Microclectron. Reliab., Vol. 33, No. 7, pp. 965-978, 1993. 0026-271419356.00+.00 Printed in Great Britain. © 1993 Pergamon Press Ltd

A COMPARISON BETWEEN RELIABILITY ANALYSES BASED PRIMARILY ON DISJOINTNESS OR STATISTICAL INDEPENDENCE:

THE CASE OF THE GENERALIZED INDRA NETWORK

ALI M. RUSHDI and ANWAR A. ABDULGHANI

Department of Electrical and Computer Engineering, King Abdulaziz University, P.O. Box 9027, Jeddah 21413, Saudi Arabia

(Received for publication 10 February 1992)

Abstract - A switching expression is readily convertible to a reliability expression if (a) all ORed terms are disjoint, and Co) all ANDed sums are statistically independent. The usual approach of system reliability analysis makes a primary use of (a) and a secondary use of Co). An alternative approach reverses the roles of (a) and Co). Symbolic reliability expressions for the source- to-terminal reliability of a generalized Indra network (GIN) with nonidentical components are derived by the two approaches. For this particular case, the second approach leads to a shorter, more elegant derivation and simpler novel results. Typical plots of the GIN reliability functions are presented and their properties are discussed.

I. INTRODUCTION

System reliability analysis pertains to calculating the probability of success

of a system in terms of the probabilities of success of its components. Usually,

this analysis depends on a use of the probability relations [1]

Pr{ A tJ B } = Pr{A} + Pr{B}, if events A and B are mutually exclusive(disjoint), (1)

Pr { A I"1 B } = Pr{A} Pr{B}, if events A and B are statistically independent. (2)

In a well known approach of system reliability analysis, the system success is

expressed as a disjoint sum of products ( disjoint sum, for short) [1,2]. Such an

approach seeks to achieve a condition whereby (1) can be applied. It also makes

use of (2), but only as far as the reliabilities of statistically independent

components are concerned. In a "dual" approach, system reliability analysis may

seek to achieve a condition whereby (2) is applicable [3-7]. Such a condition

occurs when the system success/failure is expressed as a product of statistically

965

966 A. M. RUSHD! and A. A. ABDULGHANI

independent subexpressions. Further processing of these subexpressions makes

secondary use of (1). Traditionally, system reliability analysis has relied heavily

on the first approach, with occasional use of the second approach in a few \

instances. The reason for this is probably that statistical independence is easily

lost whenever a switching expression is processed. Nevertheless, the second

approach can be effectively used to analyze series-parallel, series-like, or

parallel-like structures.

The aim of this paper is to demonstrate a case wherein the second

approach can be used to advantage. This is the case of the generalized Indra

network (GIN) [8-10] whose reduced redundancy graph (see Fig. 1 ) is an almost

parallel structure. Switching-algebraic techniques employing the aforementioned

two approaches are used for the derivation of symbolic closed-form expressions

for the source-to-terminal reliability of that network in the general case of non-

identical components. The derivation based on the first approach ( of the disjoint

sum ) is tedious, lengthy and results in a double-sum expression which reduces,

for identical components, to the expression given in [I0]. On the other hand, the

derivation based on the second approach ( of statistically independent

subexpressions ) is much shorter, more elegant and results in a new simple

Fig. 1

U, Y,1 V,

input ~ output

U, " " V,

Reduced redundancy graph of a generalized Indra network

between an input or source node and an output or terminal

node. Each link is labeled by its success.

System reliability analysis

single-sum expression. Though the two expressions obtained by the two

approaches are different in form and complexity, they yield identical results for

a wide variety of special test cases.

967

The present paper can be viewed as a sequel to two earlier papers [11,12].

The feature common to the three papers is that each of them deals with a system

of a regular structure or topology, thereby allowing a direct manipulation of the

system success function by switching-algebraic techniques. In each case, a useful

reliability expression is derived in an iterative or re, cursive form.

The organization of the balance of this paper is as follows. Section II

presents lists of assumptions and notation as well as a set of rules that are useful

in the processing of switching expressions. Two expressions for the source-to-

terminal reliability of the GIN are derived by the approach of the disjoint sum

and that of statistically independent expressions in sections 111 and IV

respectively. Section V compares results, presents typical plots of the GIN

reliability function, and discusses some of its properties.

II. ASSUMtq'IONS, NOTATION AND RULES

Assumptions

1. Both the components and the system are 2-state, i.e. either good or

failed.

2. Component states are statistically independent.

3. Component reliabilities arc not necessarily equal.

4. The system is a generalized Indra network(Gl2~l) and its

components are the switching elements of the GIN. This system is

968 A. M. Rusrim and A. A. ABDULGHANI

modeled by the reduced redundancy graph of Fig. 1, which has an

identical structure between any pair of input and output terminals

irrespective of their relative positions [10].

Notation

r

1

m

Pr{.}

Xi,X i

E{ xi}, E{ ~i} expectations of the random variables x i

number of switching elements in stage 0 of the GIN.

number of switching elements in stage m of the GIN.

index of the last stage of the GIN; i.e., the GIN has ( m + l ) stages.

probability of the event i.

Indicator variables for the successful and unsuccessful

operation of a single link i in the reduced redundancy graph of the

GIN. These are switching random variables that take only one of the

two discrete values 0 and 1; x± = 1 and "~i = 0 iff i is good, and

X i ----- 0 and "~i = 1 iff i is failed. The variables Y i j , Y i j ,

z j , a n d Z j are defined similarly. However,

x i and zj are successes of single components, while Yij is

the success of a link of (m-l) components in series.

and "~i

R

These represent the reliability and unreliability of link i:

E{ x i} = Pr{ x i = l} = l-E{~'± }.BothE{x i } andE{xi }

take real values in the closed interval [0, l]. E{ xi } is denoted by the

lower-case letter x . . 1

indicator variables for the successful and unsuccessful

operation of the system ( the GIN ), called system success and

system failure respectively.

system reliability: R=E{S}= Pr{S=I} = 1 - E{S'}.

P

IKI

c0c,n)

S y s t e m re l i ab i l i ty a n a l y s i s

reliability of a single component when component

reliabilities are identical.

the number of elements in the finite set K = the cardinality of K.

the binomial ( combinatorial ) coefficient = the number of ways of

choosing k elements from a set of n objects, when repetition is not

allowed and order does not matter.

969

R U L E S

.

.

.

A switching success(failure) expression is directly convertible, on a

one-to-one basis, to an algebraic reliability (unreliability) expression if

(a) all ORed terms are disjoint, and

(b) all ANDed sums are statistically independent.

The conversion is achieved by replacing boolean variables by their

expectations, AND operations by multiplications, and OR operations by

additions [1].

Two terms are disjoint if there is at least one variable that appears

complemented in one term and appears uncomplemented in the other,

e.g. , the two terms ABC and BC are disjoint since the literal B appears

in ABC while the literal B appears in BC [13].

If the two terms A and B in the sum ( A 0 B ) are nondisjoint, then B

can be disjointed with A by the relation

A tQ B = A t$ B (Yl Y2 " ' " Ye ')

= A U B (Yl U YlY2 U ... U YlY2 .. Ye-i Ye ) (3)

where { Yl, Y2 ,'--, ye } is the set of literals that appear in A and do

not appear in B [13]. Note that B is replaced by e terms that are disjoint

970

.

.

.

A. M. RUSHDI and A. A. ABDULGHANI

among themselves beside being disjoint with A. In the limiting case of

e = 0 t h e s e t { Y l ' Y2 ' " " We } i s e m p t y , B subsumes A and (3)

reduces to

AUB = AUB(0) = A. (4)

Given a term P and two sums of products A and B where A has no

variables in common with P, while B shares some variables with P, the

product (A N B) is disjointed with P by disjointing B with P and keeping

A unchanged.

The complements of a sum and a product are given by a product

and a disjoint sum, viz.,

n n (U A i ) = ~ Ai "

i=l i=l

(5)

n

O i=l

Ai ) = A1 U A 1 2 2 U ... U A 1 A 2 .. An_ 1 An (6)

The product of sums that have two terms each can be written as a sum of

products, namely,

n (A i U B i) = U (~ A i) (#~ Bi),

i=l k i CK i eK

(7)

where K is one of the 2 nsubsets of {1,2 ..... n}, and ~ is the set

difference {1,2,...,n} - K. If a i a n d Biare disjoint for 1 ~ i ~ n, then

the resulting sum is a disjoint one. If A i = A for 1 g i ~ n, then (7)

reduces to

n n

/'I (A U B i) = A U ( ~ B i) (8) i=l i=l

I l l . THE DISJOINT-SUM APPROACH

The source-to-terminal success of the GIN is obtained by direct path

System reliability analysis 971

enumeration in Fig. 1 as

r 1 S = U U S.. ,

i=l j=l 13 (9)

where s j . j i s the success of the path

input --4, u i - . ~ Vj ---~ output

and is given by

Sij = Xi Yij Zj , 1 <__ i <_ r, 1 <_ J <_ 1 . (i0)

The expression (10) is converted into a disjoint sum by disjointing each S:i.j

( other than s l I ) with the successes of all the preceding paths. These include

(a) The successes of the preceding (i-l) paths through vj , namely

Skj = Xk Ykj Zj , 1 <__ k < i. (Ii)

Each Ski shares the literal zj with S i j ) and has two litcrals X k and Ykj

that do not appear in s i j or in other s ' s . By virtue of rules 3 and 4, sj. 5 in kj

(9) can be replaced by

(i-l) S! I) = S.. N (Xk U X k Ykj) (12) 13 13 k=l

Note that if i = l , then s i j remains intact,i.e, s i j ( 1)__ Si j . Thanks to rule 6,

the indexed product in (12) is replaced by a disjoint sum, viz.

S (I) = S.. U ( N Xk ) ( ~ X k ) (13) ij 13 K kcK kcK Ykj '

where K is one of the 2 i - l subse t s of {1,2 . . . . . i-l}.

(b) The successes of the preceding (j-l) paths through u i , namely

S. = X. Y. Z , 1 < s < j (14) iS 1 IS S --

Each Sis has the literal x i in common with ~ i j , and has two literals z s and

972 A . M . RUSHDI and A. A. ABDULGHANI

x i s that do not appear in s (1)or in other iJ

4 leads to the replacement of s!1. ) by: 13

!

s. s . Application of rules 3 and l S

s!2 ) = ) (j-l) N (Zs O z s Xis ) ,

~3 ~3 s=l (15)

S(1) (j~l) 1 = (~s U z s Yis ) N (~. U Zs). (16) ~J s=l s

s= (j+l)

---- S ( 1 ) The expressions ~ Id z = 1, Note again that i f j = l , then s ~ ) i j " s s

j < s ~< 1, have been deliberately ANDed with the original expression of s !2) 13

to facilitate further processing and the interpretation thereof.

(c) The (i-1)(l-1) successes of the other preceding paths, namely

Sg h = Xg Ygh Zh ' 1 < g < (i - i)

, 1 < h < i, h ~ j. (17)

The expression of s ( 2 ) a s given by the combination of (16), (13), and (10) is ±j

now disjointed with Sg h . That expression involves a sum (union) over all

possible values of the set K . If gee , the literal xg appears in the

corresponding term, which means that this term is already disjoint with sg h

(rule 2). Else if ge~ , then the literal Xg appears in the corresponding term, and

as a corollary to rule 3, only the two remaining literals in Sg h (i.e Ygh

and Z h ) are to be considered in the disjoinmess process. Every term in the

expression of - (2) is ~£9 a product of sums all of which are independent of Ygh and

z h with the exception of exactly one sum that takes the form ( Zh U ZhY£h)

for 1 ~< h ~< (j-l), or the form ( zh U z h ) for (j + 1) ~< h ~< I. In accordance

with rule 4, only this particular sum is disjointed with Ygh Zh , which results

in its replacement by the corresponding expression ( zh u z h Yih Ygh ) or

(Zh U Z h 7g h) {rules 2 and 3}. The above is repeated for all g and h such that

1 ~< g ~< (i-l) and 1 ~< h ~< l, h ~ j, and has no effect on the literals ~ ,

while it results in a replacement of every literal z h by z~ 7g h provided

geK

System reliability analysis

• Finally, s !2. ) is replaced by 3-3

973

S (3). . = S. • U ( n Xk) ( n X k ~kj ) 13 13 K keK keK

(j-l) ( f% (Zs O Z s Y. ( ~ Yks )))

s=l is keK

1 ( n (Zs U Z s ( fl Yks))), s=(j+l) keK

and the system success is given by

(18)

r 1 S(3) S = u U ij ' (19)

i=l j=l

which satisfies the conditions of rule I, and hence can be converted to the

reliability expression

R =

r 1 Z x i z. 7. ( w (l-Xk)) ( w Xktl-Ykj))

i=l j=l Yij 3 K keK keK

[j-l) ( ~ ((l-z s) + Zs(l-Yis) ( ~_ (l-Yks))))

s=l keK

1 ( s= tj+l)

((l-z s) + zs(k~ ~ (l-Yks))))"

If the component relibilities are equal, then

m-i = z. = P ' Yij = P xi 3 for all i and j,

(20)

and the reliability expression (20) reduces to

R = r 1 m+l (i-l) Z Z p

i=l j=l t=0 c(t,i-l) pt (l_pm-l)t (i-p) i-l-t

(A (p, t+l)) j-I (A (p, t)) l-j, (21)

where

A(p, t) = (i - p) + p (i - pm-l)t • ( 2 2 )

974 A . M . RUSHDI and A. A. ABDULGHAN!

and t ( 0 ~< t ~< (i-l)) is the cardinality of the set ~,. There are c(t, i-l) sets of

the same cardinality t and these contribute equally to the above summation.

Equation (21) agrees with the results given in [10]. It involves a triple

summation that has been reduced [10] to a double summation by summing the

geometric progression over j. the final result for R is

R= p r (i-l) pt (i )i-l-t Z ~ c(t, i-l) -p

i=l t=0

((A(p, t))l _ (A(p, t + i))i) (23)

IV. THE APPROACH OF STATISTICALLY INDEPENDENT

EXPRESSIONS

The source-to-terminal failure of the GIN is given by the complement of

(9), namely

r 1 r 1 n ( u x i ~ zj) = . (x i ~( u Yij zj)). i=l j=l Yi~ i=l J=l

(24)

Rule 5 ( Equation (6)) is applied twice to reduce g to

r 1 " (Ri u (x iN ( n i=l j=l

(Zj U Zj Yij ))))' (25)

which can be reduced by virtue of rule 6 to the disjoint sum

1 = L; ( /% Xi) ( ._ (X i ~ (Zj U Zj Yij)))

I ieI iel j 1

1 = U ( " Xi) ( •_ Xi) ( ~_ ~ (Zj U Zj Yij)), (26)

I ieI iel ie~ j=l

where I is one of the 2:: subsets of { l ,2 , . . . ,r} . Equation (26) is actually a

Shannon's expansion [7] of ~ about x 1, x2,. . , and x r . The last part of (26)

can be simplified by interchanging the order of the ANDings and involving rule

System reliability analysis

6 ( Equation (8)), as follows:

975

1 1 I'I ¢~ tZj O Zj Yij) = n ~ (Zj O Zj Yij) ie I j=l j=l ie

1 = t% (Zj O Z. ( • Y.. )) (27)

j=l 3 ieI lJ

The substitution of (27) into (26) results in an expression that satisfies the

conditions of rule 1, and hence can be converted to the reliability expression

R = 1 - 7. ( ~ (l-xi)) ( 11"_ x i) I ieI ieI

1 ( w ((l-z i) + zi( w ))) j=l iel (l-YiJ) '

which reduces in the case of identical components to

(28)

r

R = 1 - X c(t, r) pt (l_p)r-t (A(p, t)) I, (29) t=O

where A(p,t) is given by (22), and t is the cardinality of the set

V. COMPARISON AND CONCLUSIONS

Symbolic expressions for the source-to-terminal reliability of a generalized

Indra network (GIN) has been derived by two different approaches, namely, the

approach of the disjoint sum and the approach of statistically independent

expressions. In this particular case, the second approach is preferred to the first

one for the following reasons:

1. The first approach has considerably more steps than the second one,

despite the fact that the present application of the first approach has utilized

some nonconventional "shortcuts" or "tricks ~ through the use of equations (13)

and (16).

MR 33~--£

Fig. 2

A. M. RUSHDI and A. A. ABDULGHANI

2. The expression obtained by the second approach has a single summation

while that obtained by the first approach has initially a triple summation, that

has been reduced to a double summation through extra manipulations in the

algebraic domain and not in the switching domain.

3. The second approach involves a global processing of the whole system

success expression, while the first approach works through the local processing

of each path success, and requires a particular care when handling limiting or

boundary cases.

Though the expressions obtained by the two approaches((20) and (28) for

the nonidentical case, and (23) and (29) for the identical case) differ substantially

in form and complexity, they have yielded the same symbolic and/or numerical

results for many particular test cases; a fact giving more confidence in their

being equivalent.

R

976

k R=2, L=2, g= l "--6-- R=2, L=4, M=I I I

I "-X('-- R=2, L=8, M=I -"B- R=2, L=16, M=I

P

and m = l .

A family of curves showing the increase with 1 of the

reliability R of a GIN of identical components with r=2

System reliability analysis 977

Fig. 3

R 1

Oi 0 P

i R=2, L=4, M=2 --F-- R=4, L=4, M=2 II

I -'IK-- R=8, L=4, M--2 ~ R=16, L=4, M=2

A family of curves showing the increase with

reliability R of a GIN of identical components

and m=2.

R

O I I o 1 P

r of the

with 1=4

k R=2, L=2, M=2 --t--- R=2, L=2, M=4 I

I --'&- R=2, L=2, M=8 ---8- R=2, L=2, M=IO

Fig. 4 A family of curves showing the decrease with m of the

reliability R of a GIN of identical components with r = 1 = 2.

Computer plots of the GIN reliability R versus components reliability p in

the case of identical components have been obtained, some of which are

presented in Figs. 2-4. Fig. 2 shows that R increases with 1 for fixed r and m.

Similarly, Fig. 3 indicates an increase of R with r for fixed 1 and m. On the

other hand, Fig. 4 shows that R decreases with m for fixed r and 1, as expected.

978 A.M. RUSHDI and A. A. ABDULGHANI

In most cases, the R versus p curves are S curves (type II curves ) [14], but

whenever m is considerably large, the curves are type I curves that resemble the

reliability curves of series system [14]. In all cases, the curves enjoy the usual

properties of the reliability curves of coherent systems.

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