Takine investigated the continuous time queueing system with single server and Markovian arrival streams (MAS) where service times of customers depend on the states of the underlying Markov chain immediately before and after arrivals and may differ from one another among streams. In this paper, we consider the discrete version of Takine's model. Since a MAS can be used to model the correlated traffics of several classes, we consider the discrete time queueing system with MAS as an input traffic. Moreover packets from different classes of traffic may require different service times. So service times of customers which are different from one another among streams and may depend on the states of the underlying Markov chain immediately before and after arrivals, are also considered. We obtain the distribution of the workload in the system using Neuts' matrix analytic method. Based on this result, we obtain the distributions of the actual waiting time and the sojourn time of a customer of each class and the joint distribution of the numbers of customers of each class. We also provide a recursion formula for computing the joint queue length distribution. We give an application to multimedia conference system and numerical example.