The Markovian arrival process (MAP) is used to represent the bursty and cor
related traffic arising in modem telecommunication network. In this paper.
we consider a single server finite capacity queue with general bulk service
rule in which arrivals are governed by MAP and service times are arbitrari
ly distributed. The distributions of the number of customers in the queue a
t arbitrary, post-departure and pre-arrival epochs have been obtained using
the supplementary variable and the embedded Markov chain techniques. Compu
tational procedure has been given when the service time distribution is of
phase type.