Stability Analysis Of A Repairable System With Warning Device And Vacation Strategy

With the modernization of production and living, the requirement of the higher performance and reliability of products is more and more important. But the widely application of the reliability theory, the reliability study and its related problems are receiving more and more attention.Repairable system is an important kind of system discussed in reliability theory and also a main object in reliability mathematics. Stochastic process theory is the main tool used on its research. The traditional method on reliability study is:establishing the (generalized) markov model through the supplementary variable method, studying the system reliability indices by using Laplace transform, and then discussing the optimal replacement and maintenance, etc. Many scholars has studied the system. Since the last century fifties and sixties, the study of repairable system has made great achievement.However, in traditional reliability study, because the dynamic solution of the sys-tem is difficult or even impossible to be obtained, it is ordinary to substitute the steady-state solution(indices) for the instantaneous one(ones). So the system steady state are given more attention. But the substitite is based on two points. The time-dependent solution of the system is existed and unique. The time-dependent solution converges to the steady-state solution. And in practice, it will hold at least on the guarantee of a satey coefficient or a reliable interval. So the time-dependent solution and the stability is a problem to be solved in reliability theory.Based the above reasons, we discuss a simple repairable system with a warning device and a repairman who can have delayed-multiple vacations as a research object. The existence and uniqueness as well as the regularity of the system dynamic solution is gived with the functional analysis method. By using Co semigroup theory we discuss the unique existence and the expression of system dynamic solution. Then from the point of eigenfunction, we analyze some important reliability indices of the system. Finally combining the theoretical analysis and numerical simulation methods, we discuss the system with warning device and without device in terms of availability, failure frequency and the profit.The main contents and innovations in this dissertation can be summarized as follows:In chapter1, the background, significance of the theory and practices, main con-tents and methods in this dissertation are briefly introduced.In chapter2, the system model of interest is established and the conresponding parameters and conditions are given. By Markov renewal process theory and the proba-bility analysis method, the system is first transformed into a group of integro-differential equations.In chapter3, the preliminary knowledge required are gived.In chapter4, the existence and uniqueness as well as the regularity of the system dynamic solution is gived with the functional analysis method.In chapter5, by analyzing semigroup theory and spectral theory, the unique exis-tence of the time-dependent solution and the asmyptotic stability of the solution of the system is obtain.In chapter6, from the point of eigenfunction, we analyze some main reliability indices of the system. And combining the theoretical analysis and numerical simulation methods, we discuss the system with warning device and without device in terms of availability, failure frequency and the profit.