Asymptotic Expansion of the Time-dependent Solution of the Repairable,Standby Human and Machine System
Asymptotic expansion of the time-dependent solutions of reliability models is significant not only in theory but also in practice.In this paper,we study asymptotic expansion of the time-dependent solution of the mathematical model of the repairable,standby human and machine system which was established by Sridharan et al.in 1998.When the repair rates satisfy certain conditions,firstly we prove that the underlying operator which corresponds to the model has finite eigenvalues at most in the strip region in the left half complex plane,geometric multiplicity of all eigenvalues are equal to 1,0 is an strictly dominant eigenvalue of the underlying operator.Next we prove that the adjoint operator of the underlying operator has finite eigenvalues at most in the strip region on the left half complex plane,geometric multiplicity of all eigenvalues of the adjoint operator is equal to 1,0 is an strictly dominant eigenvalue of the adjoint operator.Lastly,we prove that algebraic multiplicity of all eigenvalues of the underlying operator is equal to 1.Thus,we give asymptotic expansion of the time-dependent solution of the model.Moreover,we obtain that the time-dependent availability of the system converges to its steady-state availability,the time-dependent failure frequency converges to its steady-state failure frequency,and give concrete expression of the steady-state availability and steady-state failure frequency.Our results imply asymptotic expansion of the time-dependent solutions of several mathematical models such as the repairable computer system which were studied by several researchers.
Repairable,standby human and machine systemEigenvalueAdjoint operatorAsymptotic expansion
万方数据
维普
