Restless bandits, partial conservation laws and indexability

We show that if performance measures in a general stochastic scheduling problem satisfy partial conservation laws (PCL), which extend the generalized conservation laws (GCL) introduced by Bertsimas and Niño-Mora (1996), then the problem is solved optimally by a priority-index policy under a range of admissible linear performance objectives, with both this range and the optimal indices being determined by a one-pass adaptive-greedy algorithm that extends Klimov's: we call such scheduling problems PCL-indexable. We further apply the PCL framework to investigate the indexability property of restless bandits (two-action finite-state Markov decision chains) introduced by Whittle, obtaining the following results: (i) we present conditions on model parameters under which a single restless bandit is PCL-indexable, and hence indexable; membership of the class of PCL-indexable bandits is tested through a single run of the adaptive-greedy algorithm, which further computes the Whittle indices when the test is positive; this provides a tractable sufficient condition for indexability; (ii) we further introduce the subclass of GCL-indexable bandits (including classical bandits), which are indexable under arbitrary linear rewards. Our analysis is based on the achievable region approach to stochastic optimization, as the results follow from deriving and exploiting a new linear programming reformulation for single restless bandits.