Optimal control of a queueing system with simultaneous service requirements

We consider a system with identical servers, operating in parallel. There are two types of customers, ordinary and locking, having exponential service times with rates μOand μLand completion rewards rOand rL(Problem P1) or completion reward rOand holding cost hL(Problem P2), respectively. An ordinary customer needs a single server to be processed, and there are an infinite number of ordinary customers available in the system. Locking customers arrive in a Poisson process with rate λ and require processing by all servers simultaneously. The servers are allocated in a nonpreemptive manner to both classes. The scheduling decision in such a system consists of determining how the servers should be allocated, so that the discounted reward, as well as the long run average reward in P1 and P2, is maximized. We prove that the optimal policy is of the following simple form: Have all servers serve ordinary customers until the queue of the locking customers builds up to some threshold number k^*. Then empty the ordinary customers from the servers as fast as possible by keeping each server idle after it completes the ordinary customer currently in service. Finally, serve all available locking customers, and continue by repeating the above procedure.