The Reliability Research Of A Repairable System With Repairman Vacation

With the development of science and technology and the progress of the society, the requirement of higher performance and reliability of products is more and more important, whether in production and life or in military and scientific research. Reliability is a very important concept at the development, design and operation stages of products. Reliability analysis technology will definitely have a great influence on designing and manufacturing of reliability systems. Therefore, the reliability study is receiving more and more attention as an important research field.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 gen-eralized markov model through the supplementary variable method; on the basis, studying the system reliability indices by using Laplace and Laplace-Stieltjes transform; and then discussing the optimal replacement and maintenance strategy, etc. Many scholars has studied the system and made great achievements. But there are still some problems existed.Firstly, in traditional reliability research, Laplce transform is based on the two hy-pothese:1) the time-dependent solution of the system is existed and unique; 2) the time-dependent solution converges to the steady-state solution. They are obvious if the system is a Markov model, e.g. the work time and repair time of the system and the other corresponding distributions all follow exponential destribution. Otherwise, they need to be verified.Secondly, it is well known that in reliability study, because the dynamic solution of the system is difficult or even impossible to be obtained, it is ordinary to substitute the steady-state solution (indices) for the instantaneous one(ones). But generally, the replacement need to be based on some conditions. In other words, it will hold at least on the guarantee of a safty coefficient or a reliable interval. Therefore, the steady-state solution is not enough on system reliability study although is is necessary when considering the long-run planning of the system.Based on the above reasons, we discussed the unique existence and the expression of system dynamic solution by taking a repairable sysetm with a replaceable facility and a repairman following multiple-delayed vacation strategy as an example by using C0 semigroup theory. Moreover, in previous literatures on system stability study, the asymptotic stability has been given much more attention than exponential stability. For the improtance of the exponential stability, we discussed the property by analyzing the quasi-compactness of the system. Then we studied some important reliability indices of the system with a new method. And we analyzed the optimal vacation time of the repairman in order to get the maximum long-run expected profit by using Maple software. We also discussed the different influences on the system due to multiple vacation and single vacation. Therefore, we not only provided a solid theoretical foundation for reliability research, made up for the deficiencies in theory and application, but also expanded the scope of its applications.The main contents and innovations in this dissertation can be summarized as follows:In chapter 1, the background, significance of the theory and practices, research on-goings at Home and Abroad, existed problems, main contents and methods in this disser-tation are briefly introduced.In chapter 2, the system model of interest is established and the conresponding parameters and conditions are given. For the convenience of the study, it is translated into an abstract Cauchy problem in a Banach space.In chapter 3, the unique existence of the system time-dependent solution is discussed by using C0 semigroup theory. By analyzing the closeness, denseness and dissipative property of the system operator, the unique existence of the time-dependent solution of the system is obtained and the conservative property is also discussed.In chapter 4, the asmyptotic stability of the system is discussed by analyzing the spectra distribution of the system operator and its dual operator. The system is asmyp-totic stability because the spectral points of the system operator and its dual all lie on the left side of the Complex Plane and there is on spectrum on the imaginary axis except 0.In chapter 5, exponential stability of the system in a special case(i.e. the replacement of the facility is instaneous and perfect) is discussed. By constructing special operaters, the quasi-compactness is obtained by using Co semigroup theory and functional analysis method. And then the exponential stability is immediat.In chapter 6, some important reliability indices are discussed with a new method. Because the steady-state solution of the system is just the eigenfunction corresponding to eigenvalue 0 of the system operator, it is reasonable to consider the steady-state indices on the point of eigenfunction view. This is not only more simple than Laplace transform but also based on trict theoretical fundation. The influence on system reliability because of the repairman vacation is discussed and the optimal vacation time is analyzed by using Maple software. The different influence on the system due to multiple vacations and single vacation strategies of the repairman is also studied at the end of the paper.