Minimal Repair Redundancy for Coherent System in its Signatures Representation

The signature of a coherent system, as in [1], with independent andidentically distributed component lifetimes, as deifned by Samaniego, [2], is a vector whose i-th coordinate is the probability that the I-th component 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 distributed (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 independent and identically distributed random variables with absolutely continuous distributionF. Then the signature vectorαis defined as

The signature of a coherent system, as in [1], with independent andidentically distributed component lifetimes, as deifned by Samaniego, [2], is a vector whose i-th coordinate is the probability that the I-th component 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 distributed (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 T 1 ,…,T n , which are independent and identically distributed random variables with absolutely continuous distributionF.Then the signature vectorαis defined as 1 ( , , ) whereα i = P(T= T (i) ) and the  are the or- Clearly, under such conditions,  ,   is an almost sure (P-a.s.) partition of the probability space, with probability P, and (2) Samaniego [3], Kochar, et al. [4] and Shaked and Suarez-Llorens [5], extended thesignature concept to the case where the components lifetimes T 1 ,…,T n , of a system are exchangeable (i.e. the joint distribution function, F(t 1 ,...,t n ), of (T 1 ,…,T n ) is the same for any permutation of (t 1 ,...,t n ), an interesting and practical situation in reliability theory.
Concerning an improvement to system reliability, in its signature representation, through a redundancy operation of its components, and in view of the identically (exchangeable) distribution component lifetimes conditions, to maintain a system with its structural relation , we choose to apply the minimal repair redundancy.Intuitively the minimal repair 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

    P S t T s P T t s T s
However in the context of system signature the approach 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 information level, where the dependence (exchangeability) can be considered and the redundancy operations can be set through a compensator transform.

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, independent and identically distributed, the expected dynamic 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 simplicity 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 evolution on time recovering the dynamical system signa- , as in Sama- niego et al. [7] and the original coherent system signature in the set   n T t  as in [2].
In our general setup, we consider the vector T 1 ,…,T n of n component lifetimes which are finite and positive random variables defined in a complete probability space   , the index set of components.The lifetimes 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 P-a.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 T 1 , T 2 , … , T n , a s t h e y a p p e a r i n t i m e a n d b y where e is a fictitious mark not inE.Therefore the se- T X  defines a marked point process.
The mathematical formulation of our observations is given by a family of sub  -algebras of , denoted by    0 , where satisfies the Dellacherie ( [8]),conditions of right continuity and completeness, and T is the system lifetime 1 min max where K j , 1 ≤ j ≤ k are minimal cut sets, that is, a minimal set of components whose joint failure causes the system fail.Intuitively, at each time t the observer knows if the events or not and if they have, he knows exactly the value T (i) (T)and the mark X i .We assumed that T 1 ,…,T n 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
 is an t  -submartingale and from the Doob-Meyer de- composition we know that there exists a unique t  - predictable process is absolutely continuous by the totally inaccessibility of T i , 1 ≤ i ≤ n.We also define the lifetime T (i)j through its process The compensator process is expressed in terms of the conditional probability, given the available information 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 observations available up to, but not including, the present.
y, re,Y , as the first time fr Convenientl we define the critical level of the componentj for thei-th failu (i)j om which onwards the failure of component j lead to system failure at s the following results: notation, in the set Under the above where time of a coherent system of ordern, with component lifetimes T ,…,T which are totally in-LetT be the life accessible T 1 ,…,T n -stopping time.Then, under the above notation and at complete information level, we have with Remarks 2.3.
i) In the case of independent and identically distributed lifetimes we have rro et al., [10], asked, it is plausible to analyse the case of dependent and identically distributed lifetimes ( any way, its holds true for exchangeable 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 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 T 1 ,…,T n which are independent and identically distributed with continuous distribu-tionF.Then, where Definition 2.5.
Let T be the lifetime of a coherent system omponent lifetimes T 1 ,…, which are indepennd identically distribute dom vari les with absolutely continuous distribution F. Then the dynamic 1 1.
T n dent a d ran ab signature vector  is defined as where the order statistics of ,1 .

Minimal Repair and System Signature
It is well known that there exists a bijective relation between the space of all distributions functions and the -compensators space characterized by the so called Doléans exponential equation where is its discrete part.Therefore, to detect the effects of the independent (exchangeable) times component minimal repairs in the ordered statistics and in the signatures β i life itself, we are going to consider the minimal repair operation throug sator transform, as in Bueno [11].

T t Minimal Repair Operation
We are concerning with an improvement of the component lifetime Ti through a transformation of the t  - compensator process A i (t) of the counting process . The compensator process transform is in the form A 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 ca inima r transfor this ap- [9](see A dix).In ou se of m l repair (see [11]) the compensato m is in the form It is remarkable (Norros, [12]) that the continuous components compensator processes at its final points,  , 1 It is well known (Arjas et al. [13]) that     and the sur- 1 ln e e j j j 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 thi dependent (exchangea al repairs affects the dependent ordered statistics and the signa- where T k is a totally inaccessible t  -stopping time rep- resenting the lifetime of the componentk.Under the minimal repair transform where We observe that the -compensator of thei-th is set as: Also, the component -compensator c form: e of a minimal repair transformation of the component k, through its -compensator we have: In the cas Therefore, the effect of the componentk minimal repair compensator transform, in the compensator of the i-th failure is through the i-th term of the last summation.
, by Girsanov Theorem, under the measure and the eff i N ect of a minimal repair compensator transform, of the component k, in the compensator of the i-th failure is In view ally (exchange ility) distribution compone s conditions in the signatures defi st consider the minimal repair opera-of the identic ab nt lifetime nition we mu tions in all component lifetimes under the measure Q δ defined by the Radon Nikodym derivative and we have, using Girsanov Theorem, the resu following lt: where T j is a totally inaccessible t  -stopping time rep- resenting the lifetime of the componentj.Under minimal repair transform the where -stopping 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 component lifetimes we have: where and Proof Firstly, we note that, underQ δ , the lifetimes are independent: ),of Also, as ,1  are identical and we conclude that theT i lifetimes are identically distributed under Q δ .Therefore the signature decomposition under Q δ remains true.Furthermore  The equivalence in the third equality is justified in Norros [12] which defines theP a:s: inverse of A j (t),.As, under the hypothesis, Also, we have and uccessive Minimal Repairs More generally, we intend to define a measure n the which is obtained whenT i is deferred n times i sense of minimal repair.The measures 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   reason s follows: ppos hat we choose an w with pro dis-We a Su e t bability tributionP and starting proceeding at time 0. Su T i occurs.In order to make a minimal repair, we have fies istribution among the candidates satisfying these conditions.Indeed we choo accordingto ddenly to change our w to another, say w 0 which is indistinguishable fromw strictly before the timeT i (w) and satis-T i (w 0 ) >T i (w).Moreover, w 0 should be chosen according to an appropriated d se w 0   .,the value of the proc- is the history strictly before T i Thus choosingw 0 according to   il.Indeed, as in Section 3.1, its holds forn = 1.Suppose that its holds n fixed.We have to prove that:

S, for any random variable
As the process the forth above equality is true.The sixth equality follows from Dellacheries integration formula.
(52) and we conclude that the number of minimal repairs occurring beforeT i is modeled by a doubly stochastic Poisson process.
e realization of an equal and finite number of minimal repairs of each totally inaccessible stopping time representing the components lifet es.Asthe continuous components compensator processes at its final points, where Follows that, under the corre mpensator transform we can enunciate the Theorem.
In this paper we get resultsin a general set up in which a coherent system can be set as a stochastic process including 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 aftics in the signature representation.ansforms can help to answer such a The compensator tr question.Note that, even if the components are independent and identically distributed, the order statistics are not.We conclude that, in the general setting, we can develop some classical properties in reliability theory.

A
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 Applied probability.25, 630 -635.doi:10.2307/3213991999).Stochastic Models in-New York.doi:10.1007/b97596Life Testing: Probability models.Hold, Inc. Silver Spring, MD. 981).Point Processes and Queues: Martingale Dynamics.Springer-Verlag, New York.re of a coherent ao PauloUniversity, S~ao Paulo,