American Journal of Oper ations Research, 2011,1,815 doi: 10.4236/ajor.2011.11002 Published Online March 2011 (http://www.scirp.org/journal/ajor) Copyright © 2011 SciRes. ..............................................................................................................................................................AJOR Minimal Repair Redundancy for Coherent System in its Signatures Representation Vanderlei da Costa Bueno Institute of Mathematics and Statistics, São Paulo University,Cx. Postal 66.281 05389970, São Paulo, Brazil Email: bueno@ime.usp.br Received February 24, 2011; revised March 20, 2011; accepted March 23, 2011 Abstract In this paper we discuss how to maintain the signature representation of a coherent system through a minimal repair redundancy. In a martingale framework we use compensator transforms to identify how the compo nents minimal repairs affect the order statistics in the signature representation. Keywords: System signature; Dynamic system signature; Coherent systems; Point processes martingales. 1.Introduction The signature of a coherent system, as in [1], with in dependent andidentically distributed component life times, as deifned by Samaniego, [2], is a vector whose ith coordinate is the probability that the Ith compo nent failure is fatal for the system. The key feature of system signatures that makes them broadly useful in reliabilityanalysis is the fact that, in the context of independent and identically distrib uted (i.i.d.) absolutely continuous components lifetimes, they are distribution free measures of system quality, depending solely on the design characteristics of the system and independent of the behavior of the systems components . A detailed treatment of the theory and applications of system signatures may be found in Samaniego, [2]. This reference gives detailed justification for the i.i.d. assumption used in the definition of system signatures. By the way there are a host of applications in which the i.i.d. assumption is appropriate, ranging from batteries in lighting, to wafers or chips in a digital computer to the subsystem of spark plugs in an automobile engine. Formally the definition is: LetT be the lifetime of a coherent system of order n, with components lifetimes T1,…,Tn, which are inde pendent and identically distributed random variables with absolutely continuous distributionF. Then the sig nature vectorαis defined as 1 (, , ) n (1) whereαi= P(T= T(i)) and the are the or der statistics of n () , 1 i Ti , 1 i Tin . Clearly, under such conditions, ,1 i TT in is an almost sure (Pa.s.) partition of the probability space, with probability P, and 1 n i ii t PT PT ii i PTTPTtT T t (2) Samaniego [3], Kochar, et al. [4] and Shaked and SuarezLlorens [5], extended thesignature concept to the case where the components lifetimes T1,…,Tn, of a system are exchangeable (i.e. the joint distribution function, F(t1,...,tn), of (T1,…,Tn) is the same for any permutation of (t1,...,tn), an interesting and practical situation in reliability theory. Concerning an improvement to system reliability, in its signature representation, through a redundancy opera tion of its components, and in view of the identically (exchangeable) distribution component lifetimes condi tions, to maintain a system with its structural relation ii PT t PTt , we choose to apply the minimal repair redundancy. Intuitively the minimal re pair redundancy gives to component i an additional lifetime as it had just before the failure. Clearly, in the case of independent component lifetimes the whole system is returned to the state it had just before the failure. A minimal redundancy of a lifetimeT produces the sum T + S whereS is called spare lifetime and PStT sPTtsTs (3)
V.C. BUENO 9 If the distribution function ofT is 1 tF t , the resulting lifetime T hasthe distribution function S 1PT S tSt PT , where 0 d ln t PT StPTtPT StTs Fs Ft FtFt ( 4) However in the context of system signature the ap proach of minimal repairs is not so clear: What are the effects of the independent (exchangeable) component lifetimes minimal repair in the ordered statistics and in the signatures itself? To answer such a question we consider dynamics signatures, as in a recent work by Bueno [6], in a general set up, under a complete infor mation level, where the dependence (exchangeability) can be considered and the redundancy operations can be set through a compensator transform. 2. Dynamic System Signature We consider, as in [6], the system evolution on time under a complete information level. In this fashion, if the components lifetimes are absolutely continuous, in dependent and identically distributed, the expected dy namic system signature enjoy the special property that they are independent of both the distributionF and the time t. This fact has significance beyond the mere sim plicity and tractability of the signature vector, reflect only characteristics of the corresponding system design and may be used as proxies for system designs in the comparison of system performance. Also the dynamic system signature actualizes itself under the system evo lution on time recovering the dynamical system signa ture in the set 1ii TtT Tt , as in Sama niego et al. [7] and the original coherent system signa ture in the set n Tt as in [2]. In our general setup, we consider the vector T1,…,Tn of n component lifetimes which are finite and positive random variables defined in a complete probability space ,,P , with P(Ti≠Tj) = 1, for all in ,,ijij 1, ,En, the index set of components. The life times can be dependent but simultaneous failure are ruled out. In what follows, to simplify the notation, we assume that relations such as between random variables and measurable sets, respectively, always hold with probability one, which means that the term Pa.s., is suppressed. ,,,, The evolution of components in time define a marked point process given through the failure times and the corresponding marks. We denote by T(1)<T(2)<…< T(n) the ordered lifetimes T1,T2,…,Tn, as theyappear in time and by : i ij jT T the corresponding marks. As a con vention we set 12nn TT and 12nn Xe where e is a fictitious mark not inE. Therefore the se quence 1 ,n nn TX defines a marked point process. The mathematical formulation of our observations is given by a family of sub algebras of , denoted by 0 tt , where ,, ,1,,0 11 i Tss i tT jnjE st X (5) satisfies the Dellacherie ([8]),conditions of right conti nuity and completeness, and T is the system lifetime 1 min max j i jkiK TT (6) where Kj, 1 ≤ j ≤ k are minimal cut sets, that is, a mi nimal set of components whose joint failure causes the system fail. Intuitively, at each time t the observer knows if the events , ii tjTt TX have either occurred or not and if they have, he knows exactly the value T(i)(T)and the mark Xi. We assumed that T1,…,Tn are totally inaccessible t stopping time. In a practical sense we can think of a totally inaccessible t  stopping time as an absolutely continuous lifetime. The simple marked point process , ,1ii t ij Xj T t N is an t submartingale and from the DoobMeyer de composition we know that there exists a unique t  predictable process ,0 ij t At , called the t  compensator of t ,ij N, with ,ij A00 and such that ,ij ,ijt NA tt is an martingale. ,ijt A is absolutely continuous by the totally inaccessibility of Ti, 1 ≤ i ≤ n. We also define the lifetime T(i)j through its process , ijiji i tt tPt Ptj FT TX (7) The compensator process is expressed in terms of the conditional probability, given the available infor mation and generalize the classical notion of hazards. Intuitively, this corresponds to producing whether the failure is going to occur now, on the basis of all obser vations available up to, but not including, the present. As ,ijt N can only count on the time interval 1, ii TT , th e corresponding compensator differen Copyright © 2011 SciRes. AJOR
V.C. BUENO 10 tial ij t dA failu must vanish outside this interval. To count theithre we let 1 iij j tt NN with t  compensator process 1 iij j tt . AA The t  compensator of 1jt T , correspond e is jt Ning to t th components lif hej etim 1 . ij i t A Follows that the com j t A t pensator o is (8) which is anpredictable process and therefore unique ( see Bremaud ([9])). y, re,Y, as the first time fr f 11 nn ij ij Nt t N 1 11 1 ii nn ij TtT ij AtA t t Convenientlwe define the critical level of the componentj for theith failu(i)j om which onwards the failure of component j lead to system failure at , ii Tj TX . We consider the t compensator process 0t of the point cess At pro 1Tt t N , of thefetimeT, such tt NA is an zero mmartingale system li that ean t with  .Pt Et NA Bueno [6], T tE prove Theorem 2.1. s the following results: notation, in the set Under the above Tt, the t  compensator of 1Tt Nt , is nn t 1 11 1ii Tt T ij ijij ij t AA AY (9) where max, 0.a a Theorem 2.2. time of a coherent system of ordern, with component lifetimes T,…,T which are totally in LetT be the life 1n accessible T1,…,Tn stopping time. Then, under the above notation and at complete information level, we have 1 1 11 1 , 1ii nn t i i TtT ij t i PT TXj PT T (10) with Remarks 2.3. i) In the case of independent and identically distributed lifetimes we have 1n T. 1 1 11 1ii ni TtT t ii PT T PTt PT T (11) rro et al., [10], asked, it is plausible to analyse the case of dependent and identically dis tributed lifetimes ( any way, its holds true for ex changeable distribution). In this case we have ii) Clearly, it is not seemingly true to think the general case of dependent components in the signatures context. However, as Nava 1 1 11 1ii nit TtT t iit PT T PTt PT T (12) Clearly, in the case of exchangeability, the expression in i) is holding. Corollary 2.4. LetT be the lifetime of a coherent system of order n, with component lifetimes T1,…,Tnwhich are independ ent and identically distributed with continuous distribu tionF. Then, 1i 1i n Tt ti PTt (13) where 1 1 ii i ii PT TPTT (14) PT TPTT with T 01 0,, 0 ni T and Definition 2.5. Let T be the lifetime of a coherent system omponent lifetimes T1,…, which are indepen nd identically distributedom variles with absolutely continuous distribution F. Then the dynamic 1 1. n i i of order n, with cTn dent ad ranab signature vector is defined as ,, 1n (15) where 1 1 ii i ii PT TPTT PT TPTT and the i T are the order statistics of ,1 . i Tin Copyright © 2011 SciRes. AJOR
V.C. BUENO 11 Remarks 2.6. We observe that (16) 3. Minimal Repair and System Signature It is well known that there exists a bijective relation be tween the space of all distributions functions and the compensators space characterized by the so called Doléans exponential equation 1 1 11 1 i i ij ij n TTXj i n YTT ijij j n ij ij PT TE E PYTT FTFY 1 ij j 1j j n t 1 eπ ct tst A Ft As (17) where ct A is the continuous part of t and c tAt t A is its discrete part. Therefore, to detect the effects of the independent (exchangeable) times component minimal repairs in the ordered sta tistics and in the signatures βi life itself, we are going to consider the minimal repair operation throug sator transform, as in Bueno [11]. . he Firs h compen 3.1Tt Minimal Repair Operation We are concerning with an improvement of the com ponent lifetime Ti through a transformation of thet  compensator process Ai(t) of the counting process 1i Tt it N . The compensator process transform is in the form (18) d,1 t iii tsAsin B 0 where 0 iss is an t predictable procs. Clearly, if 01s , the hazard process es i d ii As is lower thanprocess d the hazard i s and we under stand such improvement of the components i lifetime as a redundancy operation.The main tool in proach is the Girsanov Theorem which proof is in 2. in Appen r particular cainima r transfor this ap [9](see Adix). In ouse of ml repair (see [11]) the compensatom is in the form t is A 0 dln1 1i ii i i tAstAt A As in which case (19) 1 1 ii ii A t LAT (20) it N i AT t i It is remarkable (Norros,[12]) that the continuous components compensator processes at its final points, ,1 jj jn AT , are independent and identically dis m variables with standard exponential distribution. This holds no matter how dependent the actual lifetimes are and what the history, as lo multaneous failures are ruled out. Therefore tributed rando ng as si 1 ii EAT and we have d iii di Q AT P (21) It is well known (Arjas et al.[13]) that ln jj tt AF where 1 jj t ttPt FT F and the sur j vival function of the component j after the cr transform is ompensato ln 11ln ee jjj ttt jj BAA tt FF j t F re ction. s point we can ask how the in ble) lifetimes component minim . We remark that, in the following results, the proofs are heavily based in the fact that (2 2) covering the expression of the first Se At thidependent (ex changeaal repairs af fects the dependent ordered statistics and the signa turesβi i itself t  martingales summation is amartingale and th t et  compensator is unique. Lemma 3.1.1. Let k t be the t compensator of 1k Tt k Nt where Tkis a totally inaccessible t stopping time rep resenting the lifetime of the componentk. Under the minimal repair transform 0 d 1 t k As t Bk k k A s As (23) the t or cmpensatoof theith failure, i At , un der k Q , is transformed to n 1 ii j tt BB j (24) where ij ij tt BA if and jk , 0 d 1 tik ik ik i As k tAs A (25) s Copyright © 2011 SciRes. AJOR
V.C. BUENO 12 Proof We observe that the compensator of theith is set as: t (26) Also, the componentcompensator c form: j t failure 1 11 1ii nn iTtT ij ij jj tAtA A t an be set on the 1 1 1ii nn TtT j ij 1ij tAt A t (27) e of a minimal repair transformation of the component k, through itscompensator we have: In the cas t 1 1 1 d 1 l l l l t tn t nlk lk llk T T T T As As As As As Therefore, the effect of the componentk minimal re pair compensator transform, in the compensator of the ith failure is through the ith term of the last summa tion. 1 0 dd 11 kk kk k l kk Bt As As As As (28) with 00T. 1 1 1 1, d 11 i i ii i t nk Tt Tk ij jjk k T T 1 1, d 1 i ii i t nik Tij ik jjk ik T T t B As 1Tt tAs As (29) As As At As As 1 kk EAT , by Girsanov Theorem, under the measure d d k kk Q AT P , B(i)(t) is the compensator of t 1i Tt t and the eff i Nect of a minimal repair compensator transform, of the component k, in the compensator of the ith failure is d 1 ik ik ik ik As tAs As (30) In viewally (exchangeility) distribu tion compones conditions in the signatures defist consider the minimal repair opera of the identic ab nt lifetime nition we mu tions in all component lifetimes under the measure Qδdefined by the Radon Nikodym derivative 1 d Qdπn kk k P T (31) and we have, using Girsanov Theorem, the resu following lt: Corollary 3.1.2. Let j t be the t compensator of 1j Tt jt N where Tj is a totally inaccessible t stopping time rep resenting the lifetime of the componentj. Under minimal repair transform the d,1 t j As BtAsj n 01 jj j As (32) thet compensat i At or of the ith failure, , under ans Qδ, isformed in n tr 1 ii k k tt BB (33) where d 1 ik ik i ik As k tA As otally inaccessiblestopping time representing the lifetimes of ancomp ent system with lifetimeT, which are abso independent and identically distributed. Then, under the minimal repair transformation of all compo nent lifetimes we have: s (34) Theorem 3.1.3. Let T1,T2,…,Tn be t t onent coher lutely con tinuous * 1 n i ii Tt t QQ T (35) where 1 * 1 ; ii ii i TT TT QQ TT TT QQ (36) k (37) 1 1ln 1ln ii kikikik k ik ikik i Y FY FT FT n TTF Q 1 11 n iik ik k tt QTFF t (38) and 1 ddπn kk k P Q T Proof Firstly, we note that, underQδ, the lifetimes are in dependent: 11 11 11 1 ,..., ππ 11 ππ 1 π kkkk k kk nn nn kk kk kk n nn kk kkk k tt TT kt T TtTt Q EE AT AT tt QQ TT E k (39) Copyright © 2011 SciRes. AJOR
V.C. BUENO 13 ),of Also, as,1 j Tjn are identically distributed, the t compensatorsAj(t 1 j Tt are identical and we conclude that theTi lifetimes are identically distributed under Qδ. Therefore the signature decomposition under Qδremains true. Furthermore (40) The equivalence in the third equality is justified in Norros [12] which defines theP a:s: inverse of Aj(t),.As, under the hypothesis, 1 1 1 1, 1 1 π π1 π1 1 i i i jjij jjij jjij iT n k kkT k n kk t k j nn kkj jt kkj j jj t j QT T Tij AAT AAT AAT T QTE EAT E EAT EE AT AT EAT 1 n 11 k T k n 1 1 π11 π1 ii kk T j k j nn TX AT EAT 1j n ln jij AtFt where ijij t tpt FT we have: 1l n 1ln ij ij ij ij ijij ijij TT YY FF FF ln ln 1 d 1 ij ij ij ij jjT Tx jj T T F YT Y F EAT Exx e AT (41) Therefore 1 1ln 1ln i n ij ijijijijij j TT Q FYFYFTFT (42) Also, we have (43) And i 1 1 1, 1 1 1π1 π1 1 i ii jj ij jj ij nn kk Tt Tt k j nn kkj jAT At kkj j n jj AT At j Q Tt EAT QE EATEAT EAT ln 0 1 ed 11ln jj ij ij AT At jj tx ij ij F EAT xFt Ft (44) and 1 11 ln n ij ij ij itt QTFF t (45) uccessive Minimal Repairs More generally, we intend to define a measure 3.2. S i n Q n the which is obtained whenTi is deferred n times i sense of minimal repair. The measures 0 i Q , …,i n Q 1 are defined successively by 0 i Q P and i Q , 1.. d ii i nn T Pw QQ w bability measure (46) It can be proved that, for any n, the pro i n Q i is absolutely continuous with respect to P, with Radon Nikodyn derivative 1 dd ! i n n ii P Q T n (47) reasons follows: pposhat we choose an w with pro dis We a Sue tbability tributionP and starting proceeding at time 0. Su Ti occurs. In order to make a minimal repair, we have fies istribution among the candi dates satisfying these conditions. Indeed we choo accordingto ddenly to change our w to another, say w 0 which is indistin guishable fromw strictly before the timeTi(w) and satis Ti(w0) >Ti(w). Moreover, w0 should be chosen ac cording to an appropriated d se w0 . ,the value of the proc i T P ess .t P at Ti, where ,,0 iit TAtA t T (48) is the history strictly before T i Thus choosingw0 ac cording to .i T P we may proceed further as if nothing had happened. Intuitively, ifS is an t stopping time, the difference between theσalgebras and S is that, in it is known when but its not known what else hap ime S. For ex of a sys component, i S occurs, pen at tample, if S is the failure time stemt is known in , but at this time, we do not known what component will fa for some il. Indeed, as in Section 3.1, its holds forn = 1. Suppose that its holds n fixed.We have to prove that: 1 1 1 1! i Qi SESAT En (49) n i n S,for any random variablei T measurable. Copyright © 2011 SciRes. AJOR
V.C. BUENO 14 11 1 ! nn ii ii n TT i i QQ SESE ES AT EE n 0 1 1 ! 1 1 1! i i i i n n i it t i T n EES T A n (50) As the process 0! 1d i n n EE SAt t A 0 d 1! i EE SAt n 1d1 i n itTt EE S t A 1 !i n it ES t A n is predictable, the forth above equality is true.The sixth equality fol lows from Dellacheries integration formula. t 11 11 1! 1! i nn iiii T EES ES AT AT nn (51) Furthermore we have 1 0 e 1d e 1 !! i i t nn x il t A i n iAT At ii iA t A x Ex AT nni 1 11 ! ! n i ii n Tt T l n i E QAT n (52) and we conclude that the number of minimal repairs occurring beforeTi is modeled by a doubly stochastic Poisson process. e realization of an equal and fi nite number of minimal repairs of each totally inacces sible stopping time representing the components lifetes. Asthe continuous components compensator processes at its final points, ii iit Q Tt E Next we consider th t  im ,1 ii Ti stributed random v stribution, in the n , are inde pendt and identically diariables withard exponential di case en stand of a fixed configuration ,...,mmm, wherem is the num ber of minimal repairs of componenti, we can define a product probability measure in ... 1 πi m mm i QQ (53) where 1 dd i m m i i P Q T m . Follows that, under the corrempensator transform we can enunciate the Theorem. Theorem 3.2.3. Let T1,…,Tn be totally inaccessible t stopping time representing the lifetimes of at c spondin co n componen ohere system with lifetime T. Then, under the minimal configuration m and under the probability measu Where gt  nt repair re Qm, we have 1 i i Tt t QQ T (54) n mm i 1 m TT Q TT In this paper we get resultsin a general set up in which a coherent system can be set as a stochastic process in cluding the order statistics in the signature contex fference between this results and the previous one in the signature field is the dynamic aspect. In this setting we can realize redundancy oprations and ask about the reliability component importan question is to clear out how a component operation af tics in the signature representation. ansforms can help to answer such a Applied probability. 25, 630  635.doi:10.2307/3213991 999). Stochastic Models in New York. doi:10.1007/b97596 Life Testing: Probability models. Hold, Inc. Silver Spring, MD. 981). Point Processes and Queues: Mar tingale Dynamics.SpringerVerlag, New York. re of a coherent ao PauloUniversity, S~ao Paulo, * 1 ii imm ii TT QQ TT (55) 4. Conclusions m Q t. The main di ce. The main fect the order statis The compensator tr question. Note that, even if the components are inde pendent and identically distributed, the order statistics are not. We conclude that, in the general setting, we can develop some classical properties in reliability the ory. 5. Acknowledgements This work was partially supported by FAPESP, Proc. No. 2010/522270 6. References [1] Arjas, E. and Yashin, A. (1988). A note on randonm in tensities and conditional survivalfunctions. Journal of [2] Aven, T. and Jensen, U. (1 eliability. Springer Verlag, [3] Barlow and Proschan,F. (1981). Statistical Theory of Reliability and Reinhart and Wiston , [4] Bremaud ,P. (1 [5] Bueno, V.C. (2005). Minimal standby redundancy allo cation in a Koutofn:F systemof dependent compo nents. European Journal of Operation Research. 165, 786793.doi:10.1016/j.ejor.2003.01.004 [6] Bueno, V.C. (2010). Dynamics signatu system. Submited.S~ Brazil. Copyright © 2011 SciRes. AJOR
V.C. BUENO Copyright © 2011 SciRes. AJOR 15 n to E Transactions iability. International Se . (2003). On the com ppendix 1. Stopping time n extended and positive random variable τis an [7] Kochar, S., Mukherjee, H., Samaniego, F.(1999). The signature of a coherent systemand its applicatio comparisons among systems. Naval Research Logistic. 46, 507  523. doi:10.1002/(SICI)15206750(199908)46:5<507::AID NAV4>3.0.CO;2D [8] Navarro, J., Balakrishnan, N. and Samaniego, F.J. (2008). Mixture representation ofresidual lifetimes of used systems. Journal of Applied Probability. 45, 1097 1112.doi:10.1239/jap/1231340236 [9] Norros, I. (1986). A compensator representation of multivariate life length distributions, with applications. Scand. J. Statist. 13, 99112. [10] Samaniego, F. (1985). On closure of the IFR class un pari der formation of coherent systems. IEE in Reliability. R34, 6972.doi:10.1109/TR.1985.5221935 [11] Samaniego,F.J. (2007). System signatures and their ap plications in engineering rel ries in Operation Research and Management Science, Vol 110, Springer, New York. [12] Samaniego, F.J., Navarro, J. and Balakrishnan, N. (2009). Dynamic signatures andtheir in comparing the reliability of a new and used systems. Naval Research Logistic. 56, 577596.doi:10.1002/nav.20370 [13] Shaked, M., SuarezLlorens, A son of reliability experimentsbased on the convolu tion order. Journal of American Statistical Association. 98, 693702.doi:10.1198/016214503000000602 Acompensators ,1 iti An. Let 0 it t with t , be n negative ont pre all dr all dictable processes s , an uch that for fo 0t in1 , 0 t ii Bt s i Asd, With 1,..., n tt t and . t L Then 1 πe iii nttt i i i NAB T (56) is a noen negativ t l andocal martingale a non nega tive t sup herm d er ma ore, by rtin if E t gale. [L] =probabilFurt define δ 1, the e Radon Ni ity measur m deriva A A t  stopping time if, and only if, t edictabl for all imeτis called pre if an i 0t n ; an creasing t stopping t sequence 0 nn of ists such that stopping time, τn<τ, ex t n as ; anstopping tim t e n is totally inaccessible if 0P for all p dictable t stopping time re . Fathematibasis stochastic processes applied to reliability theory or a mcal of of Avnde Qδ tive h kody ddPL Q is such that, underQδ,Bi(t) is seethe booken a Jensen [14]. the A2. Theorem A.2 (Girsanov) Let Ti, 1 ≤ i ≤ n be totally inaccessible t  stoppingtimes, the point processes unique t compen i(t). In particular, we denote sator process of N 1, ,1,,1, ,1 ii it 1 iT t N , and dd i iP QL .
