We utilize and develop elements of the recent achievable region account of Gittins indexation by Bertsimas and Niño-Mora to design index-based policies for discounted multi-armed bandits on parallel machines. The policies analyzed have expected rewards which come within an $O(\alpha)$ quantity of optimality, where $\alpha > 0$ is a discount rate. In the main, the policies make an initial once for all allocation of bandits to machines, with each machine then handling its own workload optimally. This allocation must take careful account of the index structure of the bandits. The corresponding limit policies are average-overtaking optimal.
"Index-based policies for discounted multi-armed bandits on parallel machines." Ann. Appl. Probab. 10 (3) 877 - 896, August 2000. https://doi.org/10.1214/aoap/1019487512