Journal of Applied Mathematics and Stochastic Analysis
Volume 7 (1994), Issue 2, Pages 111-124
Some performance measures for vacation models with a batch Markovian arrival process
Université de Mons-Hainaut, Place Warocqué 17, Mons B-7000, Belgium
Received 1 November 1993; Revised 1 January 1994
Copyright © 1994 Sadrac K. Matendo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider a single server infinite capacity queueing system, where
the arrival process is a batch Markovian arrival process (BMAP).
Particular BMAPs are the batch Poisson arrival process, the Markovian
arrival process (MAP), many batch arrival processes with correlated
interarrival times and batch sizes, and superpositions of these processes.
We note that the MAP includes phase-type (PH) renewal processes and
non-renewal processes such as the Markov modulated Poisson process
The server applies Kella's vacation scheme, i.e., a vacation policy
where the decision of whether to take a new vacation or not, when the
system is empty, depends on the number of vacations already taken in
the current inactive phase. This exhaustive service discipline includes the
single vacation -policy, , and the multiple vacation -policy,
. The service times are i.i.d. random variables, independent of
the interarrival times and the vacation durations. Some important
performance measures such as the distribution functions and means of
the virtual and the actual waiting times are given. Finally, a numerical
example is presented.