In a queueing network with a single server and r service nodes, a non-preemptive
non-idling policy chooses a node to service at each service completion epoch.
Under the assumptions of independent Poisson arrival processes, fixed routing
probabilities, and linear holding cost rates, we apply Whistle's method for Arm-acquiring bandits to show that for minimizing discounted cost or long-run
average cost the optimal policy is an index policy. We also give explicit
expressions for those priority indices.