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Stability Analysis of a Repairable System with Warning Device and Repairman Vacation  [PDF]
Yangli Ren,Lina Guo,Lingling Zhang
Journal of Function Spaces , 2013, DOI: 10.1155/2013/674191
Abstract: This paper considers a simple repairable system with a warning device and a repairman who can have delayed-multiple vacations. By Markov renewal process theory and the probability analysis method, the system is first transformed into a group of integrodifferential equations. Then, the existence and uniqueness as well as regularity of the system dynamic solution are discussed with the functional analysis method. Further, the asymptotic stability, especially the exponential stability of the system dynamic solution, is studied by using the strongly continuous semigroup theory or semigroup theory. The reliability indices and some applications (such as the comparisons of indices and profit of systems with and without warning device), as well as numerical examples, are presented at the end of the paper.
Gaver''''s Parallel System Attended by a Cold Standby Unit and a Repairman with Multiple Vacations
修理工可多重休假的带有一个冷贮备部件的Gaver 并联系统

YUE De-quan,ZHU Jian-ling,QIN Ya-ling,LI Chun-yan,
岳德权
,朱建玲,秦雅玲,李春艳

系统工程理论与实践 , 2006,
Abstract: This paper studies the Gaver's parallel repairable system attended by a cold standby unit and a repairman with multiple vacations.It is assumed that the operating time of units has an exponential distribution,while the repair-time and the vacation time have general continuous distributions.Using vector Markov process theory and Laplace transform method,we obtain the Laplace transform of the reliability,the mean time to first failure,the steady-state availability and the steady-state failure frequency and so on.In addition,we also investigate the parameters' effect on the steady-state availability by numerical comparison and analyze the benefit of the system.
Asymptotic Stability of a Series-Parallel Repairable System Consisting of Three-Unit with Multiple Vacations of a Repairman  [PDF]
Tursunjan Keyim, Abdukerim Haji
Journal of Applied Mathematics and Physics (JAMP) , 2017, DOI: 10.4236/jamp.2017.51018
Abstract: We study a series-parallel repairable system consisting of three units with multiple vacations of a repairman. We first show that all points on the imaginary axis except zero belong to the resolvent set of the operator and zero is an eigenvalue of the operator, and then we prove that the semigroup generated by the operator is irreducible. By combining these results with our previous result we deduce that the dynamic solution of the system converges strongly to its steady-state solution. Thus we obtain asymptotic stability of the dynamic solution of the system.
The Reliability Analysis of N-Unit Series Repairable System With One Replaceable Repair Facility and a Repairman Doing Other Work  [cached]
Xianyun Meng,Yanqin Guan,Jianying Yang,Taotao Wang
Modern Applied Science , 2009, DOI: 10.5539/mas.v2n3p113
Abstract: A model of N-unit series repairable systems with a repairman doing other work is studied and the impact on the system reliability because of a replaceable facility is also considered. It is assumed that the life of each unit, the life of the facility are exponentially distributed, the repair time of the unit, the replace time of the facility and the work time of the repairman outside the system are all generally distributed. By using approach of supplementary variables and method of generalized Markov progress, some important reliability indicates of the system and the facility are obtained.
An N-Component Series Repairable System with Repairman Doing Other Work and Priority in Repair  [cached]
Jianying Yang,Xianyun Meng,Weiyan Guo,Yanqin Guan
Modern Applied Science , 2009, DOI: 10.5539/mas.v2n6p163
Abstract: In this paper, we discussed the reliability of an N-unit series repairable system with the repairman doing other work and priority in repair. In this system, it is assumed that the working time distributions of the n components and the arrival interval time distributions of the customers which are out of the system are both exponential and the components in system is given priority in repair. It is also assumed that the repair time distributions of the n components and the service time distributions of the customers are both general continuous distributions. After repair, components are “as good as new”. Under these assumptions, using a supplementary variable technique and Laplace transform technique, some important reliability indices such as the system availability, the idle probability of the repairman and the rate of service for customers are derived. Our problem is to determine whether or not given the priority to components in repairing such that the benefit of the system is maximized.
Well-Posedness of Gaver’s Parallel System Attended by a Cold Standby Unit and a Repairman with Multiple Vacations  [PDF]
Abdukerim Haji, Bilikiz Yunus
Journal of Applied Mathematics and Physics (JAMP) , 2015, DOI: 10.4236/jamp.2015.37101
Abstract:

We investigate Gaver’s parallel system attended by a cold standby unit and a repairman with multiple vacations. By using C0-semigroup theory of linear operators in the functional analysis, we prove well-posedness and the existence of the unique positive dynamic solution of the system.

Asymptotic Stability of Gaver’s Parallel System Attended by a Cold Standby Unit and a Repairman with Multiple Vacations  [PDF]
Abdukerim Haji
World Journal of Engineering and Technology (WJET) , 2015, DOI: 10.4236/wjet.2015.33C041
Abstract:

We investigate Gaver’s parallel system attended by a cold standby unit and a repairman with multiple vacations. By analysing the spectral distribution of the system operator and taking into account the irreducibility of the semigroup generated by the system operator we prove that the dynamic solution converges strongly to the steady state solution. Thus we obtain asymptotic stability of the dynamic solution of the system.

Existence and Uniqueness of the Dynamic Solution of a Series-Parallel Repairable System Consisting of Three-Unit with Multiple Vacations of a Repairman  [PDF]
Abdukerim Haji, Tursunjan Keyim, Bilikiz Yunus
Journal of Applied Mathematics and Physics (JAMP) , 2016, DOI: 10.4236/jamp.2016.48168
Abstract:
We investigate a series-parallel repairable system consisting of three-unit with multiple vacations of a repairman. By using C0-semigroup theory of linear operators in the functional analysis, we prove that the system is well-posed and has a unique positive dynamic solution.
Research of the Optimal Management for Repairable System with Single Repairman Vacation
修理工单重休假可修系统优化管理研究

JIA Ji-shen,LIU Si-feng,DANG Yao-guo,
贾积身
,刘思峰,党耀国

系统工程理论与实践 , 2007,
Abstract: To study one unit repairable system with single repairman vacation,this paper proposes a kind of new maintenance and replacement model.Supposing that the system is repairable,under that the consecutive repair time of the system constitute an increasing geometric process,while the successive survival time of the system and the repairman every vacation time constitute a decreasing geometric process stochastically,we consider two kinds of replacement policies based on the working age T and the failure number N for the system and choose the long-run expected profit per unit time as objective function,and establish mathematic models by using renewal process and geometric process theory,the explicit expressions of the objective function are respectively derived.Under the some conditions,we derive that the policy N is better than the policy T and validate the effectiveness of the method through a numerical example also.Finally,we discuss the results.
Reliability Analysis of One Unit Repairable System With Repairman Vacation
修理工带休假的单部件可修系统的可靠性分析

TANG Ying-Hui,LIU Xiao-Yun,
唐应辉
,刘晓云

自动化学报 , 2004,
Abstract: This paper considers a one unit repairable systems with single repairman va-cation. If the system fails the repair starts immediately unless the repairman is in vaca-tion time, in which case a waiting repairable time is needed. So that the system may be in one of three states: the working state, the waiting repair state and the being repaired state. By using the total probability decomposition and the tool of the Laplace trans-form, the following reliability problems are discussed: 1) the reliability, pointwise a-vailability and the stead-state availability ; 2) the expected failure number during a gi-ven (0,t] and the stead-state failure frequency. Some important reliability results such as the pointwise availability and the stead-state availability are obtained.
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