A Linear Programming Algorithm for Computing the Stationary Distribution of Semimartingale Reflected Brownian Motion

This paper proposes a linear programming algorithm for computing the stationary distribution of semimartingale reflected Brownian motion (SRBM), which arises as an approximation of certain queueing networks operating in heavy-traffic. Our algorithm is based on a Basic Adjoint Relationship (BAR) which characterizes the stationary distribution. Approximating the state space with a finite grid of points and using a finite set of “test” functions, the BAR reduces to a set of linear equations that can be solved using standard linear programming techniques. As the set of test functions increases in complexity and the grid becomes finer, the sequence of stationary distribution estimates which arise as solutions to the algorithm converges, in a suitable sense, to the stationary distribution of the SRBM. The algorithm is seen to produce good estimates of the stationary moments as well as the entire distribution. Extension to settings with parameter uncertainty, and to Ornstein-Uhlenbeck-type diffusions arising in the many-servers heavy traffic regime, are discussed as well. Preliminary Draft Copy – Not for Distribution Short Title: Computing the stationary distribution of SRBM