含同原因故障和一个冷储备部件的可修复系统主算子性质

Properties of the Main Operator of a Repairable System with Common-cause Failure and a Cold Standby Unit

研究了含同原因故障的由两个不同型的平行部件和一个冷储备部件所组成的系统.通过选取空间并定义算子,将系统模型转化成抽象Cauchy问题.运用C_0半群和预解正算子理论,验证了系统主算子A为稠定的预解正算子,计算出了A的谱界为一c,同时得出了算子A的共轭算子及其定义域,最终利用共尾理论证得算子A的谱界和增长界相等,即为-c.

In this paper,we investigate a system consisting of two identical parallel redundant active units,with a cold standby unit and common-caused failure.By choosing appropriate space and defining operators,we transform the system into an abstract Cauchy problem.Using Co-semigroup theory and resolvent positive operators theory,we prove the main operator is a densely defined resolvent positive operator.Then we obtain the the spectral bound of the main operator A and its adjoint operator A＊.Finally,we prove the spectral bound of A is equal to its growth bound using the concept of cofinal theory.