8/12/2019 Microelectronics Reliability Volume 27 Issue 4 1987 [Doi 10.1016%2F0026-2714%2887%2990004-7] Who Kee Chu

1/3

M i c r oe l e c t r on . R e l i ab . , Vol. 27, No. 4, pp. 621 623, 1987. 0026 2714/8753.00+ .00Printed in Grea t Britain. 1987 Pergamon Journals Ltd.

A R E L IA B IL I T Y M O D E L F O R A k - O U T - O F -N : G R E D U N D A N T S Y ST E MW I T H M U L T I P L E F A I L U R E M O D E S

A N D C O M M O N C A U S E F A IL U R E SW H O K E E C H U N G

D e p a r t m e n t o f C h e m i c a l E n g in e e r i n g, U n i v e r s i ty o f O t t a w a , O t t a w a , O n t a r i o , C a n a d a K 1N 6 N 5(Received or publication 24 November 1986)

A b s t r a c t - - T h e p a p e r p r e s e n t s a r e li a b i li t y m o d e l o f a k - o u t - o f - N : G r e d u n d a n t s y s t e m w i t h M m u t u a l l yexc lus ive fa i lu re mod es an d co mm on cause fa ilu res . F a i l ed sys tem repa i r t imes a re a rb i t r a r i ly d i s t r ibu ted .The sys tem i s in a f a i l ed s ta te when ( N - k + 1) un i t s f a i l ed o r a common cause fa i lu re occur red . Lap lacet rans fo rms o f the s ta te p robab i l i t i e s and the ava i lab i l i ty o f the sys tem a re de r ived . F ina l ly , the sys temsteady-s ta te av a i lab i l i ty is a l so repor ted .

I N T R O D U C T I O NI n m o d e r n i n d u s t r i e s v e r y h i g h r e l ia b i l i ty s y s t e m s a r en e e d e d . T o i m p r o v e t h e r e l i a b il i t y o f s y s t e m s , t h ec o m p o n e n t r e d u n d a n c y p l a ys a n i m p o r t a n t r o le a n dm u s t b e s t u d i ed . T h e c o m p o n e n t f a il u re s o c c u r d u e t om a l f u n c t i o n i n g o f m e c h a n i c a l, e l e c tr i ca l a n d / o r d e s i g n( h a r d w a r e a n d / o r s o f t w a r e) p r o b l e m s .

T h i s p a p e r p r e s e n t s a m a t h e m a t i c a l m o d e l o fr e d u n d a n t s y s t e m s w i t h N i d e n t ic a l u n i ts w h o s e u n i t sm a y f a i l i n M m u t u a l l y e x c l u s iv e f a il u r e m o d e s . A l lu n i t s m a y f ai l d u e t o a c o m m o n c a us e . T h e s y s t e mr e q u i r e s a m i n i m u m o f k ~< N u n i t s i n a n o p e r a t i o n a ls t a te . T h e s y s t e m f a i le d in a n y o n e o f M f a i l u re m o d e si s r e p a i r e d b a c k t o i t s o r i g i n a l o p e r a t i o n a l s t a te . T h er e p a i r t i m e d i s t r ib u t i o n d e p e n d s o n t h e f a il u re m o d eo f t h e ( N - k + 1 )t h u n it . A c o m m o n c a u s e f a i l u r e isd e f i n e d a s a n i n s t a n c e w h e r e m u l t i p l e u n i t s fa il d u e t oa s i n g le c a u s e [ 1 - 3 ] .

A S S U M P T I O N S1 . Repa i r , com mo n cause and o th e r f ai lu res a re s ta t i s ti ca l lyi n d e p e n d e n t .2 . The sys tem cons i s t s o f N iden t ica l ac t ive un i t s andrequ i res a t l eas t k ~< N of them to be in a n o pera t io na lstate .3 . Each un i t can fa i l in M mutu a l ly exc lus ive fa i lu re modes .4 . C o m m o n c a u s e f a i l u r e s c a n o n l y o c c u r w i t h m o r e t h a no n e o p e r a t i o n a l u n i t .5 . A repa i red sys tem i s a s good as new .6 . A t t ime 0 a ll un i t s a re in an op era t ion a l s t a te .7 . U n i t a n d c o m m o n c a u s e fa i lu r e r a t e s a r e c o n s t a n t .8 . T h e f a i l u r e r a t e o f a u n i t d u e t o t h e i t h f a il u r e m o d e m a ybe d i f fe ren t f rom th a t o f the j th fa i lu re mode .9 . Repa i r t imes a re a rb i t r a r i ly d i s t r ibu ted .

N O T A T I O N0 ini t ia l s tat e ( i .e . a t t = 0)

N t h e t o t a l n u m b e r o f a c t iv e u n i t s i n t h e sy s t e mM t h e n u m b e r o f m u t u a l l y e x c lu s i ve f a i lu r e m o d e sk t h e m i n i m u m n u m b e r o f o p e ra t i o n al u n i tsr e q u i r e d f o r t h e s y s t e m t o b e c o n s i d e r e d i n a n

o p e r a t i n g s t a t e621

j the j th d egraded s ta te o f the sys tem wi th[ ( j - 1 ) / M + I ] uni t s f a i l ed wi th the l a s t{ [ ( j - 1 ) / M ] + l } t h f ai le d u n it d u e to t h ej - [ ( j - 1 ) / M ] M f ai le d m o d e , w h er e ( j - I ) / M i se q u a l t o t h e i n t e g e r o f [ ( j - 1 ) / M ] , j = c m e a n sf a i l u r e d u e t o a c o m m o n c a u s e a n dj = ( N - k )M +i , i = 1 , 2 . . .. M m e a n s s y s t e mfa i lu re due to the i th fa i lu re mode o f the( N - - K + 1)th unitPj ( t ) p r o b a b i l i t y t h a t t h e s y s t em i s i n s t a t e j a t t i m e tej(y), qj(y) r e p a i r r a t e a n d p r o b a b i l i t y d e n s it y f u n c t i o n o frepa i r t ime w hen the sys tem i s in s t a te 3 and hasan e lapsed repa i r t ime o f y , j = c andj = ( N - k ) M + i , i = 1 ,2 . . . . M

pj(y, t) probab i l i ty dens i ty (wi th respec t to repa i r t ime)tha t the fa i l ed sys tem i s in s t a te j and has a ne lapsed rep a i r t ime o f y

d c o n s t a n t c o m m o n c a u s e f a il u r e r a t e o f t h e s ys t e md o = d i f k > l o r = 0 i f k = la~ con s tan t f a i lu re ra te o f a un i t due to the i th

f a i lu r e m o d es L a p l a c e t r a n s f o r m v a r i a b l ea'r = realG j ( s ) Laplace t r ans fo rm of the pdf o f r epa i r t ime q~(y)Av(s) L a p l a c e t r a n s f o r m o f s y s t e m a v a i l a b i li t yA v sys tem s teady-s ta teava i lab i l i ty .

T H E STATE PROBABILITY OFTHE SYSTEM

T h e t r a n s i t i o n d i a g r a m o f t h e k - o u t - o f - N : Gr e d u n d a n t s y s t e m w i t h M m u t u a l l y e x c l us i v e f a i lu r em o d e s a n d c o m m o n c a u s e f a il u r e s i s d e p i c t e d i n F ig .1.

T h e s e t o f d i f f e re n t i a l e q u a t i o n s f o r t h e m o d e l i s( c 3 /O t + d + i ~ = l a N ) P o ( t )

= Pc(Y, t)e~(y) d yo

+ i~ P(N--klM+i(Y, )e IN-k)M+i(Y) dy , (1)

8/12/2019 Microelectronics Reliability Volume 27 Issue 4 1987 [Doi 10.1016%2F0026-2714%2887%2990004-7] Who Kee Chu

2/3

Je(N-k)M+iY)

a~ D ,

v

a2

eN=kM+ly

-=N_+IM(y

o~'

jM+*o,~-J-

Im

N.~

*

oN_

0i o~

JP

a~J

O'

V

ON-

t

N

N-

O~-

J

-

/~~M

aNJ

)M

N-klM

.d

d

z

-=,-Co

eoC

Fig1

8/12/2019 Microelectronics Reliability Volume 27 Issue 4 1987 [Doi 10.1016%2F0026-2714%2887%2990004-7] Who Kee Chu

3/3

Rel iabi l ity mod el for a k-out -of -N :G redu nda nt sys tem 623

(a / t 3 t+d+ ~= la~m-J )P t j - 1 )M+i ( t )M N - j + I= am P U - 2 ) M + m ( t ) ,m = l

j = 1 , 2 . . . . . N - k - 1 a n d i = 1 , 2 . . . . . M, 2)( d / t ~ t + d o + m ~ = a k ) P t N - k + l , M + , ( t )

ME k+l= am P(N-k-2)M+ m(t ) ,ra=li = 1 , 2 . . . . . M , (3 )

{ d /d t + 8/ t3y + e (N-k)M + (Y )}P(N-k)M +i(Y, t ) = 0,i = 1 , 2 . . . . M , (4 )

{ ~ / d t + a / d y + e c ( y ) } p ( y , t) = 0, (5)M

p(N_k)M+i(O, t ) = ~ akp(N_k _l )M+m (t ) ,m = li = 1 , 2 . . . . . M , (6 )

N - k -2 ) Mpc(O, t) = d ~ ~ P jM+i(t)j =O i =1M+ d e o ( t ) + d o ~ PtN-k-.u+~(t), (7)i =1

N - k M~ , ~ P j M + i ( t ) + P ~ ( t ) + P o ( t ) = 1 (8)j =O i =1a n d Pi(O) = 1 f o r i = 0 , o t he r w i s e = 0 , pj(y, 0) = 0 fora l l j a n d ( N - k ) M + i , i = 1 , 2 .. . . M .

L a p l a c e t r a n s f o r m s o f t h e s o l u t i o n s o f t h e a b o v ee q u a t i o n s a r ee~j - l )M + (s ) = Aj iDj Po(S)

f or j = l , 2 . . . . . N - k ; i = 1 , 2 . . . . . M , (9)P c( s) = [1-6~(s)]- d 1 + y , A j,O j

S j=l i

+ d o ~ (10)MP(lV -k)M+ i(S) = aki ~ A N - k , m D N - k

m = l [ 1 - G{N - k)A4+ (S)-IPo(S)/S,i = 1 , 2 . . . . M (11)

a n dP o (s ) = { [ 1 - G c ( s ) l l d ( l + N j~ = : i~__lA j i D j )

+ d o ~ ,:~ A N - k , i D N - k ]

M M+ ~ [ 1 - - G ( N - k )M + i ( $) 'la k i ~ A N - k . m D N - ki l l m l} - 1

+ s + s f _ , 1 2)j l i iw h e r e

A j i = a N - j + I s + d + a~m j ,=1j = 1 , 2 . . . . . N - k - l ; i = 1 , 2 . . . . . M ,/N - k , i = a k + l s + d o + a , i = 1 , 2 . . . . . M ,=1

DI = 1,a n d

M M MO j ~ ~ ~ . . . ~ a j - l , i~ - ,il_t = 1 i1_2 = 1 i1 =1xAj-2 , i j_2 . . .A2, i2Al , i~ .

T h e s y s t e m a v a i l a b i l it y e x p r e s s i o n i n t e r m s o fL a p l a c e t r a n s f o r m i s

(N k)MAv ( s ) = ~ P i (s ) . (13)i = 0

B y a p p l y i n g t h e l i m i t i n g t h e o r e m o f t h e L a p l a c et r a n s f o r m , w e h a v e t h e s y s t e m s t e a d y - s t a t ea v a i l a b i l i t y

N - k)MA v = l i m s ~ Pi(s). (14)s ~ O i = 0

SPECIA L CA SESL e t u s c o n s i d e r t h e f o l l o w i n g c a s es .( 1) F o r N = 2 , k = 1 , w e ha ve , f r om e q ua t i o ns 9 , 10 ,

1 1 a n d 1 2, t h e e q u a t i o n s 7 , 8 , 9 a n d 1 0 r e p o r t e d b yD h i l l o n [ 4 ,l .

(2) F o r N = 2 , k = l , M = 4 , e c ( y ) = # a n de i (Y ) = / ~ i f o r i = 5 , 6 , 7 , 8 , e q u a t i o n s ( 9 ) - (1 2 ) a r er e d u c e d t o e q u a t i o n s ( 1 5 )- ( 1 8 ) i n [ 4 ] .

R E F E R E N C E S1. B . S. Dhi l lon, On com mon -cause fa i lures- -bibl iography ,Microelectron. Reliab. 18, 533 (1979).2 . B . S . Dhi l lon, Mu l t i -s ta te device re dun dan t system wi thcomm on-cause fa i lures and on e s tandb y uni t ,Microelectron. Reliab. 20, 411 (1980).3 . Who K e e C hung , A k - ou t - o f - N : G t h re e s t a t e un i tredu nda nt system w i th comm on-cause fa i lures andreplacements, Microelectron. Reliab. 21, 589 (1981).4 . B . S . Dhi l lon, Uni f ied avai labi l i ty model ing: a redu nda ntsystem with mechanical , electr ical , software, human andcomm on-cause fa ilures , Microelectron. Reliab. 21, 653(1981).