Optimal Dynamic Assignment of Customers to Heterogeneous Servers in Parallel

The system under consideration comprises two classes of customers to be served by two stations, with parallel servers in each station. While class-1 customers can only receive service from station 1, class-2 customers can be served by either station. Arrival processes of customers form two mutually independent Poisson processes. The service time of a customer at either station is exponentially distributed with a common rate. A class-i customer, while present in the system, will incur a holding cost h_i with h₁ ≥ h₂. The objective is to dynamically assign customers to idle servers so that the expected discounted (or the long-run average) holding cost is minimized. We show that a class-j customer should be assigned to an idle server in station j, j = 1, 2, whenever possible, and a class-2 customer should be assigned to an idle server in station 1 only if (no class-1 customers are waiting, and) the length of queue 2 exceeds a critical number. Moreover, the critical number is monotonically increasing in the number of busy servers in station 1. The numerical results for some test cases are reported.