应用数学MATHEMATICA APPLICATA2020,33(3):643-651具有两类失效模式和修理工休假的重试系统可靠性分析刘思佳,胡林敏,刘朝彩(燕山大学理学院,河北秦皇岛066004)摘要：本文考虑具有两类失效模式和Bernoulli休假的可修表决重试系统,系统中每个部件或者正常工作,或者以概率p类型a失效,或者以概率1−p类型b失效.修理工修理完一个部件后,可能以概率h进行休假,也可能以概率1−h在系统中空闲.系统中没有等待空间,失效部件如果不能立即得到修理,则进入重试空间,一段时间后再进行重试,直到得到修理.利用马尔可夫过程理论和拉普拉斯变换等方法,得到了系统的稳态可用度、可靠度函数和系统首次故障前平均寿命等可靠性指标.通过数值例子分析了系统参数对可靠性指标的影响.关键词：重试;两类失效模式;Bernoulli休假;可用度;可靠度中图分类号：O213.2AMS(2000)主题分类：90B25文献标识码：A文章编号：1001-9847(2020)03-0643-091.引言对于工程系统来说,维持较高的可靠性是至关重要的.表决系统作为典型的冗余系统,能够提高系统的可靠度和可用度,因此被广泛应用于各种实际工程中.一些对表决系统经典模型及扩展模型的研究可见文[1-3].这些文献都假定部件只有正常和失效两种状态,但很多实际情况中,部件可能不止一种失效模式.例如,电子线路中就有开路失效和短路失效两类失效模式.对于具有多类失效模式的表决系统的研究还较少.Ben-Dov[4]研究了部件(例如开关和电磁继电器)具有打开失效和关闭失效两类失效模式的k/n(G)系统.Moustafa[5−6]分别研究了具有两类失效模式和多类失效模式的表决系统可靠性.TANG等[7]对基于微马尔可夫模型的多类失效模式k/n(G)系统的不可用性进行了分析.重试排队系统以其在实际问题中的广泛应用得到大量学者的关注[8−10].在系统可靠性分析中,重试系统指系统内部没有失效部件的等待空间,如果部件失效时,修理工正在忙或者休假,则失效部件进入重试空间,一段时间后再进行尝试,直到得到修理.Krishnamoor-thy和Ushakumari[11]分别对三种不同情况的k/n(G)重试系统进行了研究,得到了相应的可靠度函数.KE等[12]研究了具有温贮备部件的表决可修重试系统,给出了计算系统稳态可用度的有效算法.KUO等[13]考虑了具有混合贮备部件的表决可修重试系统,得到了系统平均寿命和稳态可用度的具体表达式.最近,YANG和Tsao[14]对修理工可工作休假的k/n(G)重试系统可靠性进行了分析,利用矩阵分析法和拉普拉斯变换等方法得到了系统的稳态可用度和平均寿命.∗收稿日期：2019-06-24基金项目：河北省自然科学基金(A2018203088),河北省教育厅高等学校科技计划重点项目资助课题(ZD2017079)作者简介：刘思佳,女,汉族,河北,研究方向：系统可靠性理论.644应用数学2020在可修系统中,修理工由于各种各样的原因需要休假.很多文献都假设只有修理完系统中的失效部件,修理工才去休假.但大多数情况下,即使系统中还有失效部件,修理工也会进行休假,典型的此类休假有Bernoulli 休假.涉及到可靠性指标的具有Bernoulli 休假的重试排队系统可见文[15-17],但在系统可靠性分析中,还没有文献对具有Bernoulli 休假及多类失效模式的重试系统模型进行研究.本文考虑部件具有两类失效模式的表决重试系统,且修理工进行Bernoulli 休假.首先,提出假设,构建模型;然后,通过对系统的整体分析,求得稳态可用度、可靠度函数和系统首次故障前平均寿命等可靠性指标;最后,通过数值例子分析系统参数的变化对可靠性指标的影响.2.模型假定本文对研究的系统模型作如下假定：1)系统由n 个同型部件和一个修理工组成,当n 个部件中有k 个或k 个以上部件正常工作时,系统正常工作,即当失效的部件数大于或等于L =n −k +1时,系统失效.特别地,本文仅研究n =5,k =4时的表决系统,此时L =2.2)每个部件有两类失效模式,记为类型a 失效和类型b 失效,失效概率分别为p 和q ,且p +q =1,p >0,q >0.所有部件的寿命均服从参数为λ的指数分布.修理工修理部件类型a 失效的时间服从参数为µa 的指数分布,修理部件类型b 失效的时间服从参数为µb 的指数分布,部件修复如新.3)修理工进行Bernoulli 休假,当修理完一个部件后,修理工或者以概率h 进行休假,或者以概率¯h在系统中保持空闲,且h +¯h =1,h >0,¯h >0.休假时间服从参数为θ的指数分布.4)系统中没有等待空间,当部件失效时,如果修理工空闲,则立即修理失效部件;如果修理工正忙或正在休假,则失效部件进入重试空间,一段时间后进行重试,直到得到修理.部件的重试时间服从参数为γ的指数分布.5)所有随机变量均相互独立.6)初始时刻所有部件都是新的,系统开始工作,修理工空闲.3.模型分析令M (t )=m 表示时刻t 重试空间中的部件数,m =0,1,2;I (m )=i 表示m 个部件的组合序列数,i =1,2,...,2m ;x mi 表示m 个部件的失效类型组合的所有可能情况,x 01=0,x 11=a ,x 12=b ,x 21=aa ,x 22=ba ,x 23=ab ,x 24=bb ;J (t )表示时刻t 修理工的状态：J (t )=0,1a,1b,2,修理工空闲,修理工修理类型a 失效的部件,修理工修理类型b 失效的部件,修理工休假.则{J (t ),M (t ),I (m ),t ≥0,m ≥0}构成一个向量马尔可夫过程,系统在时刻t 的状态概率定义如下：P 0,mx mi (t )=P {J (t )=0,M (t )=m,I (m )=i },m =0,1,2;i =1,2,...,2m ,P 1a,mx mi (t )=P {J (t )=1a,M (t )=m,I (m )=i },m =0,1;i =1,2,...,2m ,P 1b,mx mi (t )=P {J (t )=1b,M (t )=m,I (m )=i },m =0,1;i =1,2,...,2m ,P 2,mx mi (t )=P {J (t )=2,M (t )=m,I (m )=i },m =0,1,2;i =1,2,...,2m .记λm =(n −m )λ,0≤m ≤L ;γm =mγ,1≤m ≤L .进一步可得系统状态转移图如图3.1所示.在图3.1中,方框内的状态为系统的失效状态.第3期刘思佳等：具有两类失效模式和修理工休假的重试系统可靠性分析645( ⨶ 䰢1( ⨶ 2(4.系统稳态可用度令P j,mx mi =lim t →∞P j,mx mi (t )(j =0,1a,1b,2;m =0,1,2;i =1,2,...,2m ),根据模型假定和系统状态转移图,可得在稳态条件下的系统状态概率方程组为0=−λ0P 0,0+¯hµa P 1a,0+¯hµb P 1b,0+θP 2,0,(4.1)0=−[(1−δm,2)λm +γm ]P 0,mx mi +(1−δm,2)¯hµa P 1a,mx mi +(1−δm,2)¯hµb P 1b,mx mi+θP 2,mx mi ,m =1,2,i =1,2,...,2m ,(4.2)0=−(λ1+µa )P 1a,0+pλ0P 0,0+γ1P 0,1a ,(4.3)0=−µa P 1a,1a +pλ1P 0,1a +pλ1P 1a,0+γ2P 0,2aa ,(4.4)0=−µa P 1a,1b +pλ1P 0,1b +qλ1P 1a,0+γ2P 0,2ab ,(4.5)0=−(λ1+µb )P 1b,0+qλ0P 0,0+γ1P 0,1b ,(4.6)0=−µb P 1b,1a +qλ1P 0,1a +pλ1P 1b,0+γ2P 0,2ba ,(4.7)0=−µb P 1b,1b +qλ1P 0,1b +qλ1P 1b,0+γ2P 0,2bb ,(4.8)0=−(λ0+θ)P 2,0+hµa P 1a,0+hµb P 1b,0,(4.9)0=−[(1−δm,2)λm +θ]P 2,mx mi +(1−δm,2)hµa P 1a,mx mi +(1−δm,2)hµb P 1b,mx mi+pλm −1P 2,(m −1)x (m −1)i ,m =1,2,i =1,2,...2m −1,(4.10)0=−[(1−δm,2)λm +θ]P 2,mx mi +(1−δm,2)hµa P 1a,mx mi +(1−δm,2)hµb P 1b,mx mi +qλm −1P 2,(m −1)x (m −1)(i −2m −1),m =1,2,i =2m −1+1,...,2m ,(4.11)其中,δm,2={1,m =2,0,m =2.定理4.1系统稳态可用度为A T (∞)=1−4∑i =1P 0,2x 2i −2∑i =1P 1a,1x 1i −2∑i =1P 1b,1x 1i −4∑i =1P 2,2x 2i .(4.12)证方程(4.1)-(4.11)可以写成矩阵形式QP =0,(4.13)646应用数学2020其中,0=(0,0,...,0)T 20×1,P =(P 0,P 1a ,P 1b ,P 2)T,P 0=(P 0,0,P 0,1a ,P 0,1b ,P 0,2aa ,P 0,2ba ,P 0,2ab ,P 0,2bb )T,P 1a =(P 1a,0,P 1a,1a ,P 1a,1b )T,P 1b =(P 1b,0,P 1b,1a ,P 1b,1b )T,P 2=(P 2,0,P 2,1a ,P 2,1b ,P 2,2aa ,P 2,2ba ,P 2,2ab ,P 2,2bb )T,Q =Q 11Q 12Q 13Q 14Q21Q 22Q 23Q 24Q 31Q 32Q 33Q 34Q 41Q 42Q 43Q 4420×20.矩阵Q 的各分块矩阵表示如下：Q 11=−λ0−(λ1+γ1)−(λ1+γ1)−γ2−γ2−γ2−γ2,Q 12= ¯hµa 000¯hµa 000¯hµa 000000000000,Q 13= ¯hµb 000¯hµb 000¯hµb 000000000000 ,Q 14= θθθθθθθ,Q 21=pλ0γ1000000pλ10γ200000pλ100γ20 ,Q 22= −(λ1+µa )00pλ1−µa 0qλ10−µa,Q 23=03×3,Q 24=03×7,Q 31=qλ00γ100000qλ100γ20000qλ1000γ2,Q 32=03×3,Q 34=03×7,Q 33=−(λ1+µb )00pλ1−µb0qλ10−µb ,Q 41=07×7,Q 42=hµa 000hµa 000hµa 000000000000,Q 43=hµb 000hµb000hµb 000000000000,Q 44=−(λ0+θ)000000pλ0−(λ1+θ)00000qλ00−(λ1+θ)00000pλ10−θ00000pλ10−θ000qλ1000−θ000qλ1000−θ.第3期刘思佳等：具有两类失效模式和修理工休假的重试系统可靠性分析647用正则条件2m ∑i=12∑m=0P0,mxmi+2m∑i=11∑m=0P1a,mxmi+2m∑i=11∑m=0P1b,mxmi+2m∑i=12∑m=0P2,mxmi=1(4.14)替换掉(4.13)式中的任何一个等式,则可由克莱姆法则计算得到系统的稳态概率：P j,mxmi(j= 0,1a,1b,2;m=0,1,2;i=1,2,...,2m),从而得到系统稳态可用度为A T(∞)=1−4∑i=1P0,2x2i−2∑i=1P1a,1x1i−2∑i=1P1b,1x1i−4∑i=1P2,2x2i.当λ=0.1,γ=0.5,θ=0.5,h=0.5,µa=2.0,µb=3.0,p=0.5时,由MATLAB软件计算得到系统稳态可用度的数值解为A T(∞)=1−4∑i=1P0,2x2i−2∑i=1P1a,1x1i−2∑i=1P1b,1x1i−4∑i=1P2,2x2i=0.9219.5.系统首次故障前平均寿命令所有系统故障状态为马尔可夫过程吸收态，可得系统状态概率拉普拉斯变换方程组为P0,0(0)=(s+λ0)P∗0,0(s)−¯hµa P∗1a,0(s)−¯hµb P∗1b,0(s)−θP∗2,0(s),(5.1)P0,1a(0)=[s+(λ1+γ1)]P∗0,1a (s)−θP∗2,1a(s),(5.2)P0,1b(0)=[s+(λ1+γ1)]P∗0,1b (s)−θP∗2,1b(s),(5.3)P1a,0(0)=(s+λ1+µa)P∗1a,0(s)−pλ0P∗0,0(s)−γ1P∗0,1a(s),(5.4)P1a,1a(0)=sP∗1a,1a (s)−pλ1P∗0,1a(s)−pλ1P∗1a,0(s),(5.5)P1a,1b(0)=sP∗1a,1b (s)−pλ1P∗0,1b(s)−qλ1P∗1a,0(s),(5.6)P1b,0(0)=(s+λ1+µb)P∗1b,0(s)−qλ0P∗0,0(s)−γ1P∗0,1b(s),(5.7)P1b,1a(t)=sP∗1b,1a (s)−qλ1P∗0,1a(s)−pλ1P∗1b,0(s),(5.8)P1b,1b(0)=sP∗1b,1b (s)−qλ1P∗0,1b(s)−qλ1P∗1b,0(s),(5.9)P2,0(0)=(s+λ0+θ)P∗2,0(s)−hµa P∗1a,0(s)−hµb P∗1b,0(s),(5.10)P2,mxmi (0)=[s+(1−δm,2)(λm+θ)]P∗2,mx mi(s)−pλm−1P∗2,(m−1)x(m−1)i(s), m=1,2,i=1,2,...2m−1,(5.11)P2,mxmi (0)=[s+(1−δm,2)(λm+θ)]P∗2,mx mi(s)−qλm−1P∗2,(m−1)x(m−1)(i−2m−1)(s), m=1,2,i=2m−1+1,...,2m.(5.12)其中,P∗j,mx mi (s)=∫∞e−st P j,mxmi(t)d t,j=0,1a,1b,2,m=0,1,2;i=1,2,...,2m.定理5.1系统的可靠度函数为R Y(t)=2m∑i=11∑m=0P0,mxmi(t)+P1a,0(t)+P1b,0(t)+2m∑i=11∑m=0P2,mxmi(t).(5.13)证方程(5.1)-(5.12)可以写成矩阵形式:P(0)=A(s)P∗(s),(5.14)其中,P(0)=(1,0,...,0)T16×1,P∗(s)=(P∗(s),P∗1a(s),P∗1b(s),P∗2(s))T,P∗(s)=(P∗0,0(s),P∗0,1a(s),P∗0,1b(s))T,P∗1a (s)=(P∗1a,0(s),P∗1a,1a(s),P∗1a,1b(s))T,P∗1b(s)=(P∗1b,0(s),P∗1b,1a(s),P∗1b,1b(s))T,P∗2(s)=(P∗2,0(s),P∗2,1a(s),P∗2,1b(s),P∗2,2aa(s),P∗2,2ba(s),P∗2,2ab(s),P∗2,2bb(s))T,648应用数学2020A (s )=A +s I ,I 为单位矩阵,A =A 11A 12A 13A 14A 21A 22A 23A 24A 31A 32A 33A 34A 41A 42A 43A 4416×16.矩阵A 的各分块矩阵表示如下：A 11= λ0λ1+γ1λ1+γ1 ,A 12= −¯hµa 00000000 ,A 13= −¯hµb 00000000,A 14= −θ0000000−θ0000000−θ0000 ,A 21= −pλ0−γ100−pλ1000−pλ1,A 22= λ1+µa 00−pλ100−qλ100 ,A 23=03×3,A 24=03×7,A 31= −qλ00−γ10−qλ1000−qλ1,A 32=03×3,A 33= λ1+µb 00−pλ100−qλ100,A 34=03×7,A 41=07×3,A 42= −hµa 00000000000000000000 ,A 43= −hµb 00000000000000000000 ,A 44=λ0+θ000000−pλ0λ1+θ00000−qλ00λ1+θ00000−pλ10000000−pλ100000−qλ10000000−qλ100.由克莱姆法则计算得到P ∗j,mx mi(s )(j =0,1a,1b,2;m =0,1,2;i =1,2,...,2m ).令Y 为系统第一次故障时间,则系统可靠度函数为R Y (t )=1−2∑i =1P 1a,1x 1i (t )−2∑i =1P 1b,1x 1i (t )−4∑i =1P 2,2x 2i (t )=2m∑i =11∑m =0P 0,mx mi (t )+P 1a,0(t )+P 1b,0(t )+2m∑i =11∑m =0P 2,mx mi (t ),其中,P j,mx mi (t )为P ∗j,mx mi (s )的拉普拉斯逆变换.当λ=0.1,γ=0.5,θ=0.5,h =0.5,µa =2.0,µb =3.0,p =0.5,t =10时,由MATLAB 软件计算得到系统可靠度函数的数值解为R Y (10)=2m ∑i =11∑m =0P 0,mx mi (10)+P 1a,0(10)+P 1b,0(10)+2m ∑i =11∑m =0P 2,mx mi (10)=0.7319.定理5.2系统首次故障前平均寿命为MTTFF =lim s →0[2m ∑i =11∑m =0P *0,mx mi (s )+P ∗1a,0(s )+P ∗1b,0(s )+2m ∑i =11∑m =0P *2,mx mi (s )].(5.15)证MTTFF =∫∞R Y (t )d t =lims →0∫∞e −st R Y (t )d t第3期刘思佳等：具有两类失效模式和修理工休假的重试系统可靠性分析649=lims →0∫∞e −st[2m ∑i =11∑m =0P 0,mx mi (t )+P 1a,0(t )+P 1b,0(t )+2m∑i =11∑m =0P 2,mx mi (t )]d t =lims →0[2m ∑i =11∑m =0P *0,mx mi (s )+P ∗1a,0(s )+P ∗1b,0(s )+2m∑i =11∑m =0P *2,mx mi (s )].当λ=0.1,γ=0.5,θ=0.5,h =0.5,µa =2.0,µb =3.0,p =0.5时,由MATLAB 软件计算得到系统首次故障前平均寿命的数值解为MTTFF =lim s →0[2m ∑i =11∑m =0P *0,mx mi (s )+P ∗1a,0(s )+P ∗1b,0(s )+2m ∑i =11∑m =0P *2,mx mi (s )]=27.3904.6.数值例子本节主要考虑系统参数λ,γ,θ,h,µa ,µb ,p 的变化对稳态可用度A T (∞)和系统首次故障前平均寿命MTTFF 的影响.令λ=0.1,γ=0.5,θ=0.5,h =0.5,µa =2.0,µb =3.0,p =0.5,再分别改变以上各参数的值进行分析,所得结果如图5.1-5.4和表5.1-5.4所示.正如我们所期望的,A T (∞)和MTTFF 的值随λ的增大而减小,随µa 或µb 的增大而增大.从图5.1和表5.1可看出,A T (∞)和MTTFF 的变化与γ的变化成正相关,说明失效部件越快得到修理,系统的可靠性越高.另外,图5.2和表5.2显示A T (∞)和MTTFF 的值随θ的增大而增大,随h 的增大而减小,这种情况可以解释为虽然休假对于修理工来说是必不可少的,但是休假时间越少,系统可靠性越高.特别的,从图5.3、5.4和表5.3、5.4中可以看出,当µa <µb 时,A T (∞)和MTTFF 随p 的增大而减小;当µa >µb 时,A T (∞)和MTTFF 随p 的增大而增大;当µa =µb 时,A T (∞)和MTTFF 的值不随p 的改变而改变.这说明系统的可靠性很大程度上依赖于修理工的修理效率.因此,可以通过适当提高部件的重试效率、增大修理工的修理效率、减少修理工的休息时间等方法来提高系统的可靠性,同时也要兼顾系统结构和系统成本.OA T (∞)图5.1不同参数λ和γ下系统稳态可用度变化曲线TA T (∞)图5.2不同参数θ和h 下系统稳态可用度变化曲线表5.1不同参数λ和γ下系统首次故障前平均寿命变化值λγ=0.5γ=1.0γ=1.5γ=2.0γ=2.50.127.3929.3030.2130.7331.080.28.879.219.399.519.580.3 4.99 5.10 5.17 5.21 5.240.4 3.41 3.46 3.49 3.51 3.520.52.562.592.602.612.62表5.2不同参数θ和h 下系统首次故障前平均寿命变化值θh =0.1h =0.3h =0.5h =0.7h =0.90.541.1932.4627.3924.0821.751.044.4038.3434.0230.7928.281.545.6941.1537.6034.7432.392.046.3642.7739.8037.3035.162.546.7843.8141.2739.0637.14650应用数学2020P a A T (∞)图5.3不同参数µa 和p 下系统稳态可用度变化曲线P bA T (∞)图5.4不同参数µb 和p 下系统稳态可用度变化曲线表5.3不同参数µa 和p 下系统首次故障前平均寿命变化值µa p =0.1p =0.3p =0.5p =0.7p =0.91.027.8925.4023.4021.7520.372.028.9428.1427.3926.6926.033.029.3629.3629.3629.3629.364.029.5930.0630.5431.0431.575.029.7330.5031.3232.2033.13表5.4不同参数µb 和p 下系统首次故障前平均寿命变化值µb p =0.1p =0.3p =0.5p =0.7p =0.91.020.2021.1622.2423.4824.912.025.7125.7125.7125.7125.713.028.9428.1427.3926.6926.034.031.0729.6528.3827.2426.205.032.5730.6829.0427.5926.307.结论本文研究了部件具有两类失效模式且修理工Bernoulli 休假的k/n (G )(k =4,n =5)重试系统,得到了一些重要的可靠性指标,分析了系统参数对可靠性指标的影响,为设备的管理与维护提供了理论依据.在今后的研究中,我们将进一步推广该模型:1)考虑一般的具有两类失效模式和休假的k/n (G )重试系统;2)考虑修理工休假时间服从一般分布的k/n (G )重试系统.参考文献：[1]唐应辉,梁晓军.c 个修理工同步多重休假的k/n (G )表决可修系统[J].系统工程理论与实践,2013,33(9):2330-2338.[2]张元元,吴文青,唐应辉.修复率可变化的表决系统可靠性分析[J].数学的实践与认识,2017,47(6):153-162.[3]吴文青,唐应辉,张元元.两水平修理策略的k/n (G )表决系统可靠性分析[J].系统工程学报,2018,33(6):854-864.[4]BEN-DOV Y.Optimal reliability design of k -out-of-n systems subject to two kinds of failure[J].Journal of the Operational Research Society,1980,31(8):743-748.[5]MOUSTAFA M S.Transient analysis of reliability with and without repair for k -out-of-n 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[12]KE J C,YANG D Y,SHEU S H,et al.Availability of a repairable retrial system with warm standbycomponents[J].International Journal of Computer Mathematics,2013,90(11):2279-2297.[13]KUO C C,SHEU S H,KE J C,et al.Reliability-based measures for a retrial system with mixedstandby components[J].Applied Mathematical Modelling,2014,38(19-20):4640-4651.[14]YANG D Y,TSAO C L.Reliability and availability analysis of standby systems with working va-cations and retrial of failed components[J].Reliability Engineering and System Safety,2019,182: 46-55.[15]周宗好,朱翼隽,冯艳刚.具有Bernoulli休假的M/G/1重试可修的排队系统[J].运筹学学报,2008,12(1):71-82.[16]CHOUDHURY G,KE J C.An unreliable retrial queue with delaying repair and general retrial timesunder Bernoulli vacation schedule[J].Applied Mathematics and Computation,2014,230:436-450.[17]CHOUDHURY G,TADJ L,DEKA M.An unreliable server retrial queue with two phases of serviceand general retrial times under Bernoulli vacation schedule[J].Quality Technology and Quantitative Management,2015,12(4):437-464.Reliability Analysis of a Retrial System with Two FailureModes and VacationLIU Sijia,HU Linmin,LIU Zhaocai(School of Science,Yanshan University,Qinhuangdao066004,China) Abstract:This paper considers a repairable voting retrial system with two failure modes andBernoulli vacation.Each component has two failure modes,we call them failure mode a and failure mode b.The probabilities of failure modes a and b are p and1−p,respectively.After the completion of a repair, the repairman either goes for a vacation with probability h,or stays idle in system with probability1−h. There is no waiting room in the system,if a component fails,it is repaired at once when the repairman is idle,otherwise the failed component enters into an orbit and tries again for the repair.The steady-state availability,the reliability function and the mean time to systemfirst failure are derived by using vector Markov process and Laplace transform theory.Finally,some numerical experiments are conducted to show the effects of system parameters on reliability indexes.Key words:Retrial;Two failure mode;Bernoulli vacation;Availability;Reliability。
