1.School of Mathematics and Physics,Lanzhou Jiaotong University,Lanzhou 730070,China;2.Department of Mathematics and Statistics,McMaster University,Hamilton L8S 4K1,Canada

Representations for reliability functions of conditional coherent systems with INID components and ordered properties

Zhengcheng Zhang1,*and Narayanaswamy Balakrishnan2

1.School of Mathematics and Physics,Lanzhou Jiaotong University,Lanzhou 730070,China;
2.Department of Mathematics and Statistics,McMaster University,Hamilton L8S 4K1,Canada

This paper presents several useful mixture representations for the reliability function of the residual live of a coherent system with independent but non-identically distributed components. These presentations are based on order statistics,signatures and mean reliability functions.We then discuss some stochastic comparisons of residual lives between two systems based on the stochastic ordering of coefficien vectors(or components)of the two systems.These results form nice extensions of some known results for the case of independent and identically distributed components.

mixture,independent but non-identically distributed (INID),stochastic order,signature,residual life,mean reliability function,order statistics.

1.Introduction and preliminaries

Coherent systems are basic concepts in the reliability theory.Series systems,parallel systems andk-out-of-nsystems are particular cases of coherent systems.The evaluation of coherent system lifetime is a central subject in reliability engineering and survival analysis.However,it is not easy to compute the system reliability and other aging measures from the reliability of its components.A useful tool to handle coherent systems is so-called system signature,which is useful not only for computing system characteristics and for assessing the performance of a coherent structure,but also for comparing different structures in terms of their reliability characteristics. A comprehensive survey of signatures can be found in [1]which also includes applications of signatures to network reliability.SupposeX1,...,Xnare the lifetimes ofnindependent and identically distributed(IID)components of a coherent system,andXi,n(i= 1,2,...,n) denotes theith order statistic amongX1,...,Xn.Then, the lifetime(denoted byT)of the coherent system can be expressed asT=φ(X1,...,Xn),whereφis a coherent life function(see Barlow and Proschan[2],and Esary and Marshall[3]).Samaniego[4]define the signature of a coherent system as a probability vectors=(s1,...,sn) where thejth element is the probability that the system fails upon the failure of thejth component,that is,sj=P(T=Xj,n)forj=1,2,...,n,such thatUnderthe IID assumption,the signaturesis a distributionfree measure of the system’s design,so the computationof signatures reduces to a well-definecombinatorial problem.It has been shown by Kochar et al.[5]and Samaniego [4]that the reliability functionof a coherentsystem havingnIID units can be represented as a mixture ofk-out-of-nsystems with weightsskfork=1,2,...,n;that is,for anyt＞0,

It has been observed by Navarro et al.[6]that(1)holds even when the componentsof the system are exchangeable (i.e.,when the joint survival functionR(x1,...,xn)ofX1,...,Xnis symmetric inx1,...,xn).The result holds for mixed systems as well.A mixed system of ordernis a stochastic mixture of coherent systems of ordernthat selects a system over the class of coherent systems of ordernat random according to a f xed probability distribution (see Navarro et al.[7],and Samaniego[1]).However,the representation in(1)need not necessarily hold when the components are non-identically distributed,as described by Navarro et al.[7].

Navarro et al.[8]introduced the concepts of minimal and maximal signatures of coherent systems.With minimal and maximal signatures,the reliability function of a coherent system can be written as general discrete mix-tures:

where the coefficien vectors a=(a1,...,an)and b= (b1,...,bn)are,respectively,called the minimal signature (orthe dominationvector)and the maximalsignature,withbut some of the coefficient may be negative;see[8]for more details.Under the IID assumption of the components of systems,the vectors s,a and b only depend on the system structure function.Moreover,since s=aAnfor a triangular matrix An,s can be obtained from a and vice versa;see David and Nagaraja [9]for further details.

Upon using the distribution of order statistics,the expression in(1)can be readily given as

In fact,this polynomial may also be written in the formwhere d=(d1,...,dn)is generally referred to as the system’s domination vector(see Satyanarayana and Prabhakar[10]).The relationship between s and d has been described by Samaniego[7].The coeffi cients of the polynomial in(5)only depend on the system structure,and h(p)strictly increases for p∈(0,1),such that h(0)=0 and h(1)=1.

Mixture representations have been proven useful in studying the aging characteristics and comparing of the performance of competing systems.For example,several preservation theorems have been established in[11–30].

Navarro et al.[31]showed that if T=φ(X1,···,Xn) is the lifetime of a coherent system with IID component lifetimes X1,...,Xndistributed according to a common continuousdistribution F and s=(s1,...,sn)is the system signature,then the distribution of the system residual lifetime T?t,given T＞t,is a mixture of the residual lives of k-out-of-n systems.Specificall,for all t≥0 and x≥0,

From the expression in(3),we can express the residual lifetime of T?t,given T＞t,as

For establishing the main results of this paper,we firs need to introduce the following stochastic orders.Let X and Y be the lifetimes of two components,with respective distribution functions F(x)and G(x),and survival functionsLet their probability density functions be f(x)and g(x),respectively.Then,X is said to be smaller than Y in

(i)usual stochastic order(denoted by X ≤stY)iffor all x;

(ii)hazard rate order(denoted by X ≤hrY)ifis decreasing in x;

(iii)reversed hazard rate order(denoted by X ≤rhY) if F(x)/G(x)is decreasing in x;

(iv)likelihood ratio order(denoted by X ≤lrY)if f(x)/g(x)is decreasing in the union of the supports of f(x)and g(x).

Fortwo discrete probability distributionsp= (p1,...,pn)andq=(q1,...,qn),pis said to be smaller thanqin the

(i)usual stochastic order(denoted byp≤stq)iffor alli=1,2,...,n;

(ii)hazard rate order(denoted byp≤hrq)ifis decreasing ini;

(iii)reversed hazard rate order(denoted byp≤rhq)ifis decreasing ini;

(iv)likelihood ratio order(denoted byp≤lrq)ifpi/qiis decreasing ini.

For more comprehensive discussions on all properties and other details of these stochastic orderings,one may refer to Shaked and Shanthikumar[32].

Based on the expression in(6),the following results have been established by Navarro et al.[31].

Theorem 1LetT1=φ1(X1,···,Xn)andT2=φ2(X1,...,Xn)be the lifetimes of two coherent systems, both based onncomponents with IID lifetimes distributed according to a common continuous distributionF.Let, for allt≥ 0,s1(t)=(s11(t),...,s1i(t),...,s1n(t)) ands2(t)=(s21(t),...,s2i(t),...,s2n(t))denotethe respective coefficien vectors in(6).

(i)Ifs1(t)≤sts2(t),then(T1?t|T1＞t)≤st(T2?t|T2＞t);

(ii)Ifs1(t)≤hrs2(t),then(T1?t|T1＞t)≤hr(T2?t|T2＞t);

(iii)Ifs1(t)≤lrs2(t),then(T1?t|T1＞t)≤lr(T2?t|T2＞t).

The lifetimes of coherent systems with independent but non-identically distributed(INID)components have been studied subsequently by Navarro et al.[33].They obtained the signature-based representations for the reliability of these systems,as described in the following theorem.

Theorem 2LetTbe the lifetime of a coherent system having independentcomponent lifetimesX1,...,Xn, andbe the reliability function ofXi,fori=1,...,n.Assume thathandHare the system’s reliability polynomial and reliability structure function,respectively,ands=(s1,...,sn),a=(a1,...,an)andb=(b1,...,bn) are the corresponding signature,minimal signature and maximal signature vectors of ann-componentsystem with the same structure asTbut with IID component lifetimesY1,...,Yn.Then,the reliability function of the system lifetimeTcan be expressed as

whereY1,n,...,Yn,nare the orderstatistics fromY1,...,Ynhaving a common reliability function

According to the properties ofhandH,the functionsatisfie the propertiesof a reliability functionand is therefore a proper reliability function;see[33].The reliability functioncan therefore be called the mean reliability function associated with the system and the components’reliabilityfunction.Formoredetails aboutmeanfunctions, one may refer to[23].

By utilizing the representations in Theorem 2,some stochastic comparison results for two different systems with two sets of independent components have been obtained in[33](see Theorem 2.2 in[33]),which extended the existing results on signature-based results for the IID case to the INID case.By using Theorem 2.2 in[33],the followingresultbelowcan be obtainedimmediatelyandits proof is therefore omitted for the sake of brevity.

Theorem 3LetT1=φ1(X11,...,X1n)andT2=φ2(X21,...,X2n)be the lifetimes of two coherent systems with the same signatures=(s1,...,si,...,sn) but having different independent components.Leth1andh2be their reliability polynomials,andH1andH2be theirstructure reliability functions,respectively.As described in Theorem 2,supposeis the reliability function of a set of IID random variablesY11,...,Y1n,andis the reliability function of another set of IID random variablesY21,...,Y2n,for anyt≥0.

Few representation results have been obtained in the literature for the residual life of coherent systems with heterogeneous components.In this paper we study the residual life of a coherent system with INID components.In Section 2,we present mixture representations for the residual lifetime by using order statistics and signature.Some stochastic comparisons of residual lives are then made between two systems with different structures and two different sets of INID components.Some properties are thenobtained from these representations.These results form nice extensions of some known results in the literature for the IID case.

2.Main results

As the firs result,we give the mixture representation for the residual lifetime of coherent systems with INID components based on the signature and mean reliability function.

Theorem 4Supposes=(s1,...,sn)is the signature vector of ann-component system,with INID components lifetimesX1,...,Xnhaving corresponding reliability functionsLethandHbe the system’s reliability polynomialand reliability structure function,respectively.Then,for anyt>0 andx>0,

ProofFrom Theorem 3 and(12),for anyt>0 andx>0,we have

Theorem 4 shows that the residual life(T?t|T>t) of a coherent system with INID components at timetmay be represented as a mixture of the residual lifetimes (Yi,n?t|Yi,n>t)of the order statistics from a set of IID component lifetimes with coefficientsi(t),fori= 1,...,n.The coefficientsi(t)depend onHandˉFifori=1,...,n;that is,the residual life of the coherent system with INID components,givenT>t,can be regarded as a discrete mixture of the residual lifetimes ofk-out-of-nsystems formed by a set of IID random variables.

Now,we shall discuss some stochastic properties of the coefficien vectors(t)=(s1(t),...,sn(t)).It can be shown thats(t)has tail stochastic behavior when the lifetimes of thencomponentsare INID,which is an extension of the corresponding results for the case when the components are IID,presented by Navarro et al.[31].That is,ifTis a coherent system with signature vectors= (s1,...,sj,0,...,0),andsj>0 forj∈{1,2,...,n},Furthermore,

we can show thats(t1)?sts(t2)fort1?t2as known in the case when the components are IID.This shows that (T?t1|T>t1)?st(T?t2|T>t2)for allt1?t2.

Example 1Consider a system with lifetimeT= max{X1,X2}having INID component lifetimesXidistributed according toFˉi,fori=1,2,and the signature vector for this system is clearly(0,1).For anyt>0,the reliability function is given by

whereH(p1,p2)=p1+p2?p1p2.If we letp=p1=p2, thenh(p)=2p?p2.Its inverse function ish?1(x)=Hence,for 0?t,we have

Hence,the conditional reliability function is

Example 2Consider a system with lifetimeT= max{X1,min{X2,X3}}having INID component lifetimesXidistributed according toThe signature vector for this system isThen,for anyt>0,the reliability function is given by

and the conditional residual survival function is

whereH(p1,p2,p3)=p1+p2p3?p1p2p3.If letp=p1=p2=p3,thenh(p)=p+p2?p3.The inverse function of this cubic equation is complicated,and so numericalmethods need to be employed to compute the inverse function.

In what follows,we discuss some stochastic comparisons of two coherent systems with a common structure, but with different INID components.In order to obtain these results,we need the following corollary.

Corollary 1LetT1,T2be the lifetimes of two coherent systems with two sets of IID component lifetimesX1i,X2i(i=1,...,n)having respective reliability functionsAssume these two systems have a common signatureX1i,nbeing theith order statistic fromX1i,andX2i,nbeing theith order statistic fromX2i(i=1,...,n).

ProofWe only need to show that for anyt≥0 and for alli=1,...,n,

Hence,it is enough to show that,fork≤j,

which is equivalent to

Theorem 5LetT1be the lifetime of a coherent system with independent component lifetimesX1i(i= 1,...,n)having a reliability functionandT2be the lifetime of another coherent system with independent component lifetimesX2i(i= 1,...,n)having a reliability functionSuppose these two systems have a common structure,and so the same signature. Letbe the corresponding mean reliability functions.

Proof(i)From Theorem 4,for anyt＞0 andx＞0, we have

where the firs inequality follows fromthe second onefrom thefactthatis increasing iny, and the last one holds from the fact

The proof of(iv)is quite easy and therefore is omitted.

The following example shows that the conditioncannot be deleted.

Example 3LetY1andY2be distributed as two parameter Weibull with respective positive scale parametersλ1,λ2,and a common shape parameterk=2,that is,Then, for 0＜t＜1 andx＞0,we haveWe then obtain

The following example shows that the conditionis increasing inyfor anyx＞0 in(iii)also can not be deleted.

Example 4Consider two systems with respective lifetimesT1= min{X11,X12,X13}andT2= min{X21,X22,X23},whereXijis distributed as exponential with parameterλij(＞0)fori=1,2 andj= 1,2,3.Supposeis the reliability function of IID component lifetimesYi1,Yi2,Yi3,fori=1,2.It can be computed from(12),

Clearly,λ11+λ12+λ13≥λ21+λ22+λ23meansY11≤rhY21.However,it is easily seen thatis decreasing iny≥0 for anyx≥0.

In the following theorem,we present a stochastic comparison of two coherent systems with different structures and different sets of INID components.

Theorem 6LetT1=φ1(X11,···,X1n)be the lifetime of a coherent system with independent component lifetimesX1i(i= 1,...,n)having a reliability distributionwith a coefficien vectors1(t)= (s11(t),...,s1n(t)).Also,letT2=φ2(X21,...,X2n) be the lifetime of another coherent system with independent component lifetimesX2i(i= 1,...,n) having a reliability distributionwith a coeffi cient vectors2(t)= (s21(t),...,s2n(t)).Leth1andh2be the reliability polynomials,andH1andH2be the structure reliability functions,respectively.LetThen if

ProofLetY11,n,...,Y1n,nbe the order statistics obtained from the IID random variablesY11,...,Y1nhaving reliability functionbe the order statistics obtained fromY21,...,Y2nhaving reliability functionThen from(13),for anyt＞0 andx＞0,the reliability functions of the random variables (T1?t|T1＞t)and(T2?t|T2＞t)can be expressed as

FromTheorem1.A.6 of Shakedand Shanthikumar[32],s1(t)≤sts2(t)implies that

Also by Theorem 1.B.34 of Shaked and Shanthikumar[32],the conditionimplies thatand soThis means thatwhich in turn implies

as required.

Example 5Consider two coherent systems with lifetimesT1= min{X11,max{X12,X13}}andT2= max{X21,min{X22,X23}}.LetXi1,Xi2,Xi3be distributed according to reliability functions,respectively,fori=1,2.Assumeis the corresponding mean reliability function of IID component lifetimesYi1,Yi2,Yi3,fori=1,2.After some computations,it can be shown that the coefficien vectors of the systemsT1andT2are

respectively.

Using Mathematica software,it can be checked thatfor allx＞0,for each fi edt≥0;that is,

Next,we show that the expressions in(7)and(8)can be extended to coherent systems with INID components.

Theorem 7Leta= (a1,...,an)andb= (b1,...,bn)be the minimal and maximal signatures ofann-component system with independentcomponent lifetimesX1,...,Xn.Then,the reliability function of the residual lifetimeT?tat timetcan be expressed as

whereY1,n,...,Yn,nare the order statistics obtained from IID random variablesY1,...,Ynhaving a common reliability functionandwithSome of these coeffi cients may be negative.

Theorem 7 shows that the residual life(T?t|T＞t) of the coherent system with INID components at timetmay be represented as a mixture of the residual lifetimes (Y1,i?t|Y1,i＞t)of the smallest order statistics from a set of IID component lifetimes with coefficientai(t) fori=1,...,n,or a mixture of the residual lifetimes (Yi,i?t|Yi,i＞t)of the largest order statistics from a set of IID component lifetimes with coefficientbi(t)fori=1,...,n.Also,the coefficientai(t)andbi(t)depend onHand

3.Conclusions

In this papers we present several useful mixture representations for the reliability function of the residual life of a coherent system with independent but non-identically distributed components.These new results are used to compare the residual lives of coherent systems under different conditions based on the stochastic ordering of coefficien vectors(or components).The results form extensions of some known results when component lifetimes are independent and identically distributed.Furthermore,the utility of the results is illustrated in several examples in which the systems’residual reliabilities are computed and compared.For the case of systems with dependent and nonidentically distributed components,it is a very difficul problem.This will be worth further study and discussion in the future.

[1]F.J.Samaniego.System signatures and their applications in engineering reliability.New York:Springer,2007.

[2]R.E.Barlow,F.Proschan.Statistical theory of reliability and life testing.New York:Holt,Rinehart and Winston,1975.

[3]J.D.Esary,A.W.Marshall.Coherent life functions.SIAM Journal on Applied Mathematics,1970,18(4):810–814.

[4]F.J.Samaniego.On closure of the IFR class under formation of coherent systems.IEEE Trans.on Reliability,1985,34(1): 69–72.

[5]S.C.Kochar,H.Mukerjee,F.J.Samaniego.The signature of a coherent system and its application to comparisons among systems.Naval Research Logistics,1999,46(5):507–523.

[6]J.Navarro,J.M.Ruiz,C.J.Sandoval.A note on comparisons among coherent systems with dependent components using signatures.Statistics and Probability Letters,2005,72(2): 179–185.

[7]J.Navarro,F.J.Samaniego,N.Balakrishnan,et al.On the application and extension of system signatures in engineering reliability.Naval Research Logistics,2008,55(4):313–327.

[8]J.Navarro,J.M.Ruiz,C.J.Sandoval.Properties of coherent systems with dependent components.Communications in Statistics-Simulation and Computation,2007,36(1):175–191.

[9]H.A.David,H.N.Nagaraja.Order statistics.3rd ed.New Jersey:John Wiley and Sons,Hoboken,2003.

[10]A.Satyanarayana,A.Prabhakar.A new topological formula and rapid algorithm for reliability analysis of complex networks.IEEE Trans.on Reliability,1978,R-27(2):82–100.

[11]M.Asadi.On the mean past lifetime of components of a parallelsystem.Journal ofStatisticalPlanning andInference,2006, 136(4):1197–1206

[12]F.Belzunce,M.Franco,J.M.Ruiz.On partial orderings between coherent systems with different structures.Probability in the Engineering and Informational Sciences,2001,15(2): 273–293.

[13]P.J.Boland.Signature of indirect majority systems.Journal of Applied Probability,2001,38(2):597–603.

[14]S.Eryilmaz.On residual lifetime of coherent systems after therth failure.Statistical Papers,2013,54(1):243–250.

[15]S.Goliforushani,M.Asadi.Stochastic ordering among inactivity times of coherent systems.Sankhya,2011,73(2):241–262.

[16]S.Goliforushani,M.Asadi,N.Balakrishnan.On the residual and inactivity times of the components of used coherent systems.Journal of Applied Probability,2012,49(2):385–404.

[17]K.Jasinski,J.Navarro,T.Rychlik.Bounds on variances of lifetimes of coherent and mixed systems.Journal of Applied Probability,2009,46(3):894–908.

[18]B.E.Khaledi,M.Shaked.Ordering conditional lifetimes of coherent systems.Journal of Statistical Planning and Inference,2007,137(4):1173–1184.

[19]X.Li,Z.Zhang.Some stochastic comparisons of conditional coherent systems.Applied Stochastic Models in Business and Industry,2008,24(6):541–549.

[20]V.Vapnik.J.Navarro,R.Rubio.Comparisons of coherent systemsusing stochastic precedence.Test,2010,19(3):469–486.

[21]J.Navarro,T.Rychlik.Reliability and expectation bounds for coherent systems with exchangeable components.Journal of Multivariate Analysis,2007,98(1):102–113.

[22]J.Navarro,F.J.Samaniego,N.Balakrishnan.The joint signature of coherent systems with shared components.Journal of Applied Probability,2010,47(1):235–253.

[23]J.Navarro,F.Spizzichino.Comparisons of series and paral-lel systems with components sharing the same copula.Applied Stochastic Models in Business and Industry,2010,26(6): 775–791.

[24]A.Parvardeh,N.Balakrishnan.Conditional residual lifetimes of coherent systems.Statistics and Probability Letters,2013, 83(12):2664–2672.

[25]M.Shaked,A.Suarez-Llorens.On the comparison of reliability experiments based on the convolution order.Journal of the American Statistical Association,2003,98(463):693–702.

[26]Z.Zhang.Ordering conditional general coherent systems with exchangeable components.Journal of Statistical Planning and Inference,2010,140(2):454–460.

[27]Z.Zhang.Mixture representations of inactivity times of conditional coherent systems and their applications.Journal of Applied Probability,2010,47(3):876–885.

[28]Z.Zhang,X.Li.Some new results on stochastic orders and aging properties of coherent systems.IEEE Trans.on Reliability, 2010,59(4):718–724.

[29]Z.Zhang,W.Q.Meeker.Mixture representations of reliability in coherent systems and preservation results under double monitoring.Communications in Statistics-Theory and Methods,2013,42(3):385–397.

[30]Z.Zhang,W.Q.Meeker.The residual lifetime of surviving components from failed coherent systems.IEEE Trans.on Reliability,2014,63(2):534–542.

[31]J.Navarro,N.Balakrishnan,F.J.Samaniego.Mixture representations of residual lifetimes of used systems.Journal of Applied Probability,2008,45(4):1097–1112.

[32]M.Shaked,J.G.Shanthikumar.Stochastic orders.New York: Springer,2007.

[33]J.Navarro,N.Balakrishnan,F.J.Samaniego.Signaturebased representations for the reliability of systems with heterogeneous components.Journal of Applied Probability,2011, 48(3):856–867.

Biographies

Zhengcheng Zhangis a professor at the School of Mathematics and Physics,Lanzhou Jiaotong University.He obtained his Ph.D.in 2008 at Lanzhou University.He has published twenty referred journal papers on reliability theory,applied probability and statistics.Hisresearch interests include system signatures,stochastic ordering,reliability,and survival analysis.

E-mail:zhzhcheng004@163com

Narayanaswamy Balakrishnanis a professor of statistics at McMaster University,Hamilton,Ontario,Canada.He received his B.S.and M.S.degrees in statistics from the University of Madras, India,in 1976,and 1978,respectively.He finishe his Ph.D.in statistics from Indian Institute of Technology,Kanpur,India,in 1981.He is a fellow of the American Statistical Association,and a fellow ofthe Institute of Mathematical Statistics.He is currently the editor-in-chief of Communications in Statistics.His research interests include distribution theory,ordered data analysis,censoring methodology,reliability,survival analysis,and statistical quality control.

E-mail:bala@mcmaster.ca

10.1109/JSEE.2015.00144

Manuscript received September 12,2014.

*Corresponding author.

This work was supported by the National Natural Science Foundation of China(11161028;71361020).