On asymptotic reliability functions of series-parallel and parallel-series systems with identical components

Reliability Engineering and System Safety 41 (1993) 251-257

On asymptotic reliability functions of seriesparallel and parallel-series systems with identical components Krzysztof K o l o w r o c k i

Department of Mathematics, Maritime University, Morska 83, 81-962 Gdynia, Poland (Received 13 July 1992; accepted 9 March 1993)

In the investigations 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 tend to infinity and finding the limit reliability function of this system. In this paper 10-element closed classes of limit reliability functions for regular homogeneous series-parallel and parallel-series systems are fixed. These systems are such that at least the number of their series components or the number of their parallel components tends to infinity. The result is obtained under the assumption that lifetimes of the particular components are independent, identically distributed random variables. The fixed classes are more extensive than classes hitherto known of limit distributions of minimax and maximin statistics of independent random variables with the same distributions. The results can be useful in the reliability evaluation of large regular homogeneous systems and may originate reliability investigations of large nonregular nonhomogeneous systems.

1 INTRODUCTION

effort to combine these results and to put an end to the current state of the problem under discussion. The author's intention is to fix the closed classes of possible limit reliability functions for any regular homogeneous series-parallel or parallel-series system in which at least the number of its series components or the number of its parallel components tends to infinity.

Limit theorems about distributions of extreme statistics included in Refs 1, 3 and 4 and some partial results of investigations on limit distributions of minimax statistics contained in Refs 1 and 2 provided the motivation for this paper. The results obtained in Ref. 2 and also presented in Ref. 1 allow us to state that the only possible limit reliability functions of the system with independent, identical components and equal numbers of series and parallel components are ~l(x)=l-exp[-x -~] for x > 0 , o r > 0 ; R 2 ( x ) = l e x p [ - ( - x ) "] for x < 0 , o r > 0 ; and ~ 3 ( x ) = l e x p [ - e x p ( - x ) ] for x e ( - ~ , ~) for a series-parallel system; and h i ( x ) = l - ~ i ( - x ) , x~(-o% ~), i= 1, 2, 3, for a parallel-series system. In a natural way the problem of the existence of limit reliability functions for the series-parallel and parallel-series systems with unequal numbers of series and parallel components arises. This problem was partly solved in Refs 5-8. All these results were improved and generalized in Ref. 9. This work is an

2 ESSENTIAL NOTIONS Let Eij be components of a system S and Xij be the lifetimes of E~. Let Xii be independent random variables (i = 1, 2 . . . . . k; j = 1, 2 . . . . . l~). The following definitions are well known.

Definition 1 A system S is called series-parallel if its lifetime X is given by

Reliability Engineering and System Safety 0951-8320/93 / $06.00 © 1993 Elsevier Science Publishers Ltd, England.

X = max { min {X~j}} l~i<--k

251

l<_j
Krzysztof Kotowrocki

252 Definition 2

Definition 5

A system S is called parallel-series if its lifetime X is given by X = man {max {X~j}}

A reliability function R(x) is called degenerate if there exists x~c ( - ~ , z¢) such that R ( x ) = 1 for x < x , , and R(x) = 0 for x -> x.. We shall investigate limit distributions of a standardized random vriable ( X - b,)/a,, where X is the lifetime of the regular homogeneous seriesparallel system and a t > 0 , h , e ( - ~ , ~-) are some suitably chosen functions. And, since

l
1~<-iI1

Definition 3

A system S is called regular if It = 12 . . . . . where l e N.

1, = 1,

P

Definition 4

bt>x at

A regular system S is called homogeneous if random variables X 0, i = 1, 2 . . . . . k, j = 1, 2 . . . . . 1, have the same distribution function F ( x ) = P(Xo<-x ), i.e. if components E~j have the same reliability function R(x) = P(Xij > x) = 1 - F(x) for x ~ ( - ~ , ~). Now, assuming in definitions 3 and 4 that k = k. and I = l., where n tends to infinity and k~ and l. are sequences of natural numbers such that at least one of them tends to infinity, we obtain sequences of the regular homogeneous systems corresponding to the sequence (k,, 1,,). Next, replacing n by a positive real number t and assuming that k, and 1, are positive real numbers, we obtain families of the regular homogeneous systems corresponding to the pair (k,, l,). For these families of systems there exist families of reliability functions. It is easy to show that the family of reliability functions of the regular homogeneous series-parallel systems is given by =

1 -

I1 -

(R(x))"]

for

(1) x ~ (-o¢, oo), t 6 ( 0 , ~)

and the family of reliability functions of the regular homogeneous parallel-series systems is given by ~,(x) = [1 - (F(x))',] k, for

(2) x c (-o% ~),

t~ (0, ~)

Let us assume that the lifetime distributions do not necessarily have to be concentrated on the interval (0, ~). Then a reliability function does not have to satisfy the usual demand condition R ( x ) = l for x ~ ( - ~ , O ) . This is a generalization of the usual conception of a reliability function. This generalization is convenient in theoretical considerations, while at the same time the results achieved using these generalized reliability functions yield the same properties of the usual reliability functions. In particular, as we shall see later, the useless results for series-parallel systems immediately yield the useful results for parallel-series systems.

t = P ( X > a t x + b , ) --[Rt(a,x+bt)

we assume the following definition. Definition 6

A reliability function ~(x) is called an asymptotic reliability function of the family ~,(x) or an asymptotic reliability function of the regular homogeneous series-parallel system if there exist functions a, = a(t) > 0 and b, = b(t) ~ ( - ~ , ~) such that

~,(a,x + b,)----~ ~(x)

as

t---~~c for

x e (-;1~

where C~ is the set of continuity points of t~(x). A pair (a,, b,) is called a norming functions pair. Having the asymptotic ~(x) of the system, in practice, we may use the following approximate formula:

(x-b, 1 for sufficiently large t. Similarly, we define an asymptotic reliability function of the regular homogeneous parallel-series system. Since, from condition

~,(a,x+b,)---~(x)

as

t---~

for

x~Cr~

it follows that, assuming 0c, = aa,, [3, = ba, + b,, where a > 0 , b ~ ( - ~ , ~), we have

~,(cvtx + [3,) = ~,(a,(ax + b) + b,)--~ ~(ax + h) t---*~

for

as xcC~,

therefore, if ~ (x) is an asymptotic reliability function, then ~ ( a x + b ) for any a > 0 and b ~ (-~¢, ~) has the same property. This fact justifies the following definition. Definition 7

The reliability functions ~0(x) and ~(x) are said to be of the same type if there exist numbers a > 0 and b c (-0% ~) such that for all x e ( - ~ , ~¢) ~,,(x) = ~ ( a x + b)

Asymptotic reliability functions

253

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

and

Combining the results of Refs 5-8, we may state our main result concerning the only possible asymptotic reliability functions of the homogeneous seriesparallel system. Before that, we assume the following notations: x(t)<>y(t) means that x(t) is of order much greater than y(t) in the sense that x(t)/y(t)---,co as t---~co. Moreover, we define the function A(t) for t ~ (0, co), such that

for every natural v->2, where 6 > 0 and ~ = r~(t), t e ( - ~ , co), is given by

Ip(~v) -

T_Xv= VI/[I-p(t)lA(t) t

then the only possible nondegenerate asymptotic reliability function of the regular homogeneous series-parallel system is one of the following types:

-exp[-(-x)

~2(x)=

i=1

where f,-(t) for 1 - < i -< n is the superposition of the function in t taken i times, p(t) is a positive function, and n is such that

f~+l(p(t))<
for for

~]

{1

for for

0

~3(x) = 1 - e x p [ - e x p ( - x ) ]

x-O,

where c~>O

xO for

wherec~>O

x ~ ( - ~ , ~)

If

k,=t,

t,-clnt~s,

A >0

c>0,

s~(-co, co),

t c (0, ~)

then the only possible nondegenerate asymptotic reliability function of the regular homogeneous series-parallel system is one of the following types:

Theorem 1 If k, = t,

11 - e x p [ - x -~]

[~l(x)=

rt

A(t) ~ H fi(p(t))

6l In v In t[In(ln t)]

p(t)l

l , = c ( l n t ) p~'), t 6 ( 0 , ~ ) ,

c>0

where

Ea(X)=

1-exp

-exp -x ~-

R~(x) =

{,0 I

for x->0, where o: > 0

In(In t) << II, - c In tl

p(t) << (In t) x for every A > 0, and [p(~v) - p(t)l <<

(-x)~ -

where o~> 0

61nv

s

In t[In(In t)]

for every natural v_>2, where rv = T~(t), t 6 (0, co), is given by T;v

- exp - e x p

0<6:#1

and

R6(x) = 1- exp[-exp(-x~-

~)]

for

x->O

for

x <0

where ~ > O, /3>0

vl/1--p(t)

t

i

or where

s<<[l,-clntl<.
s>O,

C>O

~7(X)=

[

for xXI ~Xx, ~X 2 for

--exp--exp-

for

x>-x2,

where x~
and Ip(rv) - p(t)l

61nv

for every natural v-> 2, where 6 > 0 and rv = rv(t), t ¢ (0, oo), is given by Tv

If

In t[In(In t)]

vl/ll--p(t)]ln(Int)

k,-k,

l,>>c

for every

c>0,

then the only possible nondegenerate asymptotic reliability function of the regular homogeneous series-parallel system is one of the following types:

t ~(x)

or where

p ( t ) ~ ( I n t ) ~, ~.>0

k>0

=

{~ - [1 - e x p [ - ( - x ) - " ] ] k

for x < 0 , where a~> 0 for x - > 0

254

Krzysztof

Kotowrocki

or where

=

1 1 - [1 - exp(-x")]k

for for

x
),

p(t)~(Int)

x->O,

,,(x) = 1 - [1 - e x p ( - e x p x)] k for

wherecr>O

).>0

and

x ~ (-zc, ~) Ip(r~)

The proof in parts can be found in Refs 5-8. The complete extensive proof is given in Ref. 9.

-

61nv In t[ln(ln t)]

p(t)l

for every natural v_>2, where 6 > 0 and r~ t e ( - ~ , ~), is given by Tv

4 ASYMPTOTIC RELIABILITY FUNCTIONS OF A REGULAR HOMOGENEOUS PARALLEL-SERIES SYSTEM

VI/[I plt)lA(t)

t

It is well known j that if E(x) is an asymptotic reliability function of the regular homogeneous series-parallel systems with the reliability function of a particular component R(x), then [~(x) = 1 - E ( - x ) , x ~ Ca, is an asymptotic reliability function of the regular homogeneous parallel-series system with the reliability function of a particular component / ? ( x ) = 1 - R(-x), x e CR. This fact and Theorem 1 allow us to state our main result concerning the only possible asymptotic reliability functions of the regular homogeneous parallel-series system.

then the only possible nondegenerate asymptotic reliability function of the regular homogeneous parallel-series system is one of the following types: ~'(x)=/exp[-(-x)-"]:0

for for

1 //~2(x)= e x p ( - x " )

x <0 x->O,

for for

~3(x)=exp(-expx)

for

k, = t,

where

or>0

c~>O

s 6 (-~,

~c ),

t ~ (0, w)

then the only possible nondegenerate asymptotic reliability function of the regular homogeneous parallel-series system is one of the following types:

If c>O

where

eXp[-exp(--x°-0

In(in t) << II, - c In t[

p(t) << On

where

xc(-~,~)

lr - c ln t - s,

c>O,

l , = c ( l n t ) '~'~, t ~ ( O , ~ ) ,

x<0, x->0

If

Theorem 2

k~ = t ,

=

where o~> 0 for x~>O

t)

for every Z > 0, and 61nv IP(T~) - P ( t ) l ~

for every natural v - 2 , where r~ = ~'~(t), t 6 (0, ~), is given by 7~v

~5(x)--

In t[ln(ln t)] 0
and

vl/l--p(t)

=

t

iiexp - e x p (- ( - x ) " - s)] for c for x -> 0,

or where

,

s<
s>O,

C>O

and Ip(rv) - p(t)l

blnv In t[ln(ln t)]

~7(x)=

for every natural v->2, where 6 > 0 and r~ = z'v(t), t 6 (0, ~) is given by ~'v

t

vl/[I H(t)[In(hlt)

for x ~ 0 . where o r > 0

exp - e x p x " -

fi >

c/J

0

'[

f°r x x'

exp-exp-

for

0

for x ~x~, where x~
l

xl<-x
If k,-k,

l,>>c

for every

c>O,

k>O

Asymptotic reliability functions then the only possible nondegenerate asymptotic reliability function of the regular homogeneous parallel-series system is one of the following types: 1 for x<-O ~s(x) = [ 1 - e x p ( - x - " ) ] k for x > 0 , where o~> 0 for x < 0 , where cr > 0 for x_>O

~ 4 x ) = {~1 - e x p [ - l - x l ~ l k

255

then for each x • ( - 2 , o~) and sufficiently large t

atx+bt=bt

x+l

>0

Hence for all x • ( - 2 , 2) and sufficient large t

R(a,x + b,) = exp[-A(a,x +

b,)B]

Since lim kt(R(a,x + bt)) I, t~c

= lim kt exp[-Alt(a~x + bt) t~]

,o(x) = [1 - e x p [ - e x p ( - x ) ] ] k for x • ( - ~ , o,) 5 EXAMPLES

= } i m k t e x p -Al,(b,) B 1 +~,,x) j

We shall need the following lemma. 5

=iim,~=k, e x p [ - l n kt(1 + B l -I ~ k~) nx 8

Lemma 1

= lim kt e x p ( - l n k, - x) = e x p ( - x ) t~oc

If (i)

then, by Lemma 1, ~3(x) is the asymptotic reliability function of the system.

a reliability E(x) is given by I~(x) = 1 - e x p [ - V ( x ) ] ,

x • ( - 2 , ~).

(ii) a family ~,(x) is given by eqn (1),

Example 2

(iii) lim k, = oo, t~c

(iv) a, > O, b, • ( - 2 , 2) are some functions, then the assertion lim~t(a,x+b,)=~(x)

for

Let the regular homogeneous series-parallel system be such that

x•C~

t~c

R(x)=

is equivalent to the assertion lim kt(R(a,x + bt))/, = V(x)

for

x • Cv

where Cv is the set composed of continuity points of V(x) and points such that V(x) = 2.

1, e -l, O,

x<0 0-A, A > 0

If pairs (k,, 1,) and (a,, b,) satisfy the conditions (i) k,=t, (ii) a , = l ,

l,=lnt, b,=0,

then Example 1 l~7(x)=

Let the regular homogeneous series-parallel system be such that

R(x)=

1, exp(_AxB),

x<0 x-->0,

A>0,

B>0

If pairs (k,, 1,) and (a,, b,) satisfy the conditions (i) k , = t , l , = c ( I n t ) °, c>O, O < - p < l , (~/,)ira l (ii) b , = Ink, , at=--(b,) l-a, ABI, then l~3(x) = 1 - e x p [ - e x p ( - x ) ] ,

x e ( - 2 , o0)

is the asymptotic reliability function of the system. Motivation: Since

b,>O, lim ,= 0 a t

1, 1-exp(-1), 1.0,

x<0 O~x-A

I

is the asymptotic reliability function of the system. Motivation: Since for all x e ( - 2 , o0) and t e (0, 2)

a,x + b, = x then

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

1, e -1, 1.0,

I

x<0 O<_x
x>-A

Therefore lim k,(R(a,x + b,)) l, = oo for

x< 0

l----, zc

lim k,(R(a,x + b,))/, = 0 t~oc

for

x ->A

256

Krzysztof Kotowrocki

and for O-
If pairs (k,./,) and (a,. b,) satisfy the conditions

lira k , ( R ( a , x + b,)) ~, = lim k , ( R ( x ) ) t,

(i) l i m k , = k ,

lim/,=~.

t~:

t~

= lim k,e h,k, = 1

(ii) a, Hence, by Lemma 1, ~7(x) reliability function of the system.

is the

asymptotic

1 Al,'

b, = O,

then

Example 3

1, 1 [ 1 - e x p ( - x ) ] ~,

[~,)(x)= Let the regular homogeneous series-parallel system be such that J 1, I exp(-Ax ~ -1),

R(x)

x< 0 x >- O,

A > O,

is the asymptotic reliability function of the system. Motivation : Since a, > 0 for t ~ (0, ~), then for all t

B > O

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

l,=lnt,

(Al,),m,

R(atx) =

b,=0,

1

then

~,(a,x)=

/

l_[l_(R(a,x))l]k

1 - exp[-exp(-x~)l ,

x <:~0 x ~0

for for

x< 0 x-->0

Hence

x -> 0

is the asymptotic reliability function of the system.

lim~,(a,x)=l

Motivation: Since for all x e ( - 2 , ~c)

t~x

lim (a,x + b,) = lim a,x = lira

....

for for

x< 0

J 1. ~4(X)

1 exp(-Aarx)

and according to eqn (1)

1

(ii) a,

x -< 0 x~O

,~-. . . .

X

atx+b,
for

x<0

a,x+b,>-O

for

x->0

lim ~,(a,x) = 1 - [1 - e x p ( - x ) ] ~ t~

and then 1. R(a,x + b,) = R ( a , x ) =

exp[_A(a,x)•

_ 11,

x<0 x ->0

Therefore limk,(R(a,x+b,)) l,=~

for

x<0

and for x -> 0

- 0

(AI,) 'm

for

J-

From the above it follows that E , ( x ) is the asymptotic reliability function of the system. From examples 1 , 2 , 3 and 4 the appropriate examples of asymptotic reliability functions for the regular homogeneous parallel-series systems immediately appear.

x<0

and for x -> 0

6 CONCLUSIONS

lim k , ( R ( a , x ) ) t, = lim k, e x p [ - l , A ( a , x ) I~ - l,] = iim k, exp(-x/~ - In k,) = e x p ( - x t~) Hence, by Lemma 1, [~4(x) is the reliability function of the system.

asymptotic

Example 4 Let the regular homogeneous series-parallel system be such that R(x)=

1, exp(-Ax),

x<0 x->0,

A>0

In this paper some closed classes of asymptotic reliability functions for regular homogeneous seriesparallel and parallel-series systems have been fixed. They are 10-element classes and are more extensive than the hitherto known three-element classes of asymptotic distributions of maximum and minimum statistics of independent random variables with a common distribution function. These latter results may be obtained immediately as particular cases of Theorems 1 and 2. Similarly as particular cases of these theorems, we may consider the hitherto known theorems on limit distributions of maximin and manimax statistics of independent random variables with the same distributions.

Asymptotic reliability functions All results have been obtained under the assumptions that the lifetimes of the particular components were independent random variables and the pair (kt, 1,) had the property that at least k, or l, tended to infinity with some regularity of variation. This regularity can be diminished, because if the system has an asymptotic reliability function for the pairs (k,, l,) and (a,, b,) and it has the same asymptotic reliability function for other pairs (k,, l[) and (a,, b,), then this system also has this asymptotic reliability function for the pairs (/~,, 1]) and (6,,/~,), where (/~,,/,) = (k,, l,) and (6t,/~,) = (a,, b,) for some t and (/~,, [,) = (k;, l;) and (ti,, b,) = (a;, b;) for other t. The examples testify that there exist systems which have asymptotic reliability functions from the fixed classes. From the practical point of view it is important that k, and l, should be natural numbers. In the case when k, is not convergent, the return with k, to the natural numbers is trivial (see Lemma 1) because if we replace k, by its entire part, then ~,(x) has the same asymptotic reliability function. Also, 1, may be represented by 1, = [1,] + l, - [1,], where [l,] is the entire part of l,, and since (R(x)) I,-v,l is again a reliability function, then according to eqn (1), we may consider that the series subsystems have one component with a reliability function different from that of the remaining ones. In the case when k, is convergent, k, may be a natural number by assumption, and the return with l, to the natural numbers is trivial (see eqn (1)) because if we replace l, by its entire part, then [~,(x) has the same asymptotic reliability function. t

t

t

257

ACKNOWLEDGEMENT

The author would like to thank the reviewer for his helpful comments. REFERENCES

1. Barlow, R. F. & Proschan, F., Statistical Theory of Reliability and Life Testing: Probability Models, Holt, Rinehart & Winston, New York, 1975. 2. Chernoff, H. & Teicher, H., Limit distributions of the minimax of independent identically distributed random variables, Proc. Am. Math. Soc., 116 (1965) 474-91. 3. De Haan, L., On regular variation and its application to the weak convergence of sample extremes, Math. Centr. Tracts, 32, Amsterdam, 1970. 4. Gnedenko, B. W., Sur la distribution limite du terme maximum d'une s6rie al6atoire, Ann. Math., 44 (1943) 432-53. 5. Kolowrocki, K., On a class of limit reliability functions of some regular homogeneous series-parallel systems, Reliability Engineering & System Safety, 39 (1) (1993) 11-23. 6. Koiowrocki, K., On a class of limit reliability functions of some regular homogeneous series-parallel and parallelseries systems, Resubmitted to IEEE Trans. Reliability. 7. Kolowrocki, K., Limit reliability functions of some regular homogeneous series-parallel and parallel-series systems, Submitted to Applied Stochastic Models and Data Analysis. 8. Kolowrocki, K., Limit reliability functions of some regular homogeneous series-parallel and parallel-series systems, Proc. Int. Conf. Signals, Data, Systems, Calcutta, India, 1992. vol 1, 1992, pp 91-102. 9. Kotowrocki, K., On a class of limit reliability functions for series-parallel and parallel-series systems, monograph, 125 pp., Maritime University Press, Gdynia, Poland, 1993.