Analysis Of Operators Of Series Of Warm Redundant Repairable System

Reliability theory has been developed in the1930s, along with the request of developing industrial and complex military equipment. And machine maintenance is one of the early topics in this field. Although the reliability of the units in the system has been improved in large scale, the reliability of the whole systems itself became an important topic, when the structures and function of the systems became more and more complicated. In fact,repairable systems is a primary subject in reliability theory. Repairable systems are repairable when they fails, and the unit which fails can be repaired immediately. Warm reserve system is a special redundant reserve system.Generally, the warm reserve may also fail during reserved, it is different between the part lifetime and working life, one part works and other parts warm reserves at the beginning, When the working part failure, it will be replaced by the unfailed reserve, it has no waiting time when replace, and the fails part can be sent to repairmen immediately.We investigate the repairable system which is an1-unit system supported by an identical warm standby. Using Co-semigroup theory, we first prove the system operator is a densely defined resolvent positive operator. Then, we obtain the adjoint operator of the system operator and its domain. Furthermore, we prove that0is the growth bound of the system operator. Finally, we show that0is also the upper spectral bound of the system operator using the concept of cofinal and relative theory.The important of this article is that we analysis main operator of the series. Using Co-semigroup theory and the concept of the mean of the service rate function, we estimate the upper spectral bound of the main operator and obtain that its value is the opposite number of the mean of service rate functions. Then we show the growth bound of the main operator share the same value with its upper spectral bound by using the concept of cofinal and relative theory.