS
vi
case. Although the general stability conditions for RBM in the octant has been derived
previously, we simplify these conditions for the case we consider. In the second
part we prove that there are only two types of possible solutions for the variational
problem. In the last part, we provide a simple computational method. Also we give
an example under which a spiral path is the optimal solution.