Large deviations of the steady-state distribution of reflected processes with applications to queueing systems

Abstract

We consider a Skorohod map \mathbf{R} which takes paths in \mathbb{R}^n to paths which stay in the positive orthant \mathbb{R}_+^n. We let \mathcal{S} be the domain of definition of \mathbf{R}. A convex and lower semi-continuous function \lambda \dvtx \mathbb{R}^n \rightarrow [0,\infty] and a set A \subset \mathbb{R}_+^n are given. We are concernedl-2ptr with the calculation of the infimum of the value \int_0^t \lambda(\dot{\omega}(s))\,{\rm d}s for t \geq 0 and absolutely continuous \omega \in \mathcal{S} subject to the conditions \omega(0) = 0 and \mathbf{R}(\omega)(t) \in A. We show that such minimization problems characterize large deviation asymptotics of tail probabilities of the steady-state distribution of certain reflected processes. We approximate the infimum by a sequence of finite-dimensional minimization problems. This approximation allows to formulate an algorithm for the calculation of the infimum and to derive analytical bounds for its value. Several applications are discussed including large deviations of generalized processor sharing and large deviations of heavily loaded queueing networks.