Optimal control and performance analysis of an M[X]/M/1 queue with batches of negative customers

We consider a Markov decision process for an M^[X]/M/1 queue that is controlled by batches of negative customers. More specifically, we derive conditions that imply threshold-type optimal policies under either the total discounted cost criterion or the average cost criterion. The performance analysis of the model when it operates under a given threshold-type policy is also studied. We prove a stability condition and a complete stochastic comparison characterization for models operating under different thresholds. Exact and asymptotic results concerning the computation of the stationary distribution of the model are also derived.