We consider a Markov decision process for an MX/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.