Reflected Brownian motion in the quadrant: Tail behavior of the stationary distribution
J. M. Harrison and J. J. Hasenbein
Let Z be a two-dimensional Brownian motion
confined to the non-negative quadrant by oblique reflection at the
boundary. Such processes arise in applied probability as diffusion
approximations for two-station queueing networks. The parameters
of Z are a drift vector, a covariance matrix, and a ``direction
of reflection'' for each of the quadrant's two boundary rays.
Necessary and sufficient conditions are known for Z to be a
positive recurrent semimartingale, and those conditions are
restated here in a novel form; they are the only restrictions
imposed on the process data in our study. Under those minimal
assumptions, a large deviations principle (LDP) is known to be
valid for the stationary distribution of Z. For sufficiently
regular sets B, the LDP says that the stationary probability of
xB decays exponentially as x tends to infinity, and the
asymptotic decay rate is the minimum value achieved by a certain
function I(. ) over the set B. Avram, Dai and
Hasenbein (2001)
provided a complete and explicit solution for
the large deviations rate function I(.). In this paper we
re-express their solution in a simplified form, showing along the
way that the computation of I(.) reduces to a relatively
simple problem of least-cost travel between a point and a line.