Optimal Routing of Fixed Size Jobs to Two Parallel Servers

We consider a heterogeneous two-server system processing fixed size jobs. This includes the scheduling system, where jobs wait in a common queue, and the dispatching system, where jobs are assigned to server-specific queues upon arrival. The optimal policy with respect to the delay in both systems is a threshold policy characterized by a single parameter. In this special case, the scheduling and dispatching systems achieve the same performance with the optimal threshold. The optimal threshold depends on the arrival rate and the service rates. It can be determined by means of dynamic programming, where the required value functions can be evaluated only at the necessary points by means of efficient Monte Carlo simulations. We also give the optimal threshold at three different limits, which yield a simple closed-form expression for a near-optimal threshold. The optimal policy is illustrated and compared with several heuristic dispatching policies.