Geometric Decay In A QBD Process With A Countable Background State And Application

We consider asymptotic behavior of stationary tail probabilities in the discrete time quasi-birth-and-death process with a countable background state space. Based on the research of the geometric tail decay of the stationary distribution for the GI/G/1 type Markov chain, some expressions was refined because of a matrix geometric form of the stationary distribution. Applying the Markov renewal theorem, it is shown that certain reasonable conditions of the QBD process lead to the geometric decay of the tail probabilities as the level goes to infinity. In particular, CC -positivity of the rate matrix is characterized by the renewal blocks of transition matrix of the QBD.This dissertation contains three parts: In Chapter 1, the history, the application of the existing condition, the fundamental knowledge, the main idea and result of queueing theory is introduced. In chapter 2, the notions of birth-and-death process, quasi-birth-and-death process, GI/G/1 type model are introduced, together with the technique of this dissertation. In Chapter 3, I consider the geometric decay in a QBD process with a countable background state. The main theorem and proof is in Chapter 3. In Chapter 4,I apply the result of Chapter 3 to a discrete time joining the shortest queue model, and a two-demand queue model by elementary computations, getting the result that they have a geometric tail decay.