Analysis Of Operators Of The Repairable System Of Two-Unit Paralled With Two Failures

Repairable system is one of the most important systems in reliability theory research. More and more scholars and experts are paying attention to the theoretical study of repairable system in recent years. This paper mainly focus on the two component paralleling repairable system with two failures, repairable failures and non-repairable failures. Many systems consists of repairable part and non-repairable part, as the failures in the system consists of repairable failures and non-repairable failures. There are many reasons may cause non-repairable failures, such as the technique is so complex to be repaired, or it isn’t worth repairing considering the economy factor. Except the states of working, testing and repairing in repairable system, there is also another state of invalid which can be understood as that the system will be invalid or stops working sooner or later once non-repairable failures appears.System model, equations and initial and boundary conditions are offered firstly in the paper. With selecting state apace and introducing operators, this system is transformed into an abstract Cauchy problem in Banach space. By using Co-semigroup theory and the concept of cofinal and relative theories, both the system operators are densely defined resolvent positive operators and0is the growth bound and the upper spectral bound of the system operators are proved secondly. Finally, by using Co-semigroup theory and the concept of the mean of service rate functions, we have estimated the upper spectral bound of the main operator and are aware that the value is the opposite number of the minimum of the mean of service rate functions, and then we prove the growth bound of the main operator shares the same value with its upper spectral bound by using the concept of cofinal and relative theory.