Brownian motions reflected in the orthant
often arise as diffusion approximations
to queueing networks, and their tail probabilities
often serve as estimates for stationary buffer
overflow probabilities in the corresponding networks.
Under certain conditions, the tail probabilities of
stationary reflected Brownian motion in the
orthant are known to have certain variational representations.
We will discuss why these representations
are hard to analyze, and derive new methods for
studying these variational problems.
In particular we will relate this variational problem
to another class of variational problems with constrained dynamics,
which arise in the analyis of optimal stability of
queueing networks.
We show how explicit solutions can be derived for these
problems in certain cases.