Approximating Theory And Method For Instantaneous Indexes Of Repairable System

Based on the reliability theory and the theory of semi-group, with the application of theapproximating theory of semi-group, this paper presents the theory and method to solve theinstantaneous indexes of repairable system. Also, numerical examples of a specific systemare shown to verify the effectiveness of this method.Availability is one of the most important reliability indexes of repairable system,which can be divided into instantaneous availability and steady-state availability. With thewidespread use of high-reliability and long-life products, research on instantaneous indexesis becoming more and more important. It is because that instantaneous index is animportant basis to measure real-time reliability of a system. Meanwhile, instantaneousindex can offer us method to evaluate the availability of a system in certain time interval.But instantaneous availability, in general, is difficult to obtain the exact expression, so theresearch on it is still in initial stage.In order to obtain instantaneous index, paper’s main analysis are as follows:First of all, the research status at home and abroad on availability of a system isintroduced, including instantaneous availability and steady-state ones, and point out thesignificance of this kind of problem.Secondly, the basic concepts of reliability theory and the related concepts and theoriesof operator semi-groups are presented, which lay theoretical basis for our research. Thispart also gives a detailed introduction of one typical two-state repairable system, includingits operating principle and maintenance mode. Then we transform it to a generalizedMarkov process using supplementary variable method. Based on the above, we usemathematical model to formulate this repairable system. Further, we describe the model ofthis system as abstract Cauchy problem on a specific space, and verified the existence anduniqueness of solution of the system by using theories about generation of strongcontinuous semi-groups of bounded linear operators and perturbation by bounded linearoperators.Thirdly, in order to obtain the instantaneous availability of the system approximately,we describe the system by numerical method. Coupled with approximation theory ofsemi-groups of bounded linear operators, namely Trotter-Kato theory, it is verified that thesolution of re-described system is convergence to the solution of the original system. Then, we solve the approximating solution of the system, and the numerical simulation results arealso given to validate the effectiveness of the proposed method.Finally, after numerical solution obtained by traditional finite difference method isgiven, these two kinds of numerical methods are compared through error analysis. Wescreen out the influential factors that have effect on numerical results in order to provideguidance to improve the precision of the numerical algorithm.