This thesis is based on a work has been done by {Masakiyo Miyazawa and Yiqiang Q . Zhao{ (2004). Two - dimensional Markov process which will be considered is a GI/G/1 type queue . The first component of this process is referred to level , and the second component is called the phase . The majore question is following this fact that in general , finding closed form for stationary distribution of ergodic multidimensional Markov processes when the phase state space is infinite is not easy and even in some cases It is impossible . For finitely many backgrand states , this model was introduced. In this thesis combining , the Markov renevwal approach with the sensoring representation of the stationary distribution and Winner-Hopf factorization for a Markov additive process . As a next step the tail of stationary distribution of Markov processes and their asymptotic behavior have been investigated . Our goal in this thesis is that under assumption that the chain has stationary distribution , we investigate the asymptotic behavior of the stationary tail probabilities in the discrete time GI/G/1 type queue with a countable phase state space, that This result generalizes the corresponding result for the M/G/1 type queue in Miyazawa (2003) . These probabilities are presented in a matrix form with respect to the phase state space , and will be shown that it is solution of a Markov renewal equation . Using this fact , we consider their decay rates . Historically, decay rate problems in queues have been widely studied in literature because of their importance. In particular, the large deviation technique is popular and to obtain more qualitative results, the change of measure technique together with martingales has also been used. kashida; MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 140%; TEXT-ALIGN: justify; TEXT-KASHIDA: 0%" The Markov additive approach used in this paper is rather kashida; MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 140%; TEXT-ALIGN: justify; TEXT-KASHIDA: 0%" Applying the Markov renewal theorem , it is shown that certain reasonable conditions lead to the geometric decay of the tail probabilities as the level goes to infinity . We exemplify this result using a discrete time priority queue with a single server and two types of customers . This example shows where we have to be careful in applications of the main result. This thesis consists of five chapters . We introduce the Introduction in chapter 1 , and introduce the concepts, definitions, and requirements in chapter 2 , and will described queuing systems in chapter 3 . The main result and its proof are given in chapter 4 . Pilot examples and simulation of their are considered in chapter 5 and We finally make some remarks .