Properties Of The Main Operator Of The Gnedenko System With Signal Repairman Vacation

With the development of science and technology, the theory of reliability has penetrated into many fields, such as science and technology, applied science and management science, and it has attracted increasingly get people’s attention. In general, those systems which can be repaired or change components when they fail are called repairable systems. In real life, repairable systems are significantly practical and meaningful. Because of its popularity, people gradually realised its importance and it became an important branch of reliability theory. Repairable systems are consist of some componants and one or more repairman. For repairable systems, availability, reliability, system stability, the average working hours, failure rate, economic benefits, etc are important performance indexes.The paper studies properties of the main operator of the Gnedenko system with single repairman vacation. Firstly, we introduces the theory of reliability, its origin, its development and research today, meaning and method of repairable systems. Then we introduce the composition of the Gnedenko system and its differential equations and initial boundary value conditions, and the basic concept and theorems needed; Secondly, We use Co semigroup theory to study the main operator of the system, proving the main operator is a densed resolvent operator. By introducing the concept of the average service rate we estimate the bounds of the spectrum of the system’s main operator valuation, and prove that the upper spectral bound is the opposite of the average service rate; Thirdly, we use the concepts and related theories of cofinal under the densed resolvent operator, we show the growth bound of the main operator share the same value with its upper spectral bound; At the end of the paper in the research on the basis of main operator of the system, we prove that there exists unique nonnegative time-dependent system solutions.