一类修理工可休假的可修复系统的适定性理论及数值分析

A repairable system with repairman vacation is widely present in repairable systems and queuing systems. It is related to a variety of areas, such as aviation, aerospace, defense, finance and network communications. The well understanding of this type of systems is of both theoretical significance and real applications. This project aims at solving the existence and uniqueness as well as stability of the solution of a class of complex repairable system; especially addressing some unanswered questions, namely, the exponential stability, analytical and numerical solutions for a class of buffer-warning stated repairable systems with repairman vacation and replaceable repair facility. Besides, repairman vacation related applicable optimization problems are to be investigated in the project. The present study will provide new insights towards the advancement of reliability theory. The detailed investigations in the proposed research include: (1) development of a system model using the theory of stochastic processes and probabilistic statistics; (2) discussion of the existence, uniqueness and asymptotic stability of system (classical) solution by a means of semi-group theory and functional analysis; (3) validation of the exponential stability of system solution with semi-group theory and spectral theory; (4) impact of vacation strategy on reliability indices and total revenue of the system; (5) determination of the safety factor of system by adopting numerical analysis when the transient solution is replaced by steady-state solution; and (6) preliminary analysis of the well-posedness of the conjugate system.

带修理工休假的可修复系统广泛存在于可修复系统以及排队论系统中，涉及到航空、航天、国防、金融、网络通信等众多领域，具有重要的理论意义和应用价值。本项目以一类修理工可休假的带缓冲/预警状态的且修理设备可更换的可修复系统为研究对象，旨在解决可修复系统理论研究中复杂系统解的存在唯一性、稳定性，特别是指数稳定性、解析解、数值解等不完善和尚未解决的问题，以及与修理工休假相关的实际优化问题，为后续研究提供新的方法和思路。主要研究内容包括：(1)利用随机过程和概率统计的方法建立系统模型；(2)利用半群理论和泛函分析方法讨论系统(古典)解的存在唯一性及渐近稳定性；(3)利用半群理论和谱理论验证系统解的指数稳定性；(4)探讨休假对系统各指标因素及总收益的影响；(5)利用数值模拟确定稳态解代替瞬态解的安全系数；(6)尝试对系统共轭问题的适定性问题进行分析。

带修理工休假的可修系统广泛存在于可修复系统以及排队论系统中，涉及到航空、航天、国防、金融、网络通信等众多领域，从学科上看，它属于现代分布参数控制系统领域，是一门跨学科的前沿学科，具有重要的理论意义和广泛的应用价值。本项目以带修理工休假的可修复系统为研究对象，讨论了可修复系统理论和应用研究中复杂系统解的存在唯一性、稳定性，特别是指数稳定性、灵敏度分析等不完善和尚未解决的问题，为后续研究提供了新的方法和思路。项目主要研究内容包括：(1)利用泛函分析中的不动点理论得到了一类可修复系统古典解的存在唯一性及C1连续性；(2)利用随机过程理论和概率分析的方法建立了一类修理工可多重延误休假且带有预警装置的简单可修复系统模型，利用泛函分析方法、半群理论和谱理论得到了此类系统(古典)解的存在唯一性、渐近稳定性及指数稳定性，利用数值分析的方法验证了带预警装置的系统在可用度及总收益方面优于不带预警装置的系统；(3)通过建立适当的目标函数和允许集，利用泛函分析的方法讨论了一类带储备部件且具有共因故障和人为故障的退化可修复系统的两类优化问题，得到了为达到系统最佳可靠度的最优修复率函数，和为达到系统最大可用度和最低消耗之间最佳平衡的最优修复率函数的存在性的必要条件。