| Currently, reliability study is an important area of research. Reparable system is one of the most important systems discussed in the reliability theory. Many scholars, at home and abroad, have done lots of work about the system. However, all studies are based on two hypotheses:(1) the system concerned has a unique nonnegative time-dependent solution; (2) the solution of the system is asymptotic stability. However, whether the system is exponential stability that is still unsolved. Exponential stability of a repairable system with two identical unit and one standby unit is studied in the paper. Firstly, combining with graphs, the paper give the condition that the transient availability cannot be replaced by steady-state availability. Secondly, the paper gets the proof of existence and uniqueness of nonnegative solution of the system is given, by converting model equations into Volterra equation in Banach space. Thirdly, the operator of the system generates a positive contraction C0-semigroup in Banach space. Moreover, spectral points of system operator lie in the left plane,0 is shown to be the unique spectral point on the imaginary axis,0 is simple eigenvalue with positive eigenvector by discussing the distribution of spectral. At last the paper proves to be quasi-compactness of the system operator. Hence, we obtain that the time-dependent solution of the system converges to the steady-state solution exponentially.
|
PDF Full Text Request