Stochastic scheduling: a short history of index policies and new approaches to index generation for dynamic resource allocation

In the 1970’s John Gittins discovered that multi-armed bandits, an important class of models for the dynamic allocation of a single key resource among a set of competing projects, have optimal solutions of index form. At each decision epoch such policies allocate the resource to whichever project has the largest Gittins index. Since the 1970’s, Gittins’ index result together with a range of developments and reformulations of it have constituted an influential stream of ideas and results contributing to research into the scheduling of stochastic objects. We give a brief account of many of the most important contributions to this work and proceed to describe how index theory has recently been developed to produce strongly performing heuristic policies for the dynamic allocation of a divisible resource to a collection of stochastic projects (or bandits). A limitation on this work concerns the need for the structural requirement of indexability which is notoriously difficult to establish. We introduce a general framework for the development of index policies for dynamic resource allocation which circumvents this difficulty. We utilise this framework to generate index policies for two model classes of independent interest. Their performance is evaluated in an extensive numerical study