INDEXABILITY AND OPTIMAL INDEX POLICIES FOR A CLASS OF REINITIALISING RESTLESS BANDITS

Abstract

Motivated by a class of Partially Observable Markov Decision Processes with application in surveillance systems in which a set of imperfectly observed state processes is to be inferred from a subset of available observations through a Bayesian approach, we formulate and analyze a special family of multi-armed restless bandit problems. We consider the problem of finding an optimal policy for observing the processes that maximizes the total expected net rewards over an infinite time horizon subject to the resource availability. From the Lagrangian relaxation of the original problem, an index policy can be derived, as long as the existence of the Whittle index is ensured. We demonstrate that such a class of reinitializing bandits in which the projects' state deteriorates while active and resets to its initial state when passive until its completion possesses the structural property of indexability and we further show how to compute the index in closed form. In general, the Whittle index rule for restless bandit problems does not achieve optimality. However, we show that the proposed Whittle index rule is optimal for the problem under study in the case of stochastically heterogenous arms under the expected total criterion, and it is further recovered by a simple tractable rule referred to as the 1-limited Round Robin rule. Moreover, we illustrate the significant suboptimality of other widely used heuristic: the Myopic index rule, by computing in closed form its suboptimality gap. We present numerical studies which illustrate for the more general instances the performance advantages of the Whittle index rule over other simple heuristics.