Lookahead actions in dispatching to parallel queues

Applying the first policy iteration (FPI) to any static dispatching (task assignment) policy yields a new improved dynamic policy that takes into account the particular cost structure and the expected future arrivals. However, it is generally hard to go beyond that due to the complex state space and the resulting difficulty in computing the value function for a dynamic policy. For example, applying FPI to identical FCFS servers with Bernoulli split gives the Least-Work-Left (LWL) policy, for which no closed-form value function is known. In fact, LWL with identical servers is equivalent to an M/G/k queue, the performance measures of which have remained as open problems. The situation gets even more complicated with heterogeneous servers. In this paper, we take an intermediate approach and consider lookahead actions that concern not only the current job but also the job arriving next, after which a basic (static) policy is assumed to take over. This is important as the benefits from some decisions can only be reaped with appropriate subsequent actions. The lookahead enables sound estimates also for marginal admission costs, e.g., with respect to LWL. The superior performance of the new near-optimal dispatching policies is demonstrated numerically.