An assignment problem for a parallel queueing system with 2 heterogeneous servers

In this paper, the authors consider an optimization problem for a parallel queueing system with two heterogeneous servers. Each server has its own queue and customers arrive at each queue according to independent Poisson processes. Each service time is independent and exponentially distributed. When a customer arrives at queue 1, the customers in queue 1 can be transferred to queue 2 by paying an assignment cost which is proportional to the number of moved customers. Holding cost is a function of the pair of queue lengths of the two servers. Our objective is to minimize the expected total discounted cost. The authors use the dynamic programming approch for this problem. Considering the pair of queue lengths as a state space, they show that the optimal policy has a switch over structure under some conditions on the holding cost.