Optimal control of a server farm

We consider a server farm consisting of ample exponential servers, that serve a Poisson stream of arriving customers. Each server can be either busy, idle or off. An arriving customer will immediately occupy an idle server, if there is one, and otherwise, an off server will be turned on and start servicing this customer. Upon service completion, it can be decided to keep the server idle or to turn it off. It costs per unit time to keep servers idle, and it costs to turn an off server on. We derive structural properties of the dynamic control policy minimizing the expected total discounted cost. The structural results are extended to the average cost criterion. We also propose and analyze a simple control policy, and numerically investigate its performance. Keywords: Infinite server queue, successive approximation, switching curve