The Optimal Look-Ahead Policy for Admission to a Single Server System

This paper considers a service system with a single server, finite waiting room, and a renewal arrival process. Customers who arrive while the server is busy are lost. Upon completing service, the server chooses between two actions: either he immediately starts a new service, provided a customer is present, or he admits the newly arrived customer to the system, but delays service pending the next arrival, whereupon he again chooses between these two actions. This process continues until either the system is full or a new service is started. Once a service has been started, all customers who arrive while the server is busy are lost. We assume that at each decision epoch the server knows the arrival epoch of the first arriving customer. We show that there exists an optimal control-limit policy that minimizes the average expected idle time per customer served equivalently, maximizes the average number of customers served per unit of time. The special case of Poisson arrivals leads to an explicit expression for this delay that generalizes exisiting results.