Limit reliability functions of some non-homogeneous series-parallel and parallel-series systems

Reliability Engineering and System Safety 46 (1994) 171-177 © 1994 Elsevier Science Limited Printed in Northern Ireland. All rights reserved 0951-8320/94/$7.00

ELSEVIER

Limit reliability functions of some nonhomogeneous series-parallel and parallelseries systems Krzysztof Kolowrocki Department of Mathematics, Maritime University, Morska 83, 81-962 Gdynia, Poland

(Received 6 March 1993; revised version received 28 March 1994; accepted 28 April 1994)

In this paper, some three-element classes of limit reliability functions for series-parallel and parallel-series systems with non-identical components are considered. The series-parallel system is such that the number of its series components is of less order than the logarithm of the number of its parallel components. The parallel-series system is such that the number of its parallel components is of less order than the logarithm of the number of its series components. Moreover, an example of the considered systems and its limit reliability function are given. The results may be useful in the reliability evaluation of large systems with non-identical components. In particular they can be useful to reliability practitioners dealing with design problems.

1 INTRODUCTION

2 ESSENTIAL NOTION AND THEOREMS We denote by Eij, where i = 1 , 2 . . . . . k, j = 1, 2 . . . . . l, components of a series-parallel system S and by Xq their lifetimes. We assume that Xq are independent random variables.

In a reliability investigation of large-scale systems, the problem of the complexity of their reliability functions appears. This problem may be approximately solved by assuming that the number of system components tends to infinity and by finding the limit reliability function. The results obtained in Ref. 1 and also presented in Ref. 2 allow us to state that the only possible limit reliability functions of a system with independent, identical components and equal numbers of series and parallel components a r e : ~ l ( X ) = 1 - e x p [ - x -~] for x > 0 , where a > 0 , R 2 ( x ) = l e x p [ - ( - x ) '~] for x < 0 , where a > 0 and ~3(x) = 1-exp[-exp(-x)] for x ~ (-o% oo) for a seriesparallel system and R~(x)= 1 - R ; ( - x ) , i = 1, 2, 3 for a parallel-series system. In a natural way the problem of the existence of limit reliability functions for series-parallel and parallel-series systems with unequal numbers of series and parallel components and non-identical components arises. This problem is partly solved in Ref. 3 for the case when system components are identical. This work is an effort to widen the current state of the problem under discussion and to transfer the results of Ref. 3 to the systems with non-identical components.

2.1 Definition 1 A system S is called non-homogeneous if the set {{Xij:j = 1, 2 , . . . ,

l}:i = 1, 2 . . . . .

k}

consists of a, 1 <- a <- k, kinds of subset of random variables and the frequency of the vth kind of subset is equal to qv, where ~ = 1 qv = 1 and q~ > 0. Besides, the vth kind of subset consists of e~, 1 -< ev -< 1, kinds of random variables with distribution functions F (~" V)(x) = 1 - R(v'~)(x), v = 1, 2 . . . . . e~, where R (~' O)(x) is a reliability function, and the frequency of the vth kind of random variable in this subset is equal to p,~,, where ~eClp,,~ = 1 and p,,~ > 0. The above definition means that the system is composed of a different kinds of series-parallel subsystem. The frequency q~ means that there are q~k subsystems of the vth kind in the system. The 171

Krzysztof Kotowrocki

172

frequency p ~ means that there are p,,ol identical components in each series system of the vth kind of the vth series-parallel subsystem. Assuming as in Def. 1, k = k, and l = l,, where t tends to infinity, and k, and l, are positive real numbers, we obtain families of the systems corresponding to the pair (k,, l,). For these families of systems there exist families of reliability functions. The family of reliability functions of the nonhomogeneous series-parallel systems is given by ~,(x) = 1 - leI [1

-

x(t)<> y(t) means that x(t) is much greater than y(t). 2.3 L e m m a 1

If (i) the reliability function R(x) is given by eqn (3), (ii) the family ~,(x) is given by eqn (1), (iii) lim k, = 2, and

(R(i)(x))l'] qik',

t~oc

(iv) at > 0, bt E ( - 2 , ~) are some functions,

i=1

x e ( - m , oc),

t E (0, ~),

(1)

then the assertion

where lim~,(a,x+bt)=~(x)

ei

R(i)(x) = 1-I (RU'J)(x)) p',

]-1

i = 1, 2 . . . . ,a

xeC~

(4)

is equivalent to the assertion

The family of reliability functions of the nonhomogeneous parallel-series systems is given by

lim k, ~ qi(R¢i)(a,x + b,)) t' = V(x) t---+°c

At(X)

for

t~ao

(1')

= I~ [1 --

(=t

(F(i)(x))"j q'k',

x e (-oo, m),

t e (0, ~)

for

x E Cv.

(5)

i= 1

The proof is given in Appendix 1. (2)

2.4 L e m m a 2

where ei

F(i)(x) = 1-I (F(i'i)(x)) P',

i = 1 , 2 . . . . . a.

(2')

i=l

If we assume that the lifetime distributions do not necessarily have to be concentrated on the interval (0, ~), then a function R(x) = 1 - e x p [ - V ( x ) ] ,

x e ( - ~ , ~),

(3)

is a reliability function if and only if V(x) is a non-negative non-increasing, right-continuous function, V ( - ~ ) = 00, V ( ~ ) = 0; V(x) may also be equal to infinity in an interval. We denote the set of continuity points of the reliability function R(x) by CR and the set composed of continuity points of V(x) and points such that V(x) = oo by Cv.

A reliability function R(x) is called an asymptotic reliability function of the non-homogeneous seriesparallel system if there exist functions at>O and b, • ( - 2 , 2) such that

R,(a,x +

~(x)=l=~(-x)

for

xECA

is an asymptotic reliability function of the nonhomogeneous parallel-series system with the reliability functions of particular components / ~ ( i J ) ( x ) = l RU'J)(-x) for x E CR~i.,. At the same time, if (at, b,) is a norming functions pair in the first case, then (a,, - b t ) is such a pair in the second case. The proof is given in Appendix 2.

3 ASYMPTOTIC RELIABILITY FUNCTIONS OF A N O N - H O M O G E N E O U S SERIESPARALLEL SYSTEM

2.2 Definition 2

lim

If a reliability function •(x) is an asymptotic reliability function of the non-homogeneous series-parallel system with the reliability functions of particular components RUJ)(x), i = 1, 2 . . . . . a, j = 1, 2 . . . . . ei, then a reliability function

b,) = R(x)

for

x • Cn.

A pair (a,, b,) is called a norming functions pair. Similarly, we define an asymptotic reliability function of the non-homogeneous parallel-series system. The following notations should now be noted:

A problem of the assignation of a closed class of the possible non-degenerate asymptotic reliability functions for the homogeneous series-parallel system is solved in Ref. 3. The following theorem, slightly modified in Ref. 4, is proved therein. 3.1 T h e o r e m 1

If

k,=t,

l , = c ( l n t ) pu),

te(O,~),c>O,

Limit reliability functions o f some non-homogeneous systems where

173

where V(x) is a non-degenerate function. Moreover, the family

l, <
+ b,))"]

~lnv Ip(z~) - p(t)[ ~ l n t[ln(ln /)] for every natural v-->2, where r~ = z~(t), t e (0, oo), is given by

0<~1

is convergent to a non-degenerate function Vo(x) and and

R(x) = 1 - exp[-d(x)Vo(x)].

(12)

The proof is given in Appendix 3. __ =

~l/1-p(t),

t then the only possible non-degenerate asymptotic reliability functions of the regular homogeneous series-parallel system are ~l(X), R2(x) and R3(x). 3.2 Lemma 3

If (i) R(x) is a non-degenerate reliability function given by eqn (3), (ii) the family R,(x) is given by eqn (1),

3.3 Theorem 2

If the assumptions of theorem 1 on the pair (k,, 1,) are satisfied and there exists a non-increasing function d(x) given by eqn (7), then the only possible non-degenerate asymptotic reliability functions of the non-homogeneous series-parallel system are:

R (x) =~

1 for x -- 0, l 1 - e x p [ - d ( x ) x -~'] for x > 0,

where a > 0,

(iii) lim k, = 0% (iv) a~ > 0, b t ~ (-0% oc) are some functions, and (v) R(x) is one of the reliability functions given by eqn (1') such that there exists T~ such that for t>T~

g(O(atx + b~) <- R(atx + bt) for all x c ( - ~ , ~ ) there exists

:

and i = 1 , 2 . . . . . a and

lim Z qidi(at X + bt)

for x < Xo,

0

for x -> Xo

] t---~oo i = l

l

1 - exp[-d(x)(-x)"] 0

R~(x) = 1 - e x p [ - d ( x ) e x p ( - x ) ]

f o r x e (-0% oo).

3.3.1 Proof By lemma 3 the condition (7)

lim ~,(a,x + b,) = R(x)

where (8)

and Xo e ( - ~ , oo) is such a point that there exists T2 > 0 such that for t > T2 R(a~x + b , ) # O

for x e CR,

where ~(x) is a non-degenerate reliability function given by R(x) --- 1 - e x p [ - d ( x ) V ( x ) l , is equivalent to the condition lira kt(R(a,x + b,)) t, = V(x),

R(atx + bt) = 0 for x ->Xo,

lira Rt(atx + bt) = R(x)

for x E CR

(10)

is equivalent to the assertion lim kt(R(a,x + b,))l'd(x) = V(x)

x e Cv,

(9)

then the assertion

t---~¢~

(13)

forx
and

t~

where a > 0,

and

t~o¢

(R(i)(atx + b,)~ t' di(atx + b,) = \ R---~ +--b~/

for x < 0, for x >- 0

(6)

a

d(x)

/

for x E Cv, (11)

where V(x) is a non-increasing non-degenerate function. Since the last condition is identical with the necessary condition of lemma 1 given in Ref. 3, then by the results of this paper theorem 1 is applicable for R(x) given by eqn (13), in the sense that V(x) is one of the types existing in the formulae for Rl(x), R2(x) and R3(x). Therefore R~(x), R~(x) and R~(x) are the only non-degenerate asymptotic reliability functions of the system.

Krzysztof Kotowrocki

174

4 ASYMPTOTIC RELIABILITY FUNCTIONS OF A N O N - H O M O G E N E O U S P A R A L L E L SERIES S Y S T E M

and /~ (iJ)(x) = 1 - e(i4)(-x) = F(i4)(-x),

X ~E CR{Lt)

(21)

4.1 Theorem 3

Let the following be true: ~(x) is a non-degenerate reliability function; the family ~ ( x ) is given by eqn (2); a, > 0 , b, • ( - % oo) are functions such that lim (~,(a,x + b,) = ~(x)

for x • Cu.

Also let F(x) be one of the distribution functions given by eqn (2') such that there exists T~ > 0 such that for t > T~

F(°(a,x

+

bt) <- F(atx + b,)

(14)

then, by lemma 2, it follows that for ~, = a, and /~, -- - b , lim R,(ti,x +/~,) = ~(x)

d(x) =

~(x) = 1 - R ( - x )

for x • Cn.

(23)

Since from eqns (20) and (21) we have et

/~(i)(x) = I ] F(iJ)(-x) = F ( ° ( - x ) ,

0

for x < Xo

lim £ qidi(atx + b,)

for x >-xo

t~¢¢

(22)

where ~(x) is a non-degenerate reliability function and

for all x ~ (-oo, 2) and i = 1, 2 , . . . , a and I

for x • C~,

(24)

i=l

(15)

then, considering eqn (14), for t > 4, we get

i=1

R(°(~,x + 6,) = F(i)(a,(-x) + bt) is a non-decreasing function, where

< - F ( a , ( - x ) + b , ) = R ( ~ , x +/3,),

di(atx + b,)= ( F(i)(atX-- -+-bt)~t' \ F(a,x + b,) !

(16)

and Xo • ( - % ~) is such a point that there exists T2 > 0 such that for t > T2

F(a,x + b,) = 0 for x < x0

(17)

F(arx + b,) ~ 0 for x - Xo.

(18)

and

where /~(/)(x) is one of the reliability functions given by eqn (24). Moreover, considering eqns (15)-(18), for t > T2, we have

(~,x + b,) = F(a,(-x) + b,) = 0 for - x < Xo, i.e. for x > -Xo, and

If the assumptions of theorem 1 on the pair (kt, It) are satisfied, then the only possible non-degenerate asymptotic reliability functions of the nonhomogeneous parallel-series system are

~(x)={exp[-d(;)(-x)-~]

~(x)=

(25)

1 exp[_d(x)x,]

for x < 0 , for x >- 0, for x < 0 , forx>-0,

R(&x + b,) = F(a,(-x) + b,) ¢: 0 for - x ->Xo, i.e. for x <- -Xo, where -Xo • ( - ~ , 2). Since

a,(a,x + g)

where a > 0,

= (F(i)(at(-x) + b,)~ t' \ ~+--b3 ) = d,(a,(-x) + b,),

where a > 0

and

~;(x) = e x p [ - d ( x ) expx]

for x • ( - 2 , ~)

then a(x) = a ( - x )

4.1.1 Proof Let

{

~lim 2 qidi(atx + b,)

R,(x) = 1 - I~I [1 - (/~(°(x)lt'lq'k'

(19)

for x • ( - % ~), t e (0, oo), be a reliability family such that for i = 1, 2 . . . . . a el

i=1

t~¢

L

i=1

R(0(x) = ]-I (R(i'J)(x)Y '°,

=

x • ( - 2 , ~)

(20)

for x <- -x,,,

i= 1

0

for x > -Xo

and d(x) is a non-increasing function. Because all assumptions of theorem 2 are satisfied, then the only non-degenerate types of the asymptotic reliability function R(x) are Rl(x), Nz(x) and R3(x). Hence, and

Limit reliability functions of some non-homogeneous systems from eqn (23), ~(x) is one of the types Rl(x), R2(x) or

175

where for x E ( - % ~) ei

R(°(a,x + b,) = I-I exp[-Aij(a,x + b,)R'J]e' j=l

5 EXAMPLE

= e x p [ - ( a , x + b,) ~'

Let the non-homogeneous series-parallel system be such that

× ~_~ PijAij(a,x + b,)a°-&] . j=l

1,

x<0,

"R(i'°(x) = exp[-Aqxa~J],

x >- O, Aq > O, Bq > 0

Since for all x • (-0% ~) and Bq~B~ lim (a,x + b,) aij-8' = O,

for i = 1, 2 , . . . , a, j = 1, 2 , . . . , e~ and let

Bi--max{Bq}

fori=l,

1<--j<--e~

2 . . . . . a,

!

Ai= ~

1

PqAq f o r i = l ,

2,...,a,

then for each x E (-0% ~), i and sufficiently large t

exp[-Ai(atx + bt)B'(1 + E~i)(x))] <- R(°(a~x + b,) <- exp[-A~(a,x + b,)a'], where for all x • ( - ~ , ~)

(26)

and i

where E; means the sum over j such that Bq = Bi, lira E}i)(x ) = O. B = min {Bi}, l <_i~a

Let us assume

A = min {Ai:Bi = B}.

1,

i

exp[_Axa],

R(x)=

x --<0, x>0.

If the pairs (k,, l,) and (a,, b,) satisfy the conditions

k,=t,

l,=c(lnt) p, c > 0 ,

0-
and 1 ln ( kt ~ ,] qi]] ,/ai b, = [ --~,

, at= - ' ~1 ( b , ) l-a,

where ~; means the sum over i such that

Bi = B

Then, according to eqn (26), for all x c ( - % 0o) and i

R(°(a,x + bt) R(a,x + b,) <_exp[-Ai(atx + b,) n'] exp[-A(atx + b,) a] = e x p [ - Z ( a t x + b,)a[--~(a,x+ bt) & - B - 1]] <_ 1,

andA;=A,

then R3(x) = 1 - e x p [ - e x p ( - x ) ] ,

• ( - m , m)

i.e.

R(i)(a,x) <- R(a~),

is the asymptotic reliability function of the system.

for sufficiently large t. Moreover, 5.1 Justificaiton

• {R(i)(atx + b,)~ t' lm . . . . . . di(x) = t--.~ ~ R(atx + bt) J

Since lim b, = ~, t--,~

= lim exp[l,[A(a,x + b,) a

at l i m - = 0, t~ b t

t---+ oc

then for x • ( - ~ , ~)

-

Ai(atx + bt)&]] = 1

for i such that Bi = B and Ai = A, and lim(a,x + b , ) = l i m b , 1 + t--~

x = ~

( g ( i ) ( a t x + bt)~ t'

t ~

and for each x • ( - ~ , ~) and sufficiently large t

d,(x ) =

a,x + b, > O.

R T , x +-g5,) J lim exp[ -Alt(atx + b,) a t---+~

L

Hence for all x • ( - ~ , ~), i, j and sufficiently large t X[--~(atx+bt)B'-B--a]]=O

R(iJ)(a,x + b,) = exp[-Aq(a,x + b,)Bq and, according to eqn (1), for sufficiently large t

R,(atx + b,) = 1 - Ie[ [1 - (R(i)(a,x + b,))t,] q,k,, i=1

for i such that B~ ~ B or A~ ¢ A. Therefore,

d(x)=

di(x)= i=1

. qi f o r x E ( - o % ~ )

Krzysztof Kotowrocki

176

REFERENCES

and, further, for all x E ( - ~ , ~) lim k,(R(a,x + bt))t'd(x) t~

= lim k~ ~, q, exp[-Al,(a,x +

=lim(k,~'qi)

exp[-ln(kt~'qi)

b,)Bl -x]

= exp[-x].

Hence, by lemma 3, N3(x) is the asymptotic reliability function of the system. From example 1, by lemma 2, the appropriate example of the asymptotic reliability function for the parallel-series system is immediately clear.

1. Chernoff, H. & Teicher, H., Limit distributions of the mini-max of independent identically distributed random variables. Proc. Amer. Math. Soc., 116 (1965) 474-91. 2. Barlow, R. F. & Proschan, F., Statistical Theory of Reliability and Life Testing, Probability Models. Holt Rinehart and Winston, Inc., NY, 1975. 3. Kolowrocki, K., On a class of limit reliability functions of some regular homogeneous series-parallel systems. Reliability Engineering and System Safety, 39 (1993) 11-23. 4. Kolowrocki, K., On a Class of Limit Reliability Functions for Series-parallel and Parallel-series Systems. Monograph, Maritime University Press, Gdynia, 1993, p. 125.

A P P E N D I X 1: P R O O F OF L E M M A 1 6 CONCLUSIONS

In this paper, some classes of asymptotic reliability functions for non-homogeneous series-parallel and parallel-series systems have been fixed. They are three-element classes and are more extensive than the known classes of asymptotic distributions of maximin and minimax statistics of independent random variables with a common distribution function. These known results may be immediately obtained as particular cases of theorems proved in this work. The author's intention was to find the classes of possible limit reliability functions for any systems, i.e. for any relationship between k, and l,. The other cases have been recently solved and submitted for publication. The classes in these remaining case are different from the class obtained in this paper. Therefore the case presented in this paper is discussed separately. From a practical point of view, it is important that k, and l, should be natural numbers. The return with k, to the natural numbers is trivial because if we replace k, by its entire part, then •,(x) has the same asymptotic reliability function (see lemma 1). Since l, may be represented by l, = [I,] + 1,- [lt], where [It] is the entire part of I, and since (R(x)) t'-t~'l is again a reliability function, then according to eqn (1), we may consider that the series subsystems have one component which is different from the remaining components' reliability function. The next important question is the speed of convergence. This problem is currently under investigation.

Suppose that eqn (4) is satisfied. Then, for all x e C~ such that R(x) ~ 1, i.e. V(x) ~ ~, according to eqn (1), for i = 1, 2 . . . . . a, we have lim [1 - (R(i)(at x + bt)) It] =

(AI.1)

1,

Moreover, according to eqns (1) and (3), the condition (4) may be written in the form a

lim kt Z qiln[1 - (R(i)(atx + br)) t'] t ~

i= 1

=-V(x)

xeCv.

for

(A1.2)

From the expansion

ln(1 - x) = - x - O(x), where 0(x)<
from

eqn

(AI.1),

for i =

In[1 - (R~°(a,x + b,))',] = -(R~i)(a,x + bt)) t, - O((R~i)(a,x + b,))t'). Hence, for i = 1, 2 , . . . , a, we have

k,ln[1 - (R~°(a,(x + b,))" l = -k,(R~i)(dx + bt)) I, ×

[1

O((R~°(a'x + b,))',)]

Next, using eqn (A1.2), we have lira kt ~, qi(R~i)(atx + bt)) I' t ~

i--1

ACKNOWLEDGMENTS

= - lim k, ~ qJn[1 - (R~i)(a,x + b,)) t,] The author would like to thank the reviewers for their helpful comments.

= V(x).

Limit reliability functions of some non-homogeneous systems On the other hand, if eqn (5) is satisfied, then eqn (AI.1) also holds, and from eqn (A1.3), we get

177

families ~,(x) and R,(x) from the assumptions, and from eqn (3), we have --

lim k, 2 q~ ln[1 - (RU)(a,x + b,))~,] t~c

~,(x) = [1 - 1~

i- 1

j=l

• .

= - lim k, 2 qi(RU)(a, x + b,)) I' = - V ( x ) l ~

"] q i k t

= 1 - [1 - [1 - ]~I (R0'J)(-x))Pd']

i-- I

j=l

and therefore

= 1 - ~,(-x),

lim IZl [1 - (R")(a,x + b,))~,] ~k' = e x p [ - V ( x ) ] t~c

"] q i k t

(F",J'(x))"0',]

where

i= 1

P'J~(x) = 1 - k"J~(x)

and next for i = 1, 2 . . . . . a, j = 1, 2 . . . . . ei. Hence, if limN,(a,x+b,)=N(x)

for

x•Ca.

t~c

Besides, for all x such that ~(x) = 1, i.e. V(x) = % if eqn (AI.1) is satisfied, then from the previously performed discussion it follows that conditions

lim R,(atx + b,) = R(x)

x • CR,

then lim

lim N,(a,x + b,) = 1

for

t----~c

- b,) = lim [1 - R , ( a , ( - x )

Nr(a,x

+ b,)]

(A1.4) =l-R(-x)=fl~(x)

and lim k, 2 qi(RU)(atx + b,)) I' = ~ t~zc

(A1.5)

i -- l

are equivalent. Otherwise, if eqn (AI.1) does not hold, i.e. there exists i such that

APPENDIX

3: P R O O F

OF LEMMA

for

x•Cfi.

3

According to eqn (1), by lemma 1, condition (10) holds if and only if

lim [1 - (R")(a,x + b,))~,] ~ 1, t~zc

lim k, 2 qi(R(i)(a,x + b,)) I' = V(x)

and eqn (A1.4) is satisfied, then (more than in the case when eqn (AI.1) is satisfied) it follows that eqn (A1.5) holds. If eqn (A1.5) holds, then

t~zc

i=1

for x E C

lim k, 2 qfln[1 - (RU)(atx + bt))/,] t~c

Hence

i= I

<-lim[-kr2qi(R(i)(atx+bt))t,] =-~c t~

lim k,(R(a,x + b,))"

i= 1

in the case when RU)(a,x + b~) ~ 1 for i = 1, 2 . . . . . a and IeI [1 - (R")(a,x + b,))t,] q'k' = 0 i=1

in the case when there exists i such that RU)(a,x + b~) = 1. Hence

i= 1

which means that eqn (A1.4) holds. This completes the proof.

APPENDIX

2: P R O O F

OF LEMMA

2

From the designations (eqns (1) and (2)) of the

{R(°(a,x + b,)] I' % -Y ,x J = V(x)

for x • Cv and x
for

x->xo.

The above and eqn (7) mean that the assertions (10) and (11) are equivalent. Moreover, since according to eqns (6) and (7) for sufficiently large t 0<-d(x)<-I

lim ]eI [1 - (RU)(a,x + b,))/'] q`k' = 0. t~:c

V.

for

x•(-%~),

(A3.1)

then if the family

[k,(R(a,x + b,))',]

(A3.2)

is convergent to a degenerate function (Ref. 3), then, by eqns (11) and (A3.1), V(x) is also degenerate. But this is inconsistent with the assumption that V(x) is non-degenerate. Therefore, the family in eqn (A3.2) is convergent to a non-degenerate function Vo(x) and R(x) is given by eqn (12).