A unifying computations of Whittle's Index for Markovian bandits

TL;DR

This paper derives a unified analytical expression for Whittle's index applicable to any Markovian bandit, simplifying computation and broadening its applicability across various decision-making problems.

Contribution

It provides the first general analytical formula for Whittle's index for Markovian bandits with finite and infinite transition rates, including conditions for indexability.

Findings

Derived a unifying expression for Whittle's index

Established conditions for threshold-type solutions and indexability

Validated the approach with examples from machine repair and communication networks

Abstract

The multi-armed restless bandit framework allows to model a wide variety of decision-making problems in areas as diverse as industrial engineering, computer communication, operations research, financial engineering, communication networks etc. 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. However, the computation of Whittle's index itself is a difficult problem and hence researchers focused on calculating Whittle's index numerically or with a problem dependent approach. In our main contribution we derive an analytical expression for Whittle's index for any Markovian bandit with both finite and infinite transition rates. We derive sufficient conditions for the optimal solution of the relaxed problem to be of threshold type, and obtain conditions for the bandit to be indexable, a property assuring the existence of Whittle's index. Our solution approach provides a unifying expression for Whittle's index, which we highlight by retrieving known indices from literature as particular cases. The applicability of finite rates is illustrated with the machine repairmen problem, and that of infinite rates by an example of communication networks where transmission rates react instantaneously to packet losses.