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Willing to work in reliability theory in a general set up, under stochastically dependence conditions, we intend to characterize a not identically spare standby redundancy operation through compensator transform under a complete information level, the physic approach, that is, observing its component lifetime. We intend to optimize system reliability under standby redundancy allocation of its components, particularly, under minimal standby redundancy. To get results, we will use a coherent system representation through a signature point process.

In reliability theory the main application of redundancy is to allocate a redundant spare in a system component position in order to optimize system reliability. For instance, see [

There are two common types of redundancy used in reliability theory, namely active redundancy, which stochastically leads to consider maximum of random variables and standby redundancy, which stochastically leads to consider convolution of random variables.

For a k-out-of-n system, [

Few papers attained to the case where the components are stochastically dependent. Reference [

In this paper, we intend to analyze a not identically spare standby redundancy allocation for a coherent system of dependent components without simultaneous failures, at component level, under a coherent system signature point process representation and prove that it is optimal to perform standby redundancy on the weakest component of a coherent system in order to optimize system reliability.

In Section 2 we characterize a not identically spare standby redundancy through compensator transform for dependent components. In Section 3 we resume mathematical details of signature point process representation of a coherent system and in Section 4 we investigate the best standby redundancy allocation in a dependent components coherent system in order to optimize system reliability.

We observe that each component in standby redundancy has two phases, standby and operation under which they can fail. Depending on component failures characteristics during these phases, standby redundancy is classified into the following three types:

1) Hot standby: Each component has the same failure rate regardless of whether it is in standby or in operation. Since the failure rate of one component is unique and is not affected by the other components, the hot standby redundancy consists of stochastically independent components.

2) Warm standby: A standby component can fail, but it has smaller failure rate than the principal component.

Failure characteristics of the component are affected by the other, and warm standby induces dependent component failures.

3) Cold standby: Components does not fail when they are in standby. The components have non-zero failure rates only when they are in operation. A failure of one principal component forces a standby component to start operation and to have a non-zero failure rate. Thus, failure characteristics of one component are affected by the others, and the cold standby redundancy results in mutually dependent component failures.

In what follows, we consider to observe two lifetimes T and S, which are finite positive random variables defined in a complete probability space

satisfies Dellacherie’s conditions of right continuity and completeness. We assume that

In our general set up and in order to simplify the notation, in this paper we assume that relations such as ⫁=, ≤, <, ≠, between random variables and measurable sets, always hold with probability one, which means that the term P-a.s., is suppressed.

We recall that a positive random variable T is a

Generally, standby redundancy gives to the component an additional lifetime. In our context the standby operation of S by T is defined as the improvement of S by

Furthermore, in relation to

The compensator process is expressed in terms of conditional probability, given the available information and generalizes the classical notion of hazard. Intuitively this corresponds to produce whether the failure goes to occur now, on the basis of all observations available up to, but not including, the present.

The well known equivalence between distributions functions and compensator processes follows from [

In the case of independent lifetimes, the survival function of the improved lifetime by

Therefore the

In this fashion and preserving the independence case interpretation, we define, for dependent lifetimes, the

and

We observe that

Following this thinking, as a predictable compensator is unique we are going to find a probability measure under which

To proceed we consider the compensator transform

To prove the main Theorem of this section we are going to use the following Lemma:

Lemma 2.1 Under this section assumptions, the following process

is a nonnegative

Proof We consider the

It is sufficient to prove that the process

is a bounded

Note that, for any

where

On the set

Otherwise, on the set

As the integrand

is a

Secondly, we consider the compensator transform

and with the same argument used to prove Lemma 2.1 we can prove Lemma 2.2:

Lemma 2.2 Under this section assumptions, the following process

is a nonnegative

Now, we can write the main theorem:

Theorem 2.3 Under this section assumptions, the following process

is a nonnegative local

Proof. Using Lemma 2.1, Lemma 2.2 and the Stieltjes differentiation rule we have

As by assumption

We are looking for a probability measure Q, such that, under Q,

Under certain conditions, it is possible to find Q. Indeed assume that the process

Nikodyn derivative

where

Remark 2.4. In reference to the first paragraph of this section, in the above setting we can identify the measure

In the case of cold standby redundancy, T does not fail before S, we can consider S < T and we have

In the case where T and S are identically distributed, we have

which can be used to define a standby redundancy through compensator transform when the standby component and the component in operation are stochastically dependent but identically distributed as in [

Due its importance we present these results in this section which appear in [

The evolution of components in time define a marked point process given through the failure times and the corresponding marks. We denote

The mathematical description of our observations, the complete information level, is given by a family of sub σ algebras of, denoted by

satisfies the Dellacherie conditions of right continuity and completeness.

Intuitively, at each time t the observer knows if the event

We consider, conveniently, the lifetimes

The behavior of the point process

Theorem 3.1 Let

Proof. From the total probability rule we have

As T and

we conclude that

The above decomposition allows us to define the signature process at component level.

Definition 3.2 The vector

Remark 3.3 We note that the above representation can be set in two way. We would prefer the one which preserves the component index because, by example, we could talk about the reliability importance of component j for the system reliability at the k-th failure.

Also, as

form a partition of Ω and

Remark 3.4 Using Remark 3.3 we can calculate the system reliability as

If the component lifetimes are continuous, independent and identically distributed we have,

recovering the classical result as in [

Remark 3.5 The marked

Follows that, from Doob-Meyer decomposition, there exists an unique

The

Theorem 3.6 Let

Proof. We consider the process

It is left continuous and

is a

is a

We are concerned with the problem of where to allocate a spare component using standby redundancy in a coherent system in order to optimize system reliability improvement. We let

Definition 4.1 Consider two point processes,

which are,

Also, we are going to use the following result from [

Theorem 4.2 Consider two point processes,

for all decreasing real and right continuous function with left hand limits 𝜓, which implies

In this first subsection we resume the results from [

In a random environment where the component is affected by the behavior of other components, [

with

The result is: under the measure Q defined by the Radon Nikodin derivative

Observe that

and, in the absolutely continuous case, where

Recovering our setting, let

Theorem 4.1.1 Let be let

Proof From Theorem 3.6 we have to compare system’s compensators expectation values on the form

for

The final result follows from Theorem 4.2

In what follows we consider an unique spare with lifetime S, as in Section 2, with compensator processes

Theorem 4.2.1 Let be let

Proof. Follows, from Section 2, that the standby redundancy through compensator transform of the component i by a spare with compensator

Clearly, it is sufficient to prove for

The final result follows from Theorem 4.2.

As by hypothesis,

We can, also consider two spares with lifetimes

Corollary 4.2. Let be let

An efficient method to optimize the reliability of a coherent system is to add redundancy components to the system. Therefore it is very significant to know about the allocation which best optimizes system reliability.

In the last decade, many researchers devoted themselves to this topic, in general analyzing k-out-of-n systems and following a natural and classical approach: considering that the components lifetimes were stochastically independent and to observing the system at its level through

Few papers attempt to the case where the components are stochastically dependent without simultaneous failures. [

getting results for k-out-of-n systems.

With recent results in signature theory and its extension to a signature point process, we generalize results from k-out-of-n to coherent systems, particularly for minimal standby redundancy and standby redundancy.

It is also important to note the characterization of standby operation results with not identically spare. The discussion about this new approach and the classical one can be set comparing results of

This work was partially supported by São Paulo Research Foundation (FAPESP), grant 2015/02249-1.

da Costa Bueno, V. (2016) Standby Redundancy Allocation for a Coherent System under Its Signature Point Process Representation. American Journal of Operations Research, 6, 489-501. http://dx.doi.org/10.4236/ajor.2016.66045