Stochastic and fluid index policies for resource allocation problems

Abstract

We develop a unifying framework to obtain efficient index policies for restless multi-armed bandit problems with birth-and-death state evolution. This is a broad class of stochastic resource allocation problems whose objective is to determine efficient policies to share resources among competing projects. In a seminal work, Whittle developed a methodology to derive well-performing (Whittle's) index policies that are obtained by solving a relaxed version of the original problem. Our first main contribution is the derivation of a closed-form expression for Whittle's index as a function of the steady-state probabilities. It can be efficiently calculated, however, it requires several technical conditions to be verified, and in addition, it does not provide qualitative insights into Whittle's index. We therefore formulate a fluid version of the relaxed optimization problem and in our second main contribution we develop a fluid index policy. The latter does provide qualitative insights and is close to Whittle's index. The applicability of our approach is illustrated by two important problems: optimal class selection and optimal load balancing. Allowing state-dependent capacities we can model important phenomena: e.g. power-aware server-farms and opportunistic scheduling in wireless systems. Numerical simulations show that Whittle's index and our fluid index policy are both nearly optimal.