A new family of positive recurrent semimartingale reflecting Brownian motions in an orthant

Abstract

Abstract Semimartingale reflecting Brownian motions (SRBMs) are diffusion processes, which arise as approximations for open d-station queueing networks of various kinds. The data for such a process are a drift vector θ, a nonsingular d × d {d\times d} covariance matrix Δ, and a d × d {d\times d} reflection matrix R. The state space is the d-dimensional nonnegative orthant, in the interior of which the processes evolve according to a Brownian motions, and that reflect against the boundary in a specified manner. A standard problem is to determine under what conditions the process is positive recurrent. Necessary and sufficient conditions are formulated for some classes of reflection matrices and in two- and three-dimensional cases, but not more. In this work, we identify a new family of reflection matrices R for which the process is positive recurrent if and only if the drift θ ∈ Γ ̊ {\theta\in\mathring{\Gamma}} , where Γ ̊ {\mathring{\Gamma}} is the interior of the convex wedge generated by the opposite column vectors of R.