  Reflected diffusions in polyhedral domains are commonly used as approximate
models for stochastic processing networks in heavy traffic. Stationary
distributions of such models give useful information on the steady state
performance of the corresponding stochastic networks and thus it is important
to develop reliable and efficient algorithms for numerical computation of such
distributions. In this work we propose and analyze a Monte-Carlo scheme based
on an Euler type discretization of the reflected stochastic differential
equation using a single sequence of time discretization steps which decrease to
zero as time approaches infinity. Appropriately weighted empirical measures
constructed from the simulated discretized reflected diffusion are proposed as
approximations for the invariant probability measure of the true diffusion
model. Almost sure consistency results are established that in particular show
that weighted averages of polynomially growing continuous functionals evaluated
on the discretized simulated system converge a.s. to the corresponding
integrals with respect to the invariant measure. Proofs rely on constructing
suitable Lyapunov functions for tightness and uniform integrability and
characterizing almost sure limit points through an extension of Echeverria's
criteria for reflected diffusions. Regularity properties of the underlying
Skorohod problems play a key role in the proofs. Rates of convergence for
suitable families of test functions are also obtained. A key advantage of
Monte-Carlo methods is the ease of implementation, particularly for high
dimensional problems. A numerical example of a eight dimensional Skorohod
problem is presented to illustrate the applicability of the approach.
