: Matrix analytic methods initiated by Marcel Neuts, serve as a powerful framework to analyze large One advantage in presenting probabilistic results separate from the algorithms is that we make is clearly appear that the structural properties don't depend on whether there are finitely or infinitely many values for the phase dimension. Only when doing actual matrix computations does it become necessary to deal with a finite state space for the phase. Markov processes appear in two cases, discrete time and continuous time. Any analysis in which the system is observed, for analysis, only at specific points in time is a discrete time system is observed, for analysis, only at specific points in time is a discrete time system. As telecommunication systems are based more on digital technology these days than analog the need to use discrete time analysis, only at specific points in time is a discrete time system. As telecommunication systems are based more on digital technology these days than analog, the need to use discrete time analysis for queues has become more important. In this thesis, by considering discrete time Markov chains and using some results related to terminating renewal processes, we investigate the structure of stationary distributions for GI/M/1 and M/G/1 type models and the structure of transient distributions only for GI/M/1 type models. We also present the necessary conditions for existence of stationary distributions or ergodicity of chains and finally, we investigate the relations between those two models and existence of some duality between them, using Taylor and Van Houdt (2010).