We consider a dynamic control problem for a parallel server system commonly known as the N-system. An N-system is a two-server parallel server system with two job classes, one server that can serve both classes, and one server that can only serve one class. We assume that jobs within each class arrive according to a renewal process. The random service time of a job has a general distribution that may depend on both the job's class and the server providing the service. Each job independently reneges, or abandons the queue without receiving service, if service does not begin within an exponentially distributed amount of time. The objective is to minimize the expected infinite horizon discounted cost of holding jobs in the system and having customers abandon, by dynamically scheduling waiting jobs to available servers.
It is not possible to solve this control problem exactly, and so, we consider an asymptotic regime in which the system satisfies both a heavy traffic and a resource pooling condition. Then, we solve the limiting Brownian control problem, and interpret its solution as a policy in the original N-system. We label the servers and job classes so that server 1 can only serve class 1 and server 2 can serve both classes. The policy we propose has two thresholds. There is one threshold on the total number of jobs in the system, and one threshold on the number of class 1 jobs in the system. These thresholds are used to determine which job class server 2 should serve. We show that this proposed policy is asymptotically optimal within a specified class of admissible policies in the heavy traffic limit, and has the same limiting cost as the Brownian control problem solution.