The Research Of A Class Of Repairable System Reliability

Reliability studies as an important area of research, is being more and more atten-tion. Reliability Analysis of technological development is bound to greatly a?ect thereliability of system design and manufacturing. This is especially important in the coun-try. Especially now that the development of an international trend, the domestic productto enter the international market to compete with foreign products, we must improve theperformance and reliability of the product itself.In practical life, for the reliability of a repairable system, people are concerned abouta system designed to work properly as long as possible, and for the maintenance of it,it is to make the system designed to make it in the event of failure of the repair assoon as possible, the high reliability and high maintainability together, we can get highavailability. Therefore, in practice , in order to improve the reliability of the system,frequently taking maintenance means (such there is great economic value) or adding theredundant components.Repairable system is one of the most important systems discussed in the reliabilitytheory, and also the primary subject studied in reliability mathematics. The so-calledrepairable system, refers to the system consisted of a number of parts and one or morerepair facilities (repairman). When the system components can occurred malfunctionor deterioration, the repair facilities is going to repair through a variety of maintenancemeans, after repair a class of systems is as new as. The so-called non-repair system, isthe fault component can not make any repairs, not to run after system failure. Whenthe system components life distribution and fault repair time distribution, and the othersare exponential distribution, as long as the proper definition of the system state, such asthe system can be described by Markov process. In traditional reliability research, it iswell known that the dynamic solution of the system is di?cult or even impossible to beobtained. Therefore it is ordinary to substitute the steady-state solution of the systemfor the dynamic one approximately. Especially due to the importance of the steady-stateavailability, it is usual to replace the transient availability with the steady-state one.However, the replacement does not hold in general. However, in reliability study, allresults are obtained through the method of Laplace and Laplace-Stieltjes transformation of semi-Markov process based on two hypotheses:1)the system concerned has a uniquenonnegative time-dependent solution; 2)the solution of the system is asymptotic stabil-ity. Both hypotheses obviously hold if the repair time follows exponential distribution.However, whether they hold or not if the repair rate follows arbitrary distribution is stillan open question and should be justified. Moreover, because it is di?cult to determinethe convergence rate of the dynamic solution and when it can get to the stable state onlyby asymptotic stability, the study of the exponential stability is in demand, which is amore valuable property in application.In this dissertation, we establish and study a class of repairable system model:repairablecomputer system model with hardware and software which are modeled by coupled equa-tions of partial di?erential equations and ordinary di?erential equations; repairable sys-tem model with randomly selected repairman;redundant electronic equipment systemmodel with two units. The main innovation and results of obtained in this dissertationmay be summarized as the following:We obtain the system's mathematical model by using the method of supplementaryvariable. By pure analysis method, we proved the existence and uniqueness of a dynamicnonnegative strong solution of a repairable computer system with hardware and software.By the positive C0-semigroup property which is generated by repairable computer systemoperator and using functional method, we proved that the existence and uniqueness of thenonnegative time-dependent solution the system. And obtained some reliability indices,for example, reliability and availability etc. It is significant for the reliability of thesystems. And under assumption that the system fault repair rateμi(x) =μ, i =1, 2￡constant¤and failure rateλh =λs≡λ, we have the analytical expression of thesystem, by using the method of characteristics of the constant coe?cient linear di?erentialequation. Furthermore, we proved the monotone stability of nonnegative solution andthe expression of the steady-state solution of the system.Establishing a redundant electronic equipment system model with two units. Bypure analysis method and C0-semigroup theory,we proved the existence and uniquenessof a dynamic nonnegative strong solution of a system. then show that the operatorcorresponding to this system model generates a positive contraction C0-semigroup. We then prove that C0-semigroup {T(t)}t≥0 is a quasi-compact operator, and that spectralbound of this operator is zero. It illustrate that the transient solution of the systemexponentially converges to the nonnegative steady-state solution.We develop a repairable system mathematical model with randomly selected re-pairman by theory of distributed parameter system based on di?erent repair cost bydi?erent level repairman. we discussed the existence and uniqueness of a dynamic non-negative strong solution of a system by semigroup theory and operator theory. By usingC0-semigroup quasi-compact theory we discussed the transient solution is exponentialstability. The solution exponentially converges to the nonnegative steady-state solution.Assumed that lifetime of parts follows exponential distribution, by the probabilityanalysis and partial substitute method, we discussed several typical repairable systemreliability index, compared reliability, the average life expectancy and the relationshipbetween the components.The significance of the article is: accurately describe a class of repairable systemreliability characteristics, and proves the existence and uniqueness, exponential stabilityof solutions,obtained repairable System reliability analysis. These discussions provides atheoretical basis for the study of other complex repairable system.