Our object of study is a new class of controlled stochastic systems called generalised branching bandits which include discounted branching bandits and generalised bandit problems as special cases. These models allow us to study queueing scheduling and project scheduling problems in which the reward earned from processing a particular job is influenced by other job types present in the system. Our mode of analysis is the achievable region approach. What is novel here is that the full system may be partitioned into two subsystems, each of which satisfies its own set of conservation laws and has its own highly structured performance space. An optimal policy for the full system is constructed from two sets of Gittins indices derived from the conservation laws governing the two subsystems.

Nash's generalization of Gittins’ classic index result to so-called generalized bandit problems (GBPs) in which returns are dependent on the states of all arms (not only the one which is pulled) has proved important for applications. The index theory for special cases of this model in which all indices are positive is straightforward. However, this is not a natural restriction in practice. An earlier proposal for the general case did not yield satisfactory index-based suboptimality bounds for policies — a central feature of classical Gittins index theory. We develop such bounds via a notion of duality for GBPs which is of independent interest. The index which emerges naturally from this analysis is the reciprocal of the one proposed by Nash.

Nash's generalization of Gittins’ classic index result to so-called generalized bandit problems (GBPs) in which returns are dependent on the states of all arms (not only the one which is pulled) has proved important for applications. The index theory for special cases of this model in which all indices are positive is straightforward. However, this is not a natural restriction in practice. An earlier proposal for the general case did not yield satisfactory index-based suboptimality bounds for policies — a central feature of classical Gittins index theory. We develop such bounds via a notion of duality for GBPs which is of independent interest. The index which emerges naturally from this analysis is the reciprocal of the one proposed by Nash.