Ergodic Theory Of Markov Processes And Its Applications

In the dissertation, we consider ergodic theory and its applications of discrete-time Markov chains and continuous- time Markov processes. We study subgeometric convergence, exponential ergodicity and strong ergodicity for Markov processes, and polynomial convergence and strong ergodicity for Markov chains. Furthermore, we apply the known ergodic theory and our own new results to study some realistic models, which include several Markov chains and Markov processes derived from the queues, and a few q - processes such as birth-death processes and branching processes.In Chapter 1, we introduce the background where the problems are produced, list the organization of the paper and review the basic definitions in the dissertation.In Chapter 2, polynomial ergodicity for Markov chains is considered. We first give the criteria of polynomial ergodicity for the embedded M/G/1 queue and M/G/1-type Markov chains, and then get an estimate of polynomial convergence rates of stochastically ordered Markov chains. Finally, we investigate polynomial convergence rates of the embedded M/G/1 queue.In Chapter 3, we study subgeometric convergence of Markov processes. We first get a sufficient condition for subgeometric convergence of a class of Markov processes, and then consider polynomial convergence of both the queue length of the M/G/1 queues with vacations and M/G/1-type Markov processes. Subsequently, the explicit bounds on subgeometric convergence rates of the class of Markov processes are obtained, using the coupling method. By applying the result, we obtain the bounds on polynomial convergence rates of both waiting time of the classical M/G/1 queue and birth-death processes.Chapter 4 is devoted to studying geometric ergodicity of Markov chains. The conditions sufficient and necessary for geometric ergodicity of the embedded M/G/1 and GI/M/n queues and M/G/1-type Markov chains are given, and then the explicit values of the largest geometric convergence rate of the embedded M/G/1 queue and a special Markov chain of M/G/1-type are presented.