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1. 西北工业大学机电学院, 西安 71007;
2. 山西大同大学数学与计算机科学学院, 大同 037009;
3. 北京理工大学管理与经济学院, 北京 100081

Cold standby n-component repairable system with repairman vacation
WEN Yanqing1,2, CUI Lirong3 , LIU Baoliang2
1. School of Mechatronical Engineering, Northwestern Polytechnical University, Xi'an 71007;
2. College of Mathematics and Computer Science, Shanxi Datong University, Datong 037009, China;
3. School of Management and Economics, Beijing Institute of Technology, Beijing 100081, China
Received: 2015-04-14; Accepted: 2015-07-10; Published online: 2015-09-17 16:50
Foundation item: National Natural Science Foundation of China(71371031)
Corresponding author. Tel.: 010-68940780 E-mail: Lirongcui@bit.edu.cn
Abstract: The reliability for a cold standby n-component repairable system with a repairman having multiple vacations is studied. If all of units are in normal state, the repairman may take his vacations, and the vacation time is a random variable. During the repairman's vacation, if a unit online fails to work, one of cold standby components becomes the component online immediately the failure unit will not be repaired until the repairman comes back from his vacation. The operational time of the unit online, the repair time and the vacation time of repairman are governed by different phase-type (PH) distributions, respectively. The system is studied in a transient and stationary regime. The reliability, availability, rate of occurrence of failures and busy probability of repairman are derived by Kronecker operator and Markov process theory. A numerical example is given to illustrate the results obtained.
Key words: repairable system     multiple vacations     reliability     phase-type distribution     rate of occurrence of failures

1 系统模型建立

1) 在时刻t=0,所有部件是完好的,1个部件在线开始工作,其余n-1个部件在冷贮备。当系统开始工作后,修理工立即去休假,他遵循多重休假策略,休假时间是一个随机变量;当在线工作部件出现了故障,冷贮备部件立即替换它成为在线工作部件。

2) 当修理工从休假返回系统中,他可能面临3种情况:

① 系统中所有部件均完好,则他立即进行第2次休假。

② 系统中有且仅有1个故障的部件等待修理,其余部件均完好,则他立即修理这个故障的部件,且修复如新,当这个部件修理完成时,如果系统中再没有故障的部件,则他立即进行第2次休假,否则继续修理故障的部件,直到系统中所有部件都完好,他才进行第2次休假。

③ 系统中至少有2个故障的部件,则他立即修理先出现故障的部件,然后按照先坏先修的原则再修理其他故障的部件,直到系统中所有部件都完好,他才进行第2次休假。

3) 假设部件的工作时间是一个随机变量且分布为PH(α,T),其工作时间位相的阶数为m;部件故障后的修理时间也是一个随机变量,服从阶数为n1的PH分布,表示为PH(β,S);修理工的休假时间服从阶数为k的PH分布,表示为PH(γ,L)。其中：α、β和γ分别为工作时间、修理时间和休假时间的初始向量；T、S和L分别为工作时间、修理时间和休假时间瞬态之间的转移概率矩阵。假设以上3个分布相互独立。

S1={(0,i,l),1≤i≤m,1≤l≤k}表示1个部件在线工作,其余n-1个部件处于冷贮备,修理工正在休假。(0,i,l)表示系统中故障部件数为0,在线部件工作时间的位相为i,修理工休假时间的位相为l

S2={(r,i,l),1≤r≤n-1,1≤i≤m,1≤l≤k}表示系统中有r个部件出现了故障,而修理工正处于休假,在线部件工作时间的位相为i,修理工休假时间的位相为l

S3={(r,i,j),1≤r≤n-1,1≤i≤m,1≤j≤n1}表示系统中有r个部件出现了故障,修理工按照先坏先修的原则正在修理最先出现故障的部件,其余故障部件排队等待修理,在线部件工作时间的位相为i,修理时间的位相为j

S4={(n,l),1≤l≤k}表示系统中所有部件都出现了故障,都排队等待修理,修理工正在进行休假,修理工休假时间的位相为l

S5={(n,j),1≤j≤n1}表示系统中所有部件都出现了故障,修理工按照先坏先修的原则正在修理最先出现故障的部件,其余故障部件排队等待修理,修理时间的位相为j

2 系统的可靠性指标 2.1 系统的瞬态可靠性指标

1) 可用度

2) 可靠度

3) 故障频度

4) 修理工忙的概率

2.2 系统的稳态可靠性指标

π=(π0,π1v,π1r,π2v,π2r,…,πnv,πnr)表示稳态时系统处于各个宏状态的概率向量,则它满足矩阵方程πQ=0和正则化条件πe=1,即

1) 稳态可用度

2) 稳态故障频度

3) 修理工忙的概率

4) 系统2次故障之间的平均时间

3 数值算例

 t A(t) v 1(t) v 2(t) pW ( t) 0 1 0 0 0 5 0.9937 0.0426 0.0673 0.0750 15 0.9929 0.0622 0.0920 0.2692 20 0.9913 0.0907 0.1454 0.4081 25 0.9906 0.1098 0.1928 0.5680 50 0.9906 0.1343 0.1928 0.5680 ∞ 0.9906 0.1343 0.1928 0.5680
4 结 论

1) 把修理工可以进行多重休假的思想引入系统建模,使修理工这个人力资源得到充分利用,可以为企业节约成本。

2) 任何非负随机变量服从的分布都可以用PH分布逼近,假定部件的工作时间、故障部件的维修时间以及修理工的休假时间用不同的PH分布表示,使得模型更具有一般性。

3) 运用矩阵分析的方法推导系统的瞬态和稳态可靠性指标,与以往的概率分析法相比较,方法更简单。

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#### 文章信息

WEN Yanqing, CUI Lirong, LIU Baoliang

Cold standby n-component repairable system with repairman vacation

Journal of Beijing University of Aeronautics and Astronsutics, 2016, 42(3): 569-575.
http://dx.doi.org/10.13700/j.bh.1001-5965.2015.0226