Reflected Brownian motion in the quadrant: tail behavior of the stationary distribution

Abstract

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 they are the only restrictions imposed on the process data in our study. Under those assumptions, a large deviations principle (LDP) is conjectured for the stationary distribution of Z, and we recapitulate the cases for which it has been rigorously justified. For sufficiently regular sets B, the LDP says that the stationary probability of xB decays exponentially as x→∞, and the asymptotic decay rate is the minimum value achieved by a certain function I(⋅) over the set B. Avram, Dai and Hasenbein  (Queueing Syst.: Theory Appl. 37, 259–289, 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.