Analysis of Markov renewal processes with some non-regeneration points and their applications to reliability theory

Abstract

Markov renewal theory, a branch of probability theory, is based on the properties of regeneration points. Roughly speaking, a regeneration point is a random time in a process at which the future development of the process is a probabilistic replica of the process itself, independent of the past history of the process. In many industrial applications Markov renewal theory is very useful for modeling a repairable system because repairs are assumed to renew or upgrade the system so that the performance of the system is identical to that of a new one, independent of the past performance records. The information, such as the number of spare parts needed, percentage of system down time, the percentage of idle time of the repair facilities, etc., can be computed by using the elegant mathematics of Markov renewal theory. However, there are many multi-component systems in which the repair activities performed on a single component do not necessarily constitute regeneration points for the entire system because the history of the other components can not be completely disregarded. The Markov renewal theory has been modified in this research so that it can be generally applied to the non-regeneration points of a process. The theory presented in this document has several advantages over other existing methods in dealing with the problems of non-regeneration points. Examples for the applications of this theory are included. The methodology used in this research emphasizes the reduction of the effort to reach a solution, but not the existence of the solution for a particular repairable system. This theory can also be applied to other contexts, e.g. queueing theory or inventory theory.