Grabski F., Załęska-Fornal A. The models of non-renewal reliability systems with dependent lengths of components.

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THE MODELS OF NON-RENEWAL RELIABILITY

SYSTEMS WITH DEPENDENT LENGTHS OF

COMPONENTS

Grabski F., Załęska-Fornal A.

Naval University of Gdynia

Abstract: In that paper there are presented the models of the non-renewal reliability systems

with dependent times to failure of components. Their dependence follows from some common sources of shocks existing in the environment. It is assumed that the failure occurs only because of two independent sources common for two neighbour components. The series and parallel system with such dependent components are considered in that paper.

Key words: reliability, dependent components, series systems, parallel systems

In the Barlow & Proshan book [1975], there is defined, based on the reliability theory, multi-dimensional exponential distribution as a distribution of a random vector, the coordinates of which are dependent random variables defining life lengths of the components. Their dependence follows from some common sources of shocks existing in the environment. Using that idea we are going to present some examples of systems with dependent components, giving up the assumption that the joint survival probability is exponential and accepting the assumption that the failure occurs only because of two independent sources common for two neighbour components.

Assume that due to reliability there are n ordered components

) ,..., ,

(e1 e2 en

E  _.

Assume also that n+1 independent sources of shocks are present in the environment

) , ,..., , ( ₁ ₂ _₁  z z zn zn Z

and each component ei can be destroyed because of only two sources zi and zi+1.

Let Ui be non-negative random variable defining the time to failure of the component

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U (U1,U2,...,Un,Un1). (1)

Admit that the coordinates of that vector are independent random variables with distributions defined as follows

Gi(ui) P(Ui ui), i 1,2,...,n1.

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The life length of the component ei is a random variable satisfying

Ti min(Ui,Ui1), i 1,2,...,n . (3)

Notice that two neighbour components in the sequence (e1,e2,...,en) have one

common source of the shock – depend on the same random variable. The random variables T1, T2,…, Tn are dependent and their joint distribution is expressed by means

of the multidimensional reliability function can be easily determined:

. ) ( )) , max( ( ... )) , max( ( ) ( ) ), , max( ),..., , max( , ( ) ) , min( ,..., ) , min( , ) , (min( ) ,..., , ( ) ,..., , ( 1 1 2 1 2 1 1 1 1 2 1 2 1 1 1 2 3 2 1 2 1 2 2 1 1 2 1 n n n n n n n n n n n n n n n n t U P t t U P t t U P t U P t U t t U t t U t U P t U U t U U t U U P t T t T t T P t t t R                           Thus , ) ( ) , (max( ... )) , (max( ) ( ) ,..., , (t1 t2 tn G1 t1 G2 t1 t2 Gn tn 1 tn Gn 1 tn R    (4) where . 1 ,..., 2 , 1 ), ( ) ( 1 ) ( ) (u PU u  G u P T u i n Gi i i i i i i

The reliability functions of the components can be obtained as marginal distributions computing the limit of the function (4), when

      _ _ _ 0 ,..., 1 0 , 1 0 ,..., 0 1 ti ti tn t . Ri(ti) P(Ti ti)Gi(ti)Gi1(ti), i1,2,...,n (5)

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The bivariate reliability functions can be determined by computing the limit of (4), when           _ _ _ _ _ 0 ,..., ₁ 0 , ₁ 0 ,..., ₁ 0 , ₁ 0 ,..., 0 1 ti ti tj tj tn t . , 1 ,..., 2 , 1 , , 1 ), ( ) ( ) ( ) ( ) , ( ) , ( 1 1           n j i j i t G t G t G t G t T t T P t t Rij i j i i j j i i i i j j j j (6) , 1 ,..., 2 , 1 , , 1 ), ( )) , (max( ) ( ) , ( ) , ( 1 1 2 1             n j i j i t G t t G t G t T t T P t t Rij i j i i j j i i i i i i i (7)

It could be proved that

P(T1 t1 |T2 t2,...,Tn tn)P(T1 t1 |T2 t2) , (8) and generally 1 ,..., 2 , 1 ), | ( ) ,..., | (T t1 T1 t1 T t P T t T1 t1 i n P i i i n n i i i i . (9)

That property asserts that the life length of ei depends only on the life length of the next

component ei+1 , does not depend on the life lengths of the rest of the components. That is

a certain kind of Markov property.

If the object has a series reliability structure then, as well known, its life length T is the random variable defined by the formula

T min(T1,T2,...,Tn).

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Using (4) we can determine the reliability function: ₍ ₎ ₍ ₎_... ₍ ₎ ₍ ₎ ) ,..., , ( ) ,..., , ( ) ( ) ( 1 2 1 2 1 t G t G t G t G t t t R t T t T t T P t T P t n n n           R (11)

Let us compare that function with the reliability function of a series system in which the life lengths of the components T1, T2, …, Tn are independent and their

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reliability functions are defined by (5). Let R~(t), t0 be a reliability function of that system. It is of the following form:

) ( ) ( ... ) ( ) ( ) ( ) ( ) ( ... ) ( ) ( ) ( ) ( ) ( ... ) ( ) ( ) ,..., , ( ) ( ) ( ~ 3 2 1 3 2 2 1 2 1 2 1 t t G t G t G t G t G t G t G t G t G t G t R t R t R t T t T t T P t T P t n n n n n n R R            (12) Thus, for t0 ) ( ) ( ~ _t _R _t R  holds.

That inequality means that the reliability of a series system with dependent ( in the considering sense) life lengths is greater (or equal) than the reliability of that system with independent life lengths and the same distributions as the marginals of T1, T2, …, Tn .

Accepting the assumption about independence of the life lengths of the components even though the random variables describing the life lengths are dependent, we make an obvious mistake but that error is „safe” because the real series system has a greater reliability. That estimation is very conservative.

Example 1.

Assume that a non-negative random variable Ui , describing time to failure of the

component caused by the source zi has a Weibull distribution with parameters

1 ,..., 2 , 1 , , _i i  n i   _.

It means that, for ui_{> 0}

.

1 ,...,

2 ,

1 ,

)

(

)

(

_u



_P

_U



_u



_e



_i



_n



G

i i iu i i i i  

Then the reliability function of a series system with dependent components has a form

) ... ( 1 2 1 1 1 1 1

)

(

)

(

...

)

(

)

(

)

(

)

(

     







an n a _t t n n

t

G

t

e

G

t

G

t

G

t

T

P

t

 

R

For n=3 and 2 . 0 , 3 , 1 . 0 , 2 . 2 , 2 . 0 , 2 , 1 . 0 , 2 . 1 ₁ ₂ ₂ ₃ ₃ ₄ ₄ 1                 we get ) 2 . 0 1 . 0 2 . 0 1 . 0 ( 1.2 2 2.2 3 ) ( ) (_t _ _P _T __t __e t  t  t  t R

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0.5 1 1.5 2 0.2 0.4 0.6 0.8 1

Fig.1. The graph of the series reliability function with dependent components . The form of the reliability function R~(t), t0 of the series system with independent life lengths, the same marginals with the joint life distribution has a following form ) 2 . 0 2 . 0 4 . 0 1 . 0 ( 1.2 2 2.2 3 ) ( ) ( ~ _t _ _P _T __t _ _e t  t  t  t R

and its graph is presented on Fig.2.

Fig.2. The graph of the series reliability function with independent components. The life length of the object of a parallel structure is a random variable defined by

0.5 1 1.5 2 0.2 0.4 0.6 0.8 1

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T  max(T1,T2,...,Tn) .

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Let us compute the reliability function of that object:

}). { ... } { } ({ ) ,..., , ( 1 ) ( 1 ) ( ) ( 2 1 2 1 t T t T t T P t T t T t T P t T P t T P t n n                   R (14) Using the formula of probability of a sum of events we obtain

) ,..., , ( ) 1 ( ... ) , , ( ) , ( ) ( ) ( ) ( 2 1 1 1 , , 1 , 1 t T t T t T P t T t T t T P t T t T P t T P t T P t n n n k j ijk i i j k n j ij i i j n i i                            R

Hence and from (4), (6), (7) we get ). ( ) ( ... ) ( ) ( ) 1 ( ... ... ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 2 1 1 1 1 2 1 11 , 1 1 1 1 t G t G t G t G t G t G t G t G t G t G t G t G t G t n n n n i i i i n j ij i j j i i n i i i                  



R (15)

Particularly, for n=3 we have,

). ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 4 3 2 3 2 1 4 3 3 2 2 1 4 3 2 1 4 3 2 1 4 3 2 3 2 1 4 3 3 2 2 1 t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t              R (16)

If T1, T2, …, Tn are independent then

. )] ( ) ( 1 [ ... )] ( ) ( 1 [ )] ( ) ( 1 [ 1 )] ( 1 [ ... )] ( 1 [ )] ( 1 [ 1 ) ( )... ( ) ( 1 ) ,..., , ( 1 ) ( 1 3 2 2 1 2 1 2 1 2 1 t G t G t G t G t G t G t R t R t R t T P t T P t T P t T t T t T P t n n n n n                        R

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For n=3 . )] ( ) ( 1 [ )] ( ) ( 1 [ )] ( ) ( 1 [ 1 ) (t   G1 t G2 t G2 t G3 t G3 t G4 t R

After multiplication we get

). ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 4 3 3 2 2 1 4 3 3 2 4 3 2 1 3 2 2 1 4 3 3 2 2 1 t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t        R Notice that )]. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( [ ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 4 3 2 1 4 3 4 1 2 1 4 1 3 2 4 3 3 2 2 1 4 3 3 2 4 3 2 1 3 2 2 1 4 3 2 3 2 1 t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t G t t                R R

Let Ai , i=1, 2, 3, 4 be independent events with the following probabilities . 4 , 3 , 2 , 1 ), ( ) (A G t i  P i i

The expression in the square bracket can be rewritten as

)). ( ) (( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 4 3 2 1 4 1 4 3 2 1 4 3 4 1 2 1 4 1 A A P A A P A A P A P A P A P A P A P A P A P A P A P A P A P A P             As ) ( ) ( 1 2 3 4 4 1 A A A A A A      , so )) ( ) (( ) (A1 A4 P A1 A2 A3 A4 P      . Thus ) ( ) (t R t R 

That inequality can be proved for any n by the induction. It assures that the reliability of the parallel system with the independent components is greater (or equal) than the reliability of that system with dependent components.

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Computing the reliability of the real systems we often assume that the components life lengths are independent even though the random variables describing the life lengths are dependent. That example shows that such assumption leads towards careless conclusions. The real parallel system may has significantly lower reliability. Moreover, we come to the similar conclusions if we take under consideration more general assumption about the association of the random variables T1, T2, …, Tn [Barlow & Proshan].

Example 2.

Assume as previously that a non-negative random variable Ui , describing time to failure

of the component caused by the source zi has a Weibull distribution with parameters

1 ,..., 2 , 1 , , i i n i   _.

We take n=3. Then for ui_{> 0}

.

4 ,

3 ,

2 ,

1 ,

)

(

)

(

_u



_P

_U



_u



_e



_i



G

i i iu i i i i   As previously . 2 . 0 , 3 , 1 . 0 , 2 . 2 , 2 . 0 , 2 , 1 . 0 , 2 . 1 1 2 2 3 3 4 4 1                 1 2 3 4 5 6 0.2 0.4 0.6 0.8 1

Fig.3. The graph of the reliability function of the parallel system with dependent components.

Using (16) we obtain the reliability function of the parallel system with dependent components. For t > 0 that function has a following form:

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. ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 . 0 1 . 0 2 . 0 ( ) 1 . 0 2 . 0 1 . 0 ( ) 2 . 0 1 . 0 ( ) 1 . 0 2 . 0 ( ) 2 . 0 1 . 0 ( 4 3 2 3 2 1 4 3 3 2 2 1 3 2 . 2 2 2 . 2 2 2 . 1 3 2 . 2 2 . 2 2 2 2 . 1 _t _t _t _t _t _t _t _t _t _t _t t _e _e _e _e e t G t G t G t G t G t G t G t G t G t G t G t G t             _ _ _ _       R

Figure 3. presents its graph.

REFERENCES

Barlow R.E, Proshan F.: Statistical theory of reliability and life testing. Holt, Reinhart and Winston Inc.1975.