Stein's Method for Steady-State Approximations: Error Bounds and Engineering Solutions

Abstract

Heavy traffic limits of queueing systems have been studied in the literature using fluid and diffusion limits. Recently, a new method called the 'Drift Method' has been developed to study these limits. In the drift method, a function of the queue lengths is picked and its drift is set to zero in steady-state, to obtain bounds on the steady-state queue lengths that are tight in the heavy-traffic limit. The key is to establish an appropriate notion of state-space collapse in terms of steady-state moments of weighted queue length differences, and use this state-space collapse result when setting the drift equal to zero. These moment bounds involved in state space collapse are also obtained by drift arguments similar to the well-known Foster-Lyapunov theorem. We will apply the methodology to study routing, scheduling, and other resource allocation problems that arise in data centers and cloud computing systems.