Shared Load Reliability

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1/3

~ ) Pergamon

Microelect ron . Rel iab . ,

Vol. 37, No. 5, pp. 869-8 71, 1997

Copyright © 1997 ElsevierScience Ltd

Printed in Great B ritain.All rights reserved

0026-2714/97 $17.00+.00

P I I : S 0 0 2 6 - 2 7 1 4 (9 6 ) 0 0 1 0 0 - X

T E C H N I C A L N O T E

R E L IA B IL IT Y O F A D Y N A M I C n -U N I T S H A R E D L O A D P A R A L L E L

S Y ST E M U N D E R D I F F E R E N T F A I L U R E T I M E S

S. S O M A S U N D A R A M

Dep ar tment o f Mathemat ics , Coim batore Ins t i tute of Technology, C oimba tore 641 014, India

a n d

D . A U D S I N M O H A N A D H A S

Depa r tment o f Mathemat ics and Com puter Ap pl icat ions , P.S.G. Col lege of Technology Coim batore 641 004, India

( R e c e i v e d f o r p u b li c a ti o n 2 M a y 1996)

Ab stra ct-- A generalized form ula is derived for the rel iabili ty of a dyn am ic paral lel system of n-co mp one nts

with equally shared load fai l ing at different fai lure t imes. This formula is obtained using success modes

analys is (SMA) an d the sam e is completely verified for expon ential distr ibutio n. Copyrig ht © 1997 Elsevier

Science Ltd.

1 . I N T R O D U C T I O N

D y n a m i c o r t i m e d e p e n d e n t r e l ia b i l it y m o d e l s a r e

m o r e d i f f i c u l t t o e v a l u a t e t h a n s t a t i c m o d e l s . V a r i o u s

a p p r o a c h e s h a v e b e e n u s e d in e v a l u a t i n g d y n a m i c

m o d e l s . S a n d i e r [ 1 ] h a s c o n s i d e r e d t h e M a r k o v

a p p r o a c h w i t h r e s p e c t t o c o n s t a n t f a i l u r e r a t e .

R e l i a b i li t y c o m p u t a t i o n s f o r a t w o - u n i t s t a n d b y

r e d u n d a n t s y s t e m w i t h c o n s t a n t f a il u r e r a te a r e f o u n d

i n O s a k i a n d N a k a g a w a [ 2 ]. R a m a n a r a y a n a n [ 3 ] h as

c o n s i d e r e d t h e a n a l y s i s o f n - u n i t w a r m s t a n d b y

s y s t e m s w i t h E r l a n g f a i l u r e t i m e . A l i d r i s i [ 4 ] h a s

d i s c u s s e d t h e r e l i a b i l it y o f a d y n a m i c w a r m

s t a n d b y r e d u n d a n t s y s te m o f n - c o m p o n e n t s w i th

i m p e r f e c t s w i t c h i n g .

I n t h i s c o n f i g u r a t i o n , t h e p a r a l l e l s u b s y s t e m s

e q u a l l y s h a r e t h e l o a d a n d a s a s u b s y s t e m f a i l s , t h e

s u r v i v i n g s u b s y s t e m s m u s t s u s t a i n a n i n c r e a s e d l o a d.

T h u s a s s u c c e s s iv e s u b s y s t e m s f a il , t h e f a i l u r e r a te o f

t h e s u r v i v i n g c o m p o n e n t s i n c r e as e s r a p id l y .

A s s u m i n g t h a t w h e n f a il u r e o c c u r s t h e s u r v i v o r t h e n

fol low s p.d. f, q l ( t ) a n d t h a t t h i s p . d . f , d o e s n o t d e p e n d

o n t h e i n t e r v a l o f e l a p s e d t i m e . W e a p p l y s u c c e s s

m o d e s a n a l y s i s f o r s u c c e s si v e s u b s y s t e m s f a i l in g a t

d i f f e r e n t t i m e s . F o r t h i s s i t u a t i o n w i t h n - s u b s y s t e m s ,

t h e t o t a l p o s s i b l e n u m b e r o f m o d e s o f s u rv i v a l is

n ( n +

1 ) / 2 . W e s h a l l a l s o c o n s i d e r t h e p r o b a b i l i t y o f

e a c h m o d e s e p a r a t e ly t h e n a d d p r o b a b i l i t ie s s in c e th e

e v e n t s r e p r e s e n t e d a r e m u t u a l l y e x c lu s i v e .

R a m a k u m a r [ 51 h a s d i s c u s se d t h e p a r a ll e l s tr u c t u r e

o f a s y s t e m . A s et o f n - c o m p o n e n t s i s s a i d t o b e i n

p a r a l l e l fr o m a r e l ia b i l it y p o i n t o f v i e w i f t h e s y s t e m c a n

s u c c e e d w h e n a t l e a s t o n e c o m p o n e n t s u c c e e d s . T h e

b l o c k d i a g r a m o f s u c h a s y s t e m i s s h o w n i n F i g . (1 ).

Cause

[

~ U nit 1

~ ' r U n i t 2

r

Unitn

Fig. 1. Parallel structure.

Effect

2 . N O T A T I O N

q i ( t )

p.d.f , for t ime to failure un der

1/ i

load

),i failure rate of

1/i

load

Rso(t) rel iabil i ty of the system with no fai led comp one nts

whi le in ope rat ing condi t ion

Rsr(t rel iabil i ty of the system with r fai led com pon ents

while in opera t ing co ndit io n where (1 ~< r ~< n - l )

Rq,(t) reliabili ty of the system un der 1/ i load

Rs(t ) reliabili ty of the whole system with n-co mp one nts

3 . T H E MODEL

T h e r e l ia b i l it y o f t h e t i m e d e p e n d e n t p a r a l le l s y s t e m

o f n - c o m p o n e n t s w i t h e q u a l l y s h a re d l o a d f a i l in g

a t d i f f e r e n t t i m e s i s i n v e s t i g a t e d u s i n g s u c c e s s

m o d e s a n a l y s i s . L e t E l , E z . . . . E , b e t h e e v en t s th a t

a r e c o m p o n e n t s a n d a r e f u n c t io n i n g in o p e r a t i o n

m o d e . L e t T t ,

? '2 . . . . 7 ",

b e t h e r a n d o m v a r i a b l e s

r e p r e s e n t i n g t h e l i f e o f n - c o m p o n e n t s i n o p e r a t i o n

869


2/3

870 Technica l Note

mod e . Le t t~ < t 2 < . . . . . < t , be d i f fe ren t fa i lu re

t i m e s o f e a c h c o m p o n e n t .

T h e r e l i a b i l i t y o f t h e s y s t e m w i t h n - c o m p o n e n t s w i ll

b e c o m p u t e d f o r t h e s p e c i a l c a s e n = 2 , 3 , 4 a n d 5 ,

t h e n R s ( t ) f o r t h e g e n e r a l s y s t e m w i l l b e d e d u c e d a n d

l a t e r v e r if i ed . K a p u r a n d L a m b e r s o n [ 6 ] h a v e

c o n s i d e r e d t h e r e li a b i l i ty o f t h e t w o c o m p o n e n t

p a r a l l e l s u b s y s t e m s w i t h e q u a l l y s h a r e d l o a d .

Cas e 1

/ ' / = 2 .

R s (t ) = P r [ M o d e 1 w M o d e 2 w M o d e 3 ], i.e .

R s ( t) = P r [ M o d e 1 ] +

P r

[ M o d e 2 ] +

P r

[ M o d e 3 ]

i .e . Rs(t) =

P r

[ M o d e 1 ] + 2

P r

[ M o d e 2 ] . S i n c e t h e

c o m p o n e n t s a r e i .i .d , a n d t h e m o d e s a r e m u t u a l l y

exc lus ive events , i . e . Rs( t ) = Rso( t ) + 2Rs~( t ) where

Rso( t ) = [Rq: ( t ) ]2 :

Rs~(t)

= f l q 2 ( t l )R ~ ( t O R q l ( t - G ) d t~

R . ~ t) = f q 2 u )d u

a n d

R q ~ ( t) = f f ~ q ~ ( u ) d u .

I n t h e c a s e o f c o n s t a n t f a i l u r e r a t e s i . e. f o r e x p o n e n t i a l

d i s t r i b u t i o n w e o b t a i n

q ~ ( t ) =

)~ e- .1" , Rq ,( t) = e - .1 '~

an d i = 1, 2.

Rs(t) = e_2.12t + 2e _ Z : ~ 1 e-(2.12-.1t) t~

L 2 )-~ - a , ) 2 2 ~ - ~ ) j

Cas e 2

F o r t h r e e c o m p o n e n t s i .e . n = 3 , a s i n c a s e ( 1 )

Rs(t ) = Rso(t ) + 3Rs~(t) + 2Rs2(t)

w h e r e

Rso(t) = [Rq3(t) ' ] 3,

R s ' ( t )

= f l q3 ( t l) [ R q~( tl )] Z [ R q~( t -

t 0 1 2 d q ,

Rs~(t) =

qa( t l )q3( t2)Rq~( t2)Rql ( t -

t2 ) d t2 d t l .

1

I n t h e c a s e o f c o n s t a n t f a i l u r e r a t e s i . e. f o r e x p o n e n t i a l

d i s t r i b u t i o n

Rs(t) = e- 3.1~t

+ 3).3 e_2.12tF - 1 e - '3a~ -2.1~ , ' ]

L 3) -3 -

2)-2) ~

- - ~ ) J

+ 2)-3 e_ .1~ [ 1

k 3 2 2

-

k l ) 2 k ~ ) -I )

e-(3k 3-.1Dt e-(2.13-.1t)t 1

+ )-~ 0)-~ - ). 0 ) . ~ - ~ - T ~ ) j

C a s e 2

F o r f o u r c o m p o n e n t s i .e . n = 4 , a s b e f o r e ,

Rs(t ) = Rso(t ) + 4R s,( t) - 3Rs2(t ) + 2Rs3(t)

w h e r e

Rso(t ) = [Rq4(t)] 4,

R s ' ( t )

= f l q 4 ( t t )[ R q ' ( t l) ] 3 [ R q 3 ( t -

t 0 1 3 d t ,

Rs2(t ) =

q4( t l )q, ( te)[Rq4( t2)] 2

1

x [Rq2(t - t2)] 2 dt 2 dt l ,

f o ; J

s3(t ) =

q , ( t l ) q4 ( t 2 )q , ( t 3 )

1 2

x R~4(t3)Rqt(t - t3) dt 3 dt 2 dt v

I n t h e c a s e o f c o n s t a n t f a i l u r e r a t e s :

Rs( t ) = e -4~,z

+ 4 2 4 e - a a ~ F i 4 2 4 L 1 e , 4 .1 4 , . 1 , , z ]

3).3) (42 ~ ~ 3).3~ J '

322 e - 2.1~t[i4241 1

- 2 )-2)(324 - 222)

e-(4.14-2.12}t

e - {3 a4- 2.12)t ]

)-4(4)-4 -- 2) .2) / .4(3L4 -- 2) .2)J

+ 2).~ e - .1~'F

1

I-(424 - 21)(3 24 - )-1)(224 - )- t)

e ( 4 . t 4 . 11 ) e - ( 3 . 1 4 - . 1 t )

2 22(424 - ) - 1 ) ) - 2 ( 3 ) . 4 - ' ~ 1 )

e- (2.14 - .1)t 1

C a s e 4

F o r f i v e c o m p o n e n t s i .e . n = 5 ,

Rs( t ) = Rso( t ) + 5R s: ( t ) + 4R s~( t ) + 3Rs~( t ) + 2Rs , ( t )

Rs( t ) = [Rq, ( t ) ] 5,

Rs~(t)

= f l q s ( t l ) [ R ~ ' ( t l ) ] 4 [ R q ' ( t -

t l ) ] 4 d t l ,

f o

s~(t) =

qs( t l )qs ( t2)[Rq~( t2)] 3

1

x [ R ~ 3 ( t - t z ) ] 3 d t 2 d t ,

f o r : f :

, [ R

Rs~(t) =

qs ( t x )qs ( t 2 )qs ( t s

~ ( t 3 ) ] :

1 2

x [Rq2(t - t3)] 2 dt 3 dt2 d tt ,


3/3

Techn ica l No t e 871

L £ £ £

s , ( t ) =

q s ( t O q s ( t 2 ) q 5 ( t 3 ) q s ( t 4 )

1 2 3

x R q , ( t 4 ) R q , ( t - t 4 ) d r 4 d t 3 d t 2 d t , .

I n t h e c a s e o f c o n s t a n t f a i l u r e r a t e s f o r n = 5 we

h a v e

Rs( t ) = e - S~ , + 525 e -4~ .~' V

1

L(5 424)

e [5).5 _ 4;ta)t1

i 5 2 5 - 4 ) ., ) /

+ 4 22 e - 3 2 3 t [ _ _ _

1

L(525 - 323) (425 - M 3)

e - (425 - 3).3) e -- (525 -- 3-~3)t

l

F

/

2 s ( 4 2 5 - - 3 2 3 ) 2 5 (5 2 5 ~ 3 ) J

+ 323 e 2;.~,

1

(525 - 22 2)(425 - 22 2)(325 - 222)

e (325 - 2221/ e-(4 25- 2.~) ,

- 2 25z(32s - 2).2) + 22(4 25 - 222)

e - (525- 222 ) t

l

2 22 (52~ ~ 2 )~2) + 22~ e - ; " '

(5) .5 - 20 (4 25 - )~1)(3~.5 - ) . ,) (2 25 - 2 , )

e-(22 ~ ao, e (325-21),

3 253(22.s - ) ] i i

+

2[23(32 , -

2 0

e 2 ( 4 ; 5 - & ) ' e-(5;~5 - &)t 1

2 ) .35(4) .5 - - 2 0 -~ 3 2 3 ( 5 ) o - - ~ ~ 2 0 j '

I n g e n e r a l f o r a n y n ,

n - 1

R s ( t ) = ~ ( n + 1 - r ) R s . ( t ) + R s o ( t )

r = l

w h e r e

Rs o ( t) = [ Rq . ( t ) ] ' ,

L

s , ( t ) q , ( t ) [ R q , ( t l ) ] " - l [ R q , _ l ( t - t t ) ] " - 1 d t l ,

f o £ f ,

s r ( t) = , . . . . q , ( t O . . . . . q , ( t , )

1 r - I

x [ R q . ( t , ) ] " - '

x [ Rq . r (t - - t , ) ] n - r d t . . . . . . d t 2 d t ,

w h e r e ( l < r ~ < n - 1 ) .

I n t h e c a s e o f c o n s t a n t f a i l u r e r a te s , i n g e n e r a l ,

n I

Rs( t ) = e - z ' + ~ (n + 1 - - r)2 ,~ e - ( -r )~ . . . .

r= l

x 1 ( (n - p + 1 )2 , - (n - r )2 ,_ , )

[ , /2]

+ Y

p= l

( _ 1 ) . + p - 1 e - [ ( n - p + 1 ) z . - ( . - r ) a . - r l,

X

( r - p ) 2~- l [ (n - p + 1 )2 . - (n - r )2 ._ , ]

p = [r/21 + 1

( _ 1 ) , + p - t e - [ i , - p + l ) ~ . - ( . - , ) a . - . l , 1

w h e r e [ r / 2 ] i s t h e i n t e g r a l p a r t o f r / 2 .

4 . C O N C L U S I O N

I n th i s p a p e r a n a t t e m p t h a s b e en m a d e t o c o m p u t e

t h e s y s t e m r e l i a b i l it y o f a d y n a m i c s h a r e d l o a d p a r a l l e l

s y s t e m o f n - c o m p o n e n t s u n d e r d i f f e r e n t fa i l u r e t i m e s .

A g e n e r a l i z e d f o r m u l a f o r t h e s y s t e m r e l i a b i l i ty h a s

b e e n i n v e s t i g a t e d b y a n a l y s i n g t h e s y s t e m s u c c e s s

n o d e s . T h e o b t a i n e d f o r m u l a i s t h e n v e r i f i e d f o r th e

s p e c i al c a s e o f p a r a l l e l s y s t e m w i t h c o n s t a n t f a i l u re

ra t e s .

R E F E R E N C E S

1. Sandler , G. H.,

S y s t e m R e l i a b i l i t y E n g i n e e r i n g .

Prent ice-

Hall , Eng lew ood C liffs , NJ , 1963.

2 . Osak i , S . and Nakagawa , T. , On a two-un i t s t andby

redundant system with s tandby fa i lure . O p s R e s . , 1971, 19,

510-523.

3 . Ramanarayanan , R. , n -un i t warm s t andby sys tem wi th

Erlang fa i lure and general repair and i ts dual . I E E E T r a n s .

R e l i a b , 1978, R-28, 173-174.

4 . Al id ri s i, M. M. , The Re l i ab i li t y o f a D ynam ic warm

s t a n d b y r e d u n d a n t s y s te m o f n - c o m p o n e n t s w i t h

imperfect switching. M i c r o e l e c t r o n . R e l i a b . , 1992, 32(6),

851-859.

5 . Ramakumar , R. , E n g i n e e r i n g R e l ia b i l i t y F u n d a m e n t a l s a n d

A p p l i c a t i o n s .

Prent ice-H all , En glew ood Cliffs , NJ , 1993.

6 . K apur , K . C. and Lam berson , L. R.,

R e l i a b i l i t y i n

E ng i ne e r i ng De s i gn ,

Wiley, New York, 1977.

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Shared Load Reliability