Optimal dispatching in a tandem queue

We investigate a Markovian tandem queueing model in which service to the first queue is provided in batches. The main goal is to choose the batch sizes so as to minimize a linear cost function of the mean queue lengths. This model can be formulated as a Markov Decision Process (MDP) for which the optimal strategy has nice structural properties. In principle we can numerically compute the optimal decision in each state, but doing so can be computationally very demanding. A previously obtained approximation is computationally efficient for low and moderate loads, but for high loads also suffers from long computation times. In this paper, we exploit the structure of the optimal strategy and develop heuristic policies motivated by the analysis of a related controlled fluid problem. The fluid approach provides excellent approximations, and thus understanding, of the optimal MDP policy. The computational effort to determine the heuristic policies is much lower and, more importantly, hardly affected by the system load. The heuristic approximations can be extended to models with general service distributions, for which we numerically illustrate the accuracy.