Reliability bounds for coherent structures with ... - Science Direct

STATISTI@S & PROBABILITY I.ll" R$ Statistics & Probability Letters 22 (1995) 137-148

ELSEVIER

Reliability bounds for coherent structures with independent components J.C. Fu", M.V. K o u t r a s b'* aDepartment of Statistics, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2 bDepartment of Mathematics, University of Athens, Athens, Greece 15784 Received September 1992; revised January 1994

Abstract

In this article a minimal path upper bound and a minimal cut lower bound on the reliability of a coherent system are derived for the case of independent (but not necessarily identical) components. Coupling these bounds with the classical Esary and Proschan's (1963) bounds, some limit theorems are established for the reliability of large coherent systems under quite general conditions. Keywords: Reliability bounds; Cut sets; Path sets; Limit theorems

1. Introduction

Let us consider a coherent reliability system with independent components and denote by ! the set of all its components, and by C(P) the set of all minimal cut sets (path sets), respectively. Despite the fact that a compact formula is available for the structure function ~p of the system, namely (see Barlow and Proschan, 1975; Ross, 1985) ~P=cISI~c(1-I-[(1-xi))=l-pI:Jp(1-I-Ix')',c

,~P

the computation of system reliability R = E[cp] still remains a serious task, especially for large or complex systems. This is due to the fact that for systems with large number of cut sets (path sets), the reduction of ~p in the form of a sum of products of X~ (in order to apply the mean value operator to each) requires formidably m a n y algebraic manipulations. As a result, during the last years, special attention has been given to the development of computationally manageable approximations for system reliability.

* Work supported in part by the Natural Science and Engineering Research Council of Canada under Grant NSERC A-9216. * Corresponding author. 0167-7152/95/$9.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0167-7 1 5 2 ( 9 4 ) 0 0 0 6 0 - L

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Perhaps the most popular approach to this problem is the one proposed by Esary and Proschan (1963). Their lower and upper bounds have the following form:

LB=c~c(1-I-Iq')'i~c

UB=l-~e(1-I-[Pi)'~e

where q~ = 1 - p~, i e I are the failure probabilities of system's components. Due to their simplicity, the above-mentioned bounds have by now been incorporated into almost all recent text books on reliability (see for example Barlow and Proschan, 1975 and Ross 1985). It is worth mentioning that Esary and Proschan (1970), making use of the theory of association of random variables, Esary et al. (1967), extended the minimal cut lower bound for a variety of component maintenance policies and several typical cases of component dependence. The present paper was motivated by the observation that Esary and Proschan's bounds cannot be effectively used for approximating system reliability from both sides (above and below), because LB yields good approximations for high reliability systems, while UB has good performance in the opposite case. In Section 2 of this paper we provide a minimal cut upper bound UB* (and a minimal path lower bound LB*) for the reliability of a coherent system, which is very close to Esary and Proschan's lower bound LB (upper bound UB) at least in the case of components with high (low) reliabilities p~, i e I. In Section 3, the bounds LB, LB*, UB and UB* are evaluated for the well-known bridge structure and compared numerically with the Chen-Stein and Bonferroni bounds. In Section 4, a number of quite general limit theorems are established for large coherent structures with independent components. Finally, in Section 5 our approach is applied to specific systems yielding new results and alternative ways of proving some well-known results in a unified way and under weaker conditions. 2. Bounds on system reliability

Let I be the set of all components of a coherent system and C = {C1, C 2 , . . . , CN} the family of all its minimal cut sets. The components are assumed to operate independently. Define L*=0,

L * = { i : C i n C ~ v ~O,l 0.9 (resp. p ~< 0.1). As already mentioned, a different ordering of the minimal cut (path) sets leads to different bounds UB* (LB*). F o r example, the ordering C1 = {1, 2}, C2 = {1,3, 5},Ca = {4, 5}, C4 = {2, 3, 4} yields UB* = (1 - q2) (1 - pq3) (1 - pq2) (1 - p2q3).

•

Recently, the Chen-Stein m e t h o d (see Chen, 1975; Arratia et al., 1989; B a r b o u r et al., 1992) has become a very popular tool for the study of reliability systems (see Chryssaphinou and Papastavridis, 1990; K o u t r a s and Papastavridis, 1993). Using the setup of Arratia et al. (1989), we can easily derive some upper and lower reliability bounds for the bridge structure. M o r e specifically, introducing the binary variables Xi which take on the value 1 if and only if all components in the minimal cut set C~ fail, and the neighbourhoods of

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J.C. Fu, M.V. Koutras / Statistics & Probability Letters 22 (1995) 137-148

Table 1 Comparison of system reliability and minimal cut and path bounds (3.1), (3.2) for bridge structure (i.i.d. case) p

LB*

Exact

UB

p

LB

Exact

UB*

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

0.0052 0.0215 0.0491 0.0878 0.1365 0.1937 0.2576 0.3265

0.0052 0.0215 0.0494 0.0886 0.1387 0.1984 0.2662 0.3405

0.0052 0.0219 0.0509 0.0931 0.1483 0.2160 0.2946 0.3818

0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95

0.6182 0.7054 0.7840 0.8517 0.9069 0.9491 0.9781 0.9948

0.6595 0.7338 0.8016 0.8613 0.9114 0.9506 0.9785 0.9948

0.6735 0.7424 0.8063 0.8635 0.9122 0.9509 0.9785 0.9948

Table 2 Comparison of bounds (3.1), (3.2) with the Chen-Stein method-based bounds (3.3), (3.4) and Bonferroni bounds for bridge structure (i.i.d. case) p

(1)

(2)

(3)

Exact

(4)

(5)

(6)

0. I 0 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

0.0215 0.0877 0.1911 0.3098 0.4063 0.5520 0.7660 0.9040 0.9780

0.0207 0.0730 0.1122 0.0590 0.1759 0.3368 0.6949 0.8899 0.9771

0.0215 0.0878 0.1937 0.3265 0.4721 0.6182 0.7840 0.9069 0.9781

0.0215 0.0886 0.1984 0.3405 0.5000 0.6595 0.8016 0.9114 0.9785

0.0219 0.0931 0.2160 0.3818 0.5279 0.6735 0.8063 0.9122 0.9785

0.0229 0.1101 0.3051 0.6632 1.1759 0.9410 0.8878 0.9270 0.9793

0.0220 0.0960 0.2340 0.4480 0.5938 0.6902 0.8089 0.9123 0.9785

Note: (1) Maximum Bonferroni lower bound, (2) max (LBcs, LB~s), (3) max (LB, LB*), (4) min (UB, UB*), (5) min (UBcs, UB~s), (6) Minimum Bonferroni upper bound.

dependence Bi = {j: Ci c~ Cj # 0}, i = 1, 2, 3, 4 we obtain by virtue of Theorem 1 of Arratia et al. (1989), IR - e x p ( - 2) 1 ~< 1 - exp(2)(bl + b2), 2 where R = P ( ~ = 1Xi = 0) is the exact system reliability and t~ = ~ E [ X i ] , i= 1

b 1 "4- b2 =

] + i

E j e B i - {i}

(3.3)

E[X,Xj

. /

In the i.i.d, case, it is rather straightforward that 2=2qZ(q+l),

bl+bz=2q4(2q z+5q+5)

(3.4)

and inequality (3.3) generates an upper (UBcs) and a lower (LBcs) reliability bound, whose performance is expected to be good, at least for systems with highly reliable components. Working in the same fashion with the path sets we may easily derive an upper (UB*s) and a lower (LB*) bound providing good approximations for highly unreliable systems. As Table 2 indicates, in this specific example, the performance of the bounds (3.1), (3.2) is superior to Chen-Stein method-based bounds. Columns (1) and (6) of Table 2 give the maximum (minimum) of the lower (upper) Bonferroni bounds for system's reliability when the first two inclusion-exclusion terms of the minimal path and minimal cut representation of the system are retained (see Ross, 1985, p. 381).

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4. Limit theorems The problem addressed in this section can be summarized in the following general question: given a coherent structure and some procedure to increase the number of its components without bound, what are the proper conditions such that the limiting system reliability is non degenerate? The basic tool for the analysis conducted here is provided by Theorems 2.1 and 2.2. Let us first introduce some additional notations. Denote by I. the set of components of the nth system, by qi. = 1 - Pi. the failure probability of the ith component, i ~ I. and by C. = {C1, C2 . . . . . CN~} the set of its minimal cut sets. Throughout this paragraph we assume that the cardinality N. = IC.I of C. is increasing without bound, i.e. lim.-.oo N . = + ~ . If Li, j = 1, 2 . . . . . N . are the index sets defined by (2.2), (2.3) (notice that n is suppressed in Ci, L j), we set aj(n)=l-]q,.,

\ieLj

LB. = I-I (1

j = 1 , 2 ..... N.

bj(n)=ai(n)(I-IP,.~,

iECj

aj(n)),

-

UB*

/

--

I-I (1 -

bjtn))

1=1

i=1

q(n) = supqi.,

I, =

ieln

sup

ILjl.

(4.1)

1