Optimal threshold policies in a workload model with a variable number of service phases per job

We consider a two-stage service policy for a Poisson arrival queueing system. The idle server starts to work with ordinary service rate when a customer arrives. If the number of customers in the system reaches N, the service rate gets faster and continues until the system becomes empty. Otherwise, the server finishes the busy period with ordinary service rate. After assigning various operating costs to the system, we show that there exists a unique fast service rate minimizing the long-run average cost per unit time.