Optimal service assignment in a finite-source queue

A dynamic scheduling problem is considered for a finite-source M/M/1 system with several types of customers. A preemptive service-assignment policy is found that maximizes the expected present value of rewards received minus costs incurred over an infinite planning horizon. A policy that always selects customers according to a fixed priority order is called an index policy. It is shown that, when all the average thinking times are the same, there exists an index policy that is optimal over the class of policies considered, which contains policies that are not index policies