由可修、可靠的人与机器构成的系统的时间依赖解的渐近展开  

Asymptotic Expansion of the Time-dependent Solution of the Repairable,Standby Human and Machine System

作　　者：陈海新 艾尼·吾甫尔[1] CHEN Haixin;GUPUR Geni(College of Mathematics and Systems Science,Xinjiang University,Urumqi 830017,China)

机构地区：[1]新疆大学数学与系统科学学院,新疆乌鲁木齐830017

出　　处：《应用数学》2025年第1期79-94,共16页Mathematica Applicata

基　　金：国家自然科学基金(11961062)。

摘　　要：可靠性模型的时间依赖解的渐近展开不仅在理论上而且在实际中具有重要的意义.本文研究于1998年Sridharan等建立的由可修、可靠的人与机器构成的系统的数学模型的时间依赖解的渐近展开.当修复率满足一定的条件时,首先证明该模型的主算子在左半复平面中的带形区域内至多有有限个特征值,并且这些特征值的几何重数为1,0是该主算子的严格占优特征值,然后证明该模型主算子的共轭算子在左半复平面中的带形区域内至多有有限多个特征值,并且这些特征值的几何重数为1,0是该主算子的共轭算子的严格占优特征值,最后证明所有特征值的代数重数为1.由此推出该模型的时间依赖解的渐近展开.此外,还得到该系统的瞬时可用度收敛于该系统的稳态可用度,该系统的瞬时故障频度收敛于该系统的稳态故障频度,并给出稳态可用度和稳态故障频度的具体表达式.本文的结果蕴含几位其他学者研究的可修复计算机系统等模型的时间依赖解的渐近展开.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 timedependent 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 steadystate 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.

关 键 词：可修、可靠的人与机器构成的系统 特征值 共轭算子 渐近展开 

分 类 号：O177[理学—数学]