Asymptotically optimal priority policies for indexable and nonindexable restless bandits

TL;DR

This paper develops and proves the asymptotic optimality of a class of priority policies for controlling multi-class restless bandits, including nonindexable cases, with applications to queueing systems.

Contribution

It introduces a unified class of priority policies that are asymptotically optimal for both indexable and nonindexable restless bandits, extending previous results to more complex settings.

Findings

Proves asymptotic optimality of the proposed policies under certain conditions.

Shows Whittle's index policy is included in the class for indexable bandits.

Numerical evidence indicates near-optimal performance outside the asymptotic regime.

Abstract

We study the asymptotic optimal control of multi-class restless bandits. A restless bandit is a controllable stochastic process whose state evolution depends on whether or not the bandit is made active. Since finding the optimal control is typically intractable, we propose a class of priority policies that are proved to be asymptotically optimal under a global attractor property and a technical condition. We consider both a fixed population of bandits as well as a dynamic population where bandits can depart and arrive. As an example of a dynamic population of bandits, we analyze a multi-class $\mathit{M/M/S+M}$ queue for which we show asymptotic optimality of an index policy. We combine fluid-scaling techniques with linear programming results to prove that when bandits are indexable, Whittle's index policy is included in our class of priority policies. We thereby generalize a result of Weber and Weiss [J. Appl. Probab. 27 (1990) 637-648] about asymptotic optimality of Whittle's index policy to settings with (i) several classes of bandits, (ii) arrivals of new bandits and (iii) multiple actions. Indexability of the bandits is not required for our results to hold. For nonindexable bandits, we describe how to select priority policies from the class of asymptotically optimal policies and present numerical evidence that, outside the asymptotic regime, the performance of our proposed priority policies is nearly optimal.