Limit Distribution Of Markov Skeleton Processes

This paper mainly devotes to study limit theory of Markov skeleton processes through renewal method. The author obtains limit distribution,law of large numbers,central limit theorem of Markov skeleton processes, and applies the obtained results in the research of semi-Markov processes, queuing system, inventory theory and reliability.The full text is composed of seven parts. Below introduces the structure of this paper simply:Chapter one begins with the history from the definition and study of Markov processes to the introduction of Markov skeleton processes. By the way, the author introduces the existing work and most work of this paper.Chapter two introduces elementary knowledge which are needed in this paper, mainly includes: the definition of Markov skeleton processes, the backward and forward equation, the regularity and finite dimensions distribution of Markov skeleton processes. Most of proofs are omitted in this chapter except for the fourth quarter in which the author has made some supplements.Chapter three is the main result of this paper, namely limit distribution of Markov skeleton processes. The author begins with the definition of limit distribution and Doob skeleton processes. In section two, the author gives necessary and sufficient conditions forthe existence of limit distribution, removes the original requirement of absolutely continuous condition, and gives the concrete formula of limit distribution. Furthermore, the author proves the limit distribution is a probability distribution. In the third section, the author obtains some important results of Markov skeleton processes, such as: law of large numbers, central limit theorem and so on.Chapter fourth gives several kinds of important cases of Markov skeleton processes which are very familiar to us and commonly used, such as: Markov processes, semi-Markov processes, piecewise deterministic Markov processes, Doob processes, regenerative processes, semi-regenerative processes. According to those definitions and strict proofs, all these processes may be Markov skeleton processes directly besides semi-regenerative processes which need supplementary variable.In the fifth chapter, to manifest the theory value of Markov skeleton processes, the author applies the theory of Markov skeleton processes to study semi-Markov processes. In first section, the author has obtained the instantaneous distribution of semi-Markov processes, namely the backward and forward equations. The results are consistent with the results of Levy, but the method we used is Markov skeleton's method. Moreover, Levy obtained the forward equations only by conjecture, while the author gives the concrete proof. In second section, the author applies the instantaneous distribution of semi-Markov process in the research of M/G/1 and GI/M/1 queuing system. In third section, the author obtains limit distribution of semi-Markov processes by limit distribution of Markov skeleton processes. Furthermore, the author extends limit distribution of semi-Markov processes to a more widespread kind of stochastic processes.In the sixth chapter, the author applies the theory of Markov skeleton processes to study GI/G/1 queuing system, which manifests the application value of Markov skeleton process. The author mainly studies busy time, queue length and waiting time of GI/G/1 queuing system, and obtains instantaneous distribution, limit distribution and central limit theorem respectively. Professor Hou Zhenting has already conducted some research in this part and obtained some results. The author makes some further consummation and supplement, and obtained some new results.In the seventh chapter, the author applies the theory of Markov skeleton processes in the research of inventory theory and reliability. The first section obtained instantaneous distribution and limit distribution of a perishable model. The second section studies limit distribution of an incorruptible model. In the third section, the author obtains instantaneous distribution and limit distribution of serial systems. In the fourth section, the author obtains instantaneous distribution and limit distribution of parallel systems.