The BMAP/G/1 vacation queue with queue-length dependent vacation schedule

We treat a single-server vacation queue with queue-length dependent vacatio n schedules. This subsumes the single-server vacation queue with exhaustive service discipline and the vacation queue with Bernoulli schedule as speci al cases. The lengths of vacation times depend on the number of customers i n the system at the beginning of a vacation. The arrival process is a batch Markovian arrival process (BMAP). We derive the queue-length distribution at departure epochs. By using a semi-Markov process technique, we obtain th e Laplace-Stieltjes transform of the transient queue-length distribution at an arbitrary time point and its limiting distribution.