Reliability Analysis Of Parallel Repairable System With Vacation

This paper mainly studies the reliability of two components parallel repairable sys-tem.The two components in the system are the same components and the repairman has a single vacation.Firstly,it is assumed that the life of two components satisfies the exponentially dis-tribution and the repair time of the component and the vacation time of the repairman both follow the general distribution while the repairman take a vacation at the beginning.A mathematical model via partial differential equations is driven,and then the system is formulated into a suitable Banach space.Next the existence of positive solution,the non-negative steady-state solution and the exponential stability of the system is proved by the functional analysis method and C₀-semigroup theory of bounded linear operators.Based on above results,some reliability indices of the system are deduced,while the reliability indexes of the repairman on duty is obtained at the initial time,and the reliability indexes of the two situations of the repairman are analyzed and compared.Secondly,suppose that the parallel repairable system is a degenerate system,and the system consists of two components that cannot be repaired“as good as new”after failures.In addition,the repairman has a single vacation,the fault component of system may not be repaired immediately,namely,if a component fails and the repairman is on vacation,the repair of the component will be delayed,if a component fails and the repairman is on duty,the fault component can be repaired immediately.Under these assumptions,a replacement policy N based on the failed times of component 1 is studied.The explicit expression of the system average cost rate C（N）is given by using the renewal reward theorem,and the optimal replacement policy N^*by minimizing the C（N）is obtained,which means the two components of the system will be replaced at the same time if the failures of component1 reach N^*.To show the advantage of a parallel system,a replacement policy N of the cold standby system consisting of the two similar components is also considered.The numerical results of both systems are given by the numerical analysis and the comparison of numerical results shows the advantages of the parallel system.Thirdly,taking the drive system of industrial robot as an example,the mathematical model of the system is established under the assumption that the repair time and vacation time both follow the general distribution,and then the reliability of the drive system is studied by using semigroup theory.Finally,in the previous assumptions,it is assumed that the repair equipment will not fail during the repair,but this is not the case in fact,the repair equipment may fail during repair and the system will replace a new one after fails.In this section,we propose a more general repairable system that is consisting of a component,a repairman and repair equipment.Suppose that the repair equipment may fail during repair and the equipment will be replaced a new one after fails.In addition,the components cannot be repaired immediately after failure and cannot be repaired”as good as new”.In particular,the repair time follows the general distribution and the replacement time of the repair equip-ment is considered.Under the above assumptions,a partial differential equation model is established by means of geometric process and supplementary variable technique.By Laplace transform,we get the availability of the system,from the expression one can see that the availability of the system will tend to zero after a long time running.That is,the system becomes completely unavailable.Therefore,we further study a replacement policy N based on the failed times of the system.In practice,the system failure is divided into different types,such as serious faults and minor faults.If there is a slight fault in the system,the system will be as good as new after appropriate repair and the system will not deteriorative;If there is a serious faults in the system,it cannot be repaired”as good as new”after failures.To be more realistic,we further propose an extended degenerate system and study the optimal replacement policy of the system and the numerical analysis is given to illustrate the theoretical results.