Systems With Repairaman Vacation Under Poisson Shocks

Reliability analysis of the system and the optimal replacement policy is the importantcontent in the reliability mathematical theory. Researching the reliability of the systemunder the external shocks is very important that cannot be ignored in the study of thereliability theory. The cold redundant repairable system, redundant repairable system andparallel repairable system are extremely important and common modes in the study ofsystem reliability. Based on the references, consider the Poisson shocks, the repairmanvacation and repair of non-new, we derived the system reliability index, and the optimalreplacement policy of the system has been given. Finally, a numerical example is given todemonstrate the effectiveness of the method.Firstly, a warm-standby system under Poisson shocks has been researched. Theworking unit can fail due to the shocks. The arrival time of the shocks follow a Poissonprocess with the intensity. Whenever the magnitude of a shock is larger than thethreshold of the working unit, it will get failed. The paper assumes that the standby timeobeys exponential distribution, the repair time on the unit and the repairman’s vacationtime obey the general distribution. By using the supplementary variable method, theMarkov process theory, the Laplace transform, the paper derives a number of reliabilityindices: system reliability, the steady-state availability, the steady-state failure frequencyand so on. Lastly, the parameters’ effect on the steady-state availability is investigated bynumerical comparison.Secondly, a parallel-degenerated system with two different components and arepairman who may take vacation has been studied. The arrival time of the shocks followa Poisson process with the intensity. When the magnitude of a shock is larger than thethreshold of the working component, the component will fail. The system considers therepair of component1is not as good as new while the repair of component2is as good asnew. The system’s availability, reliability, average working time to the first failure andother reliability indices are obtained by using the geometric process theory, thesupplementary variable and Laplace transform.Finally, repair-replacement for a cold standby system with a repairman who may take single vacation has been studied. This system is composed of two dissimilar componentsand component1is the main component which is given priority in use. Intrinsic factorssuch as aging, deterioration or extrinsic factors like shocks may take the system down. Thearriving of the shocks follows a Poisson process. Assume component1cannot be repaired“as good as new” while component2follows a perfect repair. With the above assumptions,by using geometric process, the supplementary variable, Laplace transform technique, anumber of important reliability indexes has been derived. Furthermore, we also study areplacement policy N under which the system will be replaced whenever the number offailures of component1reaches N. The explicit expression for the expected cost rate ofthis system and the optimal replacement policy N are obtained. Lastly, a numericalexample is provided to validate the theoretical results of this system.