The optimal control of N servers and customers with simultaneous service requirements

There are N identical servers in the system, operating in parallel. There are two types of customers, the Ordinary customers and the Locking customers, having exponential service times with rates ??O and ??L and completion rewards r0 and rL (Problem P1) or completion reward r0 and holding cost hL, respectively (Problem P2). 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 ??, and require processing by all N servers simultaneously. The servers are allocated in a non-preemptive way to both classes. The scheduling decision in such a system consists of determining how the N 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 N 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 N servers as fast as possible by keeping each server idle after it completes the Ordinary customer currently in service. Then serve all available Locking customers, and continue by repeating the above procedure.