ABSTRACT

In a relay system of dependent components, the failure to close reliability measure is given as a Girsanov transform of the failure to open reliability measure.

1. Introduction

As in Barlow and Proschan [1], a complex coherent reliability system is completely characterized by its structure function assuming values in the set and where is a stochastic process assuming values in. The stochastic process, represents the state of the i-th component. if the component i is working at time t and i if the component i is in a failed state at time t. Also, the system state has such an interpretation, is increasing in each coordinate and each component is relevant, that is, there is a time t and a configuration of in which the functioning of the component i is fundamental for the functioning of the system.

A relay system is subject to two kinds of failure: failure to close and failure to open. Similarly, circuits construct from these relays are subject to the same kinds of failure. If the i-th relay responds correctly to a command to close, (that is, closes), and otherwise, i = 1; 2 and if the circuit responds correctly to a command to close (that is, closes) if, and only if, at least one of its components responds correctly to a command to close, and 0 otherwise, then,

is a parallel system. Next, let if the i-th relay responds correctly to a command to open (that is open), and otherwise, i = 1; 2. Let if the circuit responds correctly to a command to open (that is, open), and 0 otherwise. Then

a series system, the dual of, is the system representing the correct system response to a command to open.

Generally, given a structure function, its dual is defined by

where

.

The concept of dual structure is useful in analyzing system of components subject to two kinds of failures, such as relays system and safety monitoring systems. It is interesting and useful to note that both failure to close and failure to open can be analyzed using the same structure function. In this paper, under dependence conditions, we analyze the dual structure probability measure of a parallel system, through a transform of the failures counting processes compensators in the original probability space. In Section 2 we analyse the problem for a parallel system of two components. In Section 3 we generalize the results and in Section 4 we discuss some reliability preservation properties.

Without loss of generality, firstly, we consider a parallel system of two components. We observe two component lifetimes T and S, which are positive random variables defined in a complete probability space through the family of sub—algebras of, where

,

satisfies Dellacherie’s conditions.

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. In what follows we assume that S and T are totally inaccessible -stopping time and that, that is, the lifetimes can be dependent but simultaneous failures are ruled out.

The parallel operation of S and T is defined by the maximum between S and T and denoted by

If we denote the survival functions of S and T as

and respectively, it follows from Arjas and Yashin [2], that, under some conditions, the -compensator processes of and are given by

and

.

We assume such conditions and as S and T are totally inaccessible-stopping time the compensator processes are continuous.

Now we calculate

and therefore, in the set the -compensator of is

We intend to define compensator transformation of and to and in the way that the above expression is the sum which characterizes uniquely the lifetime of a series system and, therefore, the dual of such a parallel system. As this operation is symmetric on S and T, the idea is to combine compensator transformations in and.

Firstly, 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 the above hypothesis the following process

is a nonnegative local -martingale and

Proof We consider the localization sequence, the -stopping time defined By

It is sufficient to prove that the process

is a bounded -martingale.

For any -stopping time we can write

The procedure is easy: On the set we have

Otherwise, on the set,

As the integrand

is an -predictable process and is an -martingale, is an -martingale with

Secondly, we consider the compensator transform

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

Lemma 2.2 Under the above hypothesis the following process

is a nonnegative-martingale with

Observe that the same expression for the -compensator of is obtained through the transformation:

Then, we propose the compensator transforms:

and

to prove the main Theorem:

Theorem 2.3 Under the above hypothesis the following process

is a nonnegative local -martingale and.

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

As by assumption, and are continuous and we have

and therefore in an local -martingale with and the theorem is proved.

Now, we are looking for a probability measure Q, such that, under Q, becomes the - compensator of with respect to this modified probability measure.

Under certain conditions, it is possible to find Q. Indeed, assume that the process is uniformly integrable. Then it follows from well known results on point process martingales (Girsanov Theorem,Bremaud [3]) that the desired measure Q is given by the Radon Nikodyn derivative

Remark:

In the case where T and S are identically distributed, we have and the compensator transform is given by

which is used in Bueno and Carmo [4], to de_ne active redundancy operation when the component and the spare are dependent but identically distributed.

Parallel operations are very important in reliability theory: the performance of a parallel system are always better than the performance of any coherent system with the same components; it is used in replacement models, to optimize system reliability through active redundancy. However, if the component are stochastically dependent the reliability of a coherent system is a difficult and tedious calculation involving multivariate distributions. The calculation becomes more tractable under the assumption of a series system, in which case the reliability is the survival function of a multivariate positive random vector. It is also can be easily done through the compensator processes. We can show this argument easily:

Corollary 2.4

Under the hypothesis of Theorem 2.3, and under Q such that

we have

Proof As the compensators are deterministic before any failure, we can write:

As an application we calculate the Barlow and Proschan [1] reliability importance of a component for the system. In the independent and absolutely continuous case, it is the probability that the component causes the system failure. For dependent components, this quantity is

where

This expression is an extension of the Barlow and Proschan reliability importance, by Bueno and Menezes [5] where the importance of the component S for the system is

Now, we ask how we can use Corollary 2.4 to calculate the Barlow and Proscha relibility importance of component S for the system reliability. In a series representation we have:

In the above we use that

and

We expected such a relation since that the reliability of the dual system is and the measure Q is relative to the random vector To clarify this argument, suppose that we can define the reverse times, and such that the events and are equivalent to and respectively, then

2. A General Parallel System

The structural relationship between the lifetime, of a parallel system and its components lifetime is given by

We intend to define a compensator transform to characterize the parallel system as a series system. As in Section 2, the idea is to combine compensator transformations on the compensator process of the lifetime counting failure process of component

For define the compensator transform

where

and

Under the above hypothesis and notation we have

The main results, which follows from an adaptation of Girsanov Theorem, is

Theorem 3.1

Under the above hypothesis the following process

is a nonnegative -martingale with

Remarks:

1) We have

2) If the components are dependent but identically distributed, we have

and

Therefore

Remark:

As in Barlow and Proschan [1], we assume the series parallel decomposition of a coherent system:

where are minimal cut sets, that is, a minimal set of components whose joint failure causes the system to fail.

We can also define, for each the minimal parallel cut structure

and we can write

Therefore, using the compensator series transformation for each we get the series transformation for the system.

3. Preservations Results

In many reliability situations, we encounter structures of coherent systems where components share a load, so that a failure of one component results in increased load on each of the remaining components. Furthermore, the components in a coherent system could be subjected to the same set of stress. In such cases, the random variables of interest are not independent but rather associated.

Therefore, it is very interesting to verify whether the association properties of the lifetimes under P are preserved under Q.

We introduce the association definition. For this we give the concept of an upper set in

A Borel set is called an upper set if for any implies that. In the univariate case, is equal to either or.

Definition 4.1

The random variables (or the corresponding random vector) are associated, if for all upper sets and of

In particular, this definition, formulated by Esary, Proschan and Walkup [6], Esary and Proschan [7], is useful to produce upper and lower bounds for system reliability. The measure Q preserves this property from the measure P.

Theorem 4.2

If T is associated under, then also under.

Proof

We consider the upper sets in, and, and the uniformly integrable -martingale,

and

It follows that there exists a unique -predictable process, such that the covariance process is increasing, right continuous, with such that

is an -martingale.

Since a martingale has a constant expectation, we have

and

It follows that is associated if, and only if,

Now,

for a.s.

Classes of non-parametric distributions , such as increasing ( decreasing) failure rate (IFR (DFR)) distributions, new better (worse) than used (NBU (NWU)) distributions and others, have been extensively investigated in Reliability Theory. They can be used to achieve the benefit of a maintenance operation or to derive bounds on system reliability.

Several extensions of these concepts appeared in the literature, e.g. Harris [8], Barlow and Proschan [3], Marshall [9] and others. However, they all have in common that they don’t order the lifetime vectors in the sense of stochastic order as the univariate concept does. Arjas [10], considered to observe the components, continuously in time, based on a family of sub s-algebras

. Arjas, introduced the notion of increasing failure rate distribution (new better than used distribution) relative to IFR, (NBU) generalizing the conventional definition of IFR (NBU) and extending these classes into a multivariate form, denoted by MIFR, (MNBU).

In order to introduce the concepts of Arjas [10], we define the residual lifetime of at time as

Let

Definition 4.3

(Arjas, [10]) We say that is multivariate increasing failure rate relative to , denoted by MIFR, if for all and all open upper sets,

Remark: Arjas [1] proved that the class of upper sets in the above definition can be restrict to the class, of finite unions of open upper sets with corner point, where are positive rational numbers. An open upper set with corner point

is defined by

We want to prove that the MIFR class under is preserved under.

Theorem 4.4

If is multivariate IFR relative to, under, then also under.

Proof First, we show that is IFR, under. By hypothesis, we have

for all

Now

However is a measurable random variable and can be written as a suitable approximating step function

, where are constants and are measurable.

As, using the dominated convergence theorem we have

which is decreasing by hypothesis.

Herefore is IFR under. In order to prove that the vector is increasing failure rate relative to we use the relation

and the equivalent definition of MIFR in terms of the open upper sets with corner point as in the above remark.

As is MIFR,

.

The proof follows as above.

4. Conclusions

Relay system are very concerned in reliability theory, however, such a modeling is complicated under stochastically dependence conditions. The purpose of this paper is to provide a way to work with this situation using a point process martingale approach.

Some preservation important results in association and non parametric distributions classes useful in reliability theory are proved and an application in component importance is analised.

5. Acknowledgments

This work was partially supported by FAPESP, Proc. No. 2010/52227-0.

REFERENCES