All Kinds Of Ergodicity Of Markov Chains Derived From In Queuing Systems

For all kinds of stochastic models, it is of considerably significance to study stochastic stability, and stationary distribution is our focus. To many Markov models, the existence and the expression of the stationary distribution have been studied nearly perfectly. At present there is a strong demand for studying all kinds of ergodicity(that is the convergence rate towards stationary distribution, consist of l-ergodicity, geometric ergodicity, uniformly polynomial ergodicity, strong ergodicity). While we pay attention to the convergence rate towards stationary distribution, many scholars study the stochastic stability in others way, that is the tail behavior of the stationary distribution, and have got many good results. The criterion of geometric ergodicity and light tailed of the stationary distribution for some models are very similar although, intuitively, which the convergence rate towards stationary distribution and the tail behavior of the stationary distribution seem to have no relation. Is there the relation between geometric ergodicity and light tailed stationary distribution? Recently, Jiezhong Zou and Yiqiang Zhao obtained that, to matrix-analytic models with finite phase space, the geometric ergodicity is equivalent to the stationary distribution with light tailed about level In this paper, we study all kinds of ergodicity and the tail of the stationary distribution and obtain there is the equivalent relation between them to some matrix-analytic models.By the methods and techniques of imbedded chains, we change the two dimensions matrix-analytic models to one dimension. Thus we conquer the difficulties of the multiply phase and make the things easy. By these techniques, we give the necessary and sufficient conditions of geometric er-godicity and /-ergodicity of matrix-analytic models with infinite phase space. To matrix-analytic models with finite phase space, we prove that geometric ergodicity is equivalent to the stationary distribution with light tailed about level and Z-ergodicity is equivalent to the stationary distribution with /-order type of tail about level. To matrix-analytic models with finite phase space, we only need to control transition of level. But to the case with infinite phase space, we need not only control transition of level, but also control transition of phase, so we can see that it is not enough only to have the tail behavior about level, we conque the difficulty by using the methods of imbedded chains, to matrix-analytic models with infinite phase space, we give that geometric ergodicity is equivalent to the stationary distribution with light tailed about level and phase; /-ergodicity is equivalent to the stationary distribution with /-order type of tail about level and phase.At last, we discuss the random walk type of Markov chains, define ||.||, and give that Geometric ergodicity is equivalent to 3 so > 0, such that / es"^x^7r(dx) < -foe; and /-ergodicity is equivalent to / \\x\\l~ln(dx) < +oc, where ix is a stationary distribution.