Scheduling a repairman in a finite source system

In this paper we consider a closed queueing system in which N customers circulate between two stations. One station contains customers which are functioning, the other is a single server queue containing failed customers. Each functioning customer can fail, independently of the other customers, and when that happens it arrives at the queue. The repairman or server can repair the broken customers, though only one at a time. If a customer is repaired, it joins the station with functioning customers (which we call the source). The system incurs a holding cost cj for every unit of time that customer j, 1 ≤ j ≤ N , is not functioning. The failure and service times are all independent and exponentially distributed with parameters λj and μj for customer j, respectively. We are interested in finding the preemptive repair policy that minimizes the total average holding costs. If we choose cj = 1 for every customer j, this results in minimizing the average queue length, a problem which is well studied in the literature. As alternative objective we also consider minimizing the utilization factor of the server. An important class of policies are the list policies. We call a policy a list policy if all customers are ordered (the list) and served according to this order. The list policy that repairs the failed