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

Yang Woo Shin
Department of Statistics, Changwon National University, 9 Sarimdong, Changwon 641 - 773, Korea
and
Charles E. M. Pearce
Applied Mathematics Department, The University of Adelaide, Adelaide SA 5005, Australia

Abstract:

We treat a single-server vacation queue with queue-length dependent vacation schedules. This subsumes the single-server vacation queue with exhaustive service discipline and the vacation queue with Bernoulli schedule as special cases. The lengths of vacation times depend on the number of customers in 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 the Laplace-Stieltjes transform of the transient queue-length distribution at an arbitrary time point and its limiting distribution.