Optimal Routing to two Parallel Heterogeneous Servers with Resequencing

Customers arrive to a single service queue according to a Poisson process with rate δ, from which they are routed to two parallel heterogeneous and exponential servers whose rates are μ1 – μ2. Customers are released from the system after service completion, according to their arrival order—a requirement introducing additional resequencing delays. Customers which are delayed due to resequencing are waiting in a resequencing queue. We consider the optimal routing problem under the class of fixed-position routing policies, that route customers to the faster server from the head of the service queue, and to the slower server from position J. The cost function is taken as the long-run average holding cost of the customers in the system. We show that an optimal stationary policy exists and is of the following type: the faster service is kept active as long as the service queue is not empty. The decision whether or not to route a customer to the slower server is independent of the state of the resequencing queue. If the position J is greater than [formula omitted], then customers are routed to the slower server if and only if the length of the service queue is at least m* (a threshold policy). We also show that the routing position J0 is “optimal” in the sense that every policy can be improved by dispatching a customer from position J0 (if not empty), rather than from position J.