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Common-cause failure analysis of a k-out-of-n:G system with repairable units
Microelectron. R&b.,
Vol. 34, No. 3, pp. 429-442, 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0026-2714/94 $6.00+.00
Pergamon
COMMON-CAUSE
FAILURE
ANALYSIS REPAIRABLE
B. S. DHILLON Department
of Mechanical
Engineering,
OF A k-OUT-OF-n UNITS
:G SYSTEM
WITH
and 0. C. ANUDE
University Canada
of Ottawa,
Ottawa,
Ontario
KIN
6N5,
(Received for publication 23 February 1993)
ABSTRACT A newly developed of-n
: G system
is presented.
generalized composed
Also, reported
time to failure
expression
for the mean time to failure
of repairable
units and subject
are some special
of a k - out -
to common-cause
case system
reliability
failures
and variance
of
formulas.
INTRODUCTION
The importance
of incorporating of complex
engineering
In fact, as equipment
redundancy
increases,
ures potentially example,
may become
in the reliability
uration,
Moody
contributors
Broadly
to system
speaking,
clude multiple emergency
assessment
This paper
standby.
study
failure
jet engines
in redundant
is concerned
This system
examples
active
is a more general
units
diagram start
mechanism
failure
version
functioning
of operable technique
mode.
The system
condition
for system
mean
k, needed
success.
429
failures
one warm reported
system Laplace
in
1. At time
with the warm
normally
time to failure,
end results.
with
system
The system
The standby
of parallel
for system
have been used to obtain
in-
is shown in Figure
standby
is considered
to be
(k 5 n). The standby
and/or
as well as due to non-common-cause
reliability,
fail
of a k - out - of -
its switch
can fail due to a common-cause
in terms of the number units,
analysis
components
instantaneously condition.
unit fails.
items failures
loss of power station
of the parallel
for this system
in perfect
when multiple
of valves and multiple
identical
as soon as one active
Expressions
config-
were the main
[4].
is activated
are developed
fail-
[I]. For
trip breaker
of common-cause
maintenance
as long as at least k units are operating
working
not chance
failures
due to bird ingestion,
operational
normal
in reliability
be over-looked.
unavailability
reactor
is said to occur
structures
of repairable
Ref. [S]. The state-transition
in its standby
failures,
common-cause
with the common-cause
composed
and its switching
failures simply
to system
of a nuclear
that
due to inadequate
corrosion
t = 0, all n active
cannot
common-cause
contributor
cause [3]. Typical
loss of aircraft
n : G system
common-cause systems
unavailability.
a common-cause
feedwater
due to stress
the major
and Follen [2] concluded
due to a single underlying
MR Y/I--D
and quantifying
and safety assessment
failure
can fail from the
failures.
variance
of time to failure
units n and the least number transforms
and the Markov
430
B.S. DHILLON and O. C. ANUDE
r=n-k+2 m=n-k+l
Fig. 1. State-space diagram of the system.
ASSUMPTIONS The following assumptions are associated with this system model:
i)
The system has n + 1 identical and repairable units (n active and one on standby).
ii)
The common-cause and hardware failures are statistically independent. The system is operable as long as at least k units are operating normally (k < n)
iii)
A common-cause failure can occur from any of the operable states of the system. All failure and repair rates are constant.
iv)
v) vi) vii) viii) ix)
x)
The switching mechanism for the standby is considered automatic and instantaneous. The standby may fail in its standby mode, in addition, to the failure of its switching mechanism. The system, once in a failed state, is non-repairable A repaired unit is as good as new All common-cause failures are completely coupled
NOTATION The following symbols are used in this article: t n
k
Ai A°
time Number of parallel system units Least number of active units needed for system success Laplace transform variable Constant failure rate of a unit Constant failure rate of switching mechanism and/or standby itself
Consecutive-k-out-of-n:G system
431
Constant critical common-cause failure rate
~e0
from system state O. he/
Constant common-cause failure rate from system
J
jth state of the system; for j = O, 1,2,3, ....,n - k + 4
up-state i; for i = 1,2,3, ....,n - k + 2 Constant repair rate of switching mechanism
#s
and/or standby itself Constant repair rate from system up-state i; f o r i = l , 3 , 4 , ...... n - k + 2 The probability that the system is in state j at time t;
#i
Pj(t)
f o r j = 0,1,2,3, ..... , n - k + 4 Common-cause failure Derivative of the Laplace transform of the reliability function, Rn(s), with respect to s Inverse Laplaxe Transform
C.C.F R'(s)
In Figure 1, the system state is denoted by a numeral and/or a letter at the bottom of each box, ANALYSIS The set of differential equations a~sociated with this system model shown in Figure 1 is:
dPo(t) d----'-~+(nAl + $, + A~o)Po(t) = pl P1(t) + p, P2(t) dPl(t)
d-""~ 31- (n/~l "[- ~cl + ]~1) Pl(t) = n~l Po(t) "~/23 e3(t)
de2(t)
d-----t--+ (n)h + At2 + #.) P2(t) = Ao Po(t) + #3 P3(t)
dP3(t) d~
(1)
(2) (3)
+ [(n - 1)A1 + Ac3 + 2p3] P3(t) = nA1 Pl(t) + hA1 Pz(t) + ~4 P4(t) (4)
dPr(t)
d----i--+ [k~ + ~ . + ~,~]P~(t) = (k + 1);~ P~_~(t)
(5) (6)
,~P~+l(t) _ k~l P~(t) dt r
dPr+2(t) - ~ Aej Pj(t) dt j=0
(7)
At time t = 0, P0(0) = 1 and all other initial state probabilities are equal to zero. The system reliability, Rk/n(t), is given by:
r
r
Rk/.(t) = ~ P~(t) = L - ~ [ ~ Pj(,)] j=O
(S)
j=O
The mean time to failure, MTTFk/., of the system is:
r
MTTFk/. = lim S'~ P3(s)
(9)
432
B.S. DHILLONand O. C. ANUDE The variance of the time to failure, a~ln, is expressed by the following relationship:
a~/,~ = - 2 lim R'k..~(s ) - ( M T T Fk/n) 2
(10)
Special Cases i) For k = 1, and n = 3 in Figure 1, the set of differential equations describing the system becomes:
dPo(t)
d---~- + (3~, + A, + Aco)Po(t) = #1 Pl(t) + #, P2(t)
dPx(t)
d----~+ (3A1 + Acl + .1) P,(t) = 3A, Po(t) + P3 P3(t)
(11) (12)
dP2(t) dt t-(3Al+Xc2+ps)P2(t)=XsPo(t)+p3P3(t)
(13)
dP3(!) =(2Al+A~3+21~3)P3(t)=3A1Pl(t)+3;hP2(t)+~qP4(t)
(14)
dP,(t) dt
(15)
dt
(A, + At4 + I~4)P4(t) = 2)h Pz(t)
-
dPs(t) d-----~- = ~1 P4(t)
dP6(*) dt
= ~ Acj Pj(t) j=o
(16) (17)
At t i m e r = 0 , P0(0) = 1 and all other initial state probabilities are equal to zero. Using Laplace transforms, Equations (11)-(17) yield:
Ao(~) P0(~) - Ao(s)
(18)
A,(~)
(19)
A2(s)
(20)
A3(s)
(21)
A4(s) P4(~) = :4D(~)
(22)
As(s) P s ( s ) - s "AD(~)
(23)
A6(s) P6(s)- s*AD(s)
(24)
P,(s) - AD(S) P2(~) - AD(S)
P3(s)
-
AD(S)
where,
A0(s) -- (s + C3(4))(s + C3(1))[(s + C3(2))(s + C3(3)) - 3 AI~3] 3 AI #3 (s + C3(2))(s + C3(4)) - 2 A, #4 (s + C3(1))(s + C3(2))
Al(s)
=
3A, (s + C3(2))[(s + C3(3))(s + C3(4)) - 2)q/x4] + 3 AI A~~/3(s ~- C3(4)) - 9 X~#3 (s + C3(4))
Consecutive-k-out-of-n:G system
A~(,)
433
As (s + C3(1))[(s + C3(3))(s + C3(4)) - 2A, P4] + 9A~p3(s + C3(4)) - 3Ai A,#3(s + C3(4))
A3(s) = 9 A~ (s + C3(2))(8 + C3(4)) + 3 A, A, (s + C3(1))(s + C3(4)) A4(s) =
18 ~13( , -}- C3(2)) Jr 6 )~]2"~s (8 + C3(1))
As(~) =
~, A4(s) 4
As(s)
=
j=O
Ao(s)
=
(s + C3(0)) Ao(s) - #, A I ( s ) - It, A2(s)
and, 6'3(0) = 3 A, + A,o + A, C3(I) = 3AI + Ae] +/q C3(2) = 3 A] + At2 + #, C3(3) = 2AI + Ae3 + 2#3 C3(4) = Az + At4 + #4 From Equation (8), the system reliability is: 4
R,/~(t) = ~PAt)
j=o =
e °at [B1 Jr Bs + Bll -k B16 + B21] +
e s==[B2 + Br + B12 + B l r + B22] + e szt
[B3 + 138 + BI3 "1-BlS + B23] +
e s ' t [134 e"t
+ B9 Jr B14 -}- B19 + B24] -r-
[Bs -{- BlO -k B,s + B2o + B25]
(25)
where symbols sl to ss are real and unique roots of the polynomial function A D ( S ) . The symbols B1 to B2s are defined in Appendix A1. A plot of (25) and corresponding tabular values for specified values of system parameters are shown in Figure 2 and Table 1 respectively.
1.00.90.8"
\
0.7"
\,
\ N
0.6-
k=2
~'~.
0
E
0.50.40.3-
~.
=
0.20.1-~
~ 0.0
0.0
~
-
3-- ~ 4 =O.O0003/~r i
1500.0
i
3000.0 Time in hours, t
i
4500.0
Fig. 2. System reliability plots for three different values of k.
i
6000.0
434
B. S, DHILLON and O. C. ANUDE T a b l e 1: S y s t e m values
Reliability
for three
different
o f k ( n = 3)
Time (in hours)
k = 1
k = 2
k = 3
0.00
1.0000
1.0000
1.0000
1500
0.9264
0.7800
0.4708
3000
0.7589
0.4456
0.1607
4500
0.5682
0.2310
0.0534
6000
0.4067
0.1164
0.0177
oo
0.0000
0.0000
0.0000
Using Equation (9), the system mean time to failure is: 4
MTTF1/3
= lim ~ Pi(s) Y~/3 Z1/3
(26)
where,
Y1/3
Ca(l) C3(4)[63(2) C3(3) - 3 h, P3] - 2 h, P4 C3(1) C3(2) 3 hi/z3 63(2) C3(4) + 3 hi C3(2) [63(3) C3(4) - 2 hi P4] + h, 63(1)[63(3)63(4) - 2 hi P4] + 9 h~ 63(2)63(4) + 3 h, h. 63(1) 63(4) + 18 h~ C3(2) + 6 hl h. C3(1)
Z1/3 = C3(0) [63(1) C3(4)(63(2) C3(3) - 3 hi #3) - 3 h, P3 C3(2) C3(4) 2 hi P4 Ca(l) C3(2)] - pl [3 A1 C3(2) (63(3) C3(4) - 2 hi p4) + 3 hi P3 63(4) (As - 3 hi)] - p, [h, 63(1) (63(3) C3(4) - 2 A1P4) -
3 hi ~,3 63(4)
(hs -
3 hi)]
Z1/3 = An(s = O) From Equation (10), the variance of the time to failure is:
a~/3 -- - 2 1 i m E 1 / 3 ( s ) - ( M T T F1/3) 2 $~0
=
"
1 2 ~12 [2(x2x3 - xlx4) - x2]
(27)
Symbols zl to x4 are defined in Appendix A1. ii) For k = 2 and n = 3 in Figure 1, the set of system differential equations becomes:
dPo(t) q- C3(0) Po(t) = #1 Pl(t) + #, P2(t) dt
(28)
dPl(t) 4_ 63(1 ) PI(Q -- 3hi P0(~) -{-/-L3P3(t) dt
(29)
dP2(t) dt [- 63(2)P2(t) = )% Po(t) + P3 P3(t)
(30)
Consccutive-k-out-of-n:G system
435
dP3(t) + C3(3)P3(t) = 3 )il P, (t) ~- 3 )il P2(t)
(31)
dP4(t) = 2A1 P3(t) dt
(32)
dPs(t) 3 dt - ~ )icj Pj(t)
(33)
dt
j=o
At time t = 0, Po(0) = 1 and all other initial state probabilities are equal to zero. Using Laplace transforms, from Equations (28)-(33), the following relationships result: AT(s)
(34)
As(s) P,(s) = A S ( s )
(35)
Ag(s)
(36)
P3(s) = AIo(s) AS(S)
(37)
Po(s)- As(s)
P2(s) = As(s)
P4(s) - s Ps(s) -
All(S) •
(38)
AS(s)
A,2(~)
(39)
s * AE(S)
where, AT(s)
=
(s + Ca(l)) [(s + C3(2))(s + C3(3)) - 3 )il ]-/31-- 3 )il ~-/3(8 "4"63(2))
As(s) = 3)ii [(s + 63(2))(s -I- 63(3)) - 3 )ii It3] + 3 )il )is Its A9(s) =
)is (s + C3(1))(s + C3(3)) - 3 )il ]/,3()is - 3 At)
Aio(s)
=
3AiAs(s+C3(1))+9A2(s+63(2))
All(s)
=
2A1Am(s)
A12(8)
=
)ic0 Av(s) "4-)ic, As(s) "4-)ic2 A g ( s ) "~- )ic3 A,o(s)
AE(S)
=
( s + C 3 ( O ) ) n T ( s ) - It, A s ( s ) - ItsA9(s)
The system reliability is given by: 3
R2/3(t) = ~ Pj(t) j=0 :
e s6t [B26 + B3o + B34 + B3a] + e sTt [B27 + B31 + B3s + B39] + e sst [B2s + B32 + B36 + B40] + e sgt [B29 + B33 + B37 + B41] (40)
where symbols s6 to s9 are real and unique roots of the polynomial function A s ( s ) . Symbols B~s to B41 are defined in Appendix A2. A plot of (40) and corresponding tabular values for specified values of system parameters are shown in Figure 2 and Table 1 respectively. The mean time to failure of this system is: 3
MTTF2/3
=
lira ~ ' P j ( s )
s ~ 0 j=O
Y213 Z2/3
(41)
436
B.S. DHILLON and O. C. ANUDE where,
Y2/3
C3(1) [(73(2) C3(3) - 3 A1 ,u3] - 3 A1 #3 C3(2) + 3 A1 [C3(2) 6'3(3) - 3 ~1,0,3] + 3 A1 As#3 + As C3(1) C3(3) - 3 A 1 ,a3 [As - 3 A,] + 3 A, A, C3(i) + 9 A]2 C3(2)
Z2/3
=
C3(0) [C3(1) (C3(2) C3(3) - 3 A1 ju3) - 3 ~1/z3 C3(2)] #1 [3 ,kl C3(2) C3(3) + 3 A1 #3 (A, - 3 A1)] #, [),, C3(1) C3(3) - 3 Ax #z(A, - 3 A1)]
Z2/3 = AE(S = O) Using Equation (10), the formular for variance of the time to failure is:
a~/3
=
-21imR~t3(s ) - (MTTF2/3) 2
=
1 x-~ [2(=6x7 - x5=8) - ~ ]
$40
/
(42)
Symbols z5 to xs are defined in Appendix A2. iii) For k = 3, and n = 3 in Figure 1, we obtain the following set of system differential equations:
dPo(t) + C3(0) Po(t) -= #1 Pl(t) + #, P2(t) dt
(43)
dPl(t) + C3(1 ) Pl(t) = 3A1 Po(t) dt
(44)
dP2(t) + C3(2)P~(t) = A, Po(t) dt
(45)
dP3(t) dt - 3A1 Pl(t) + 3A1 P2(t)
(46)
2
dP,(t) _
dt
Z ~J P~(t)
(47)
j=o
Equations (43)-(47) using Laplace transforms lead to: A13(s) AF(S)
(48)
AI,(S) P I ( S ) - AF(S)
(49)
P°(s)-
Als(s)
P 2 ( s ) - AF(S) Alo(s) P3(s) = s • AF(S ) A17(s) P4(s) = s • AF(s) where, A,3(s) = (s + C3(1))(s + C3(2)) Al~(s) = 3 ~1 (s + C3(2)) A15(s) : A,(s + c3(1))
(~o) (51)
(52)
437
Consecutive-k-out-ofon: G system
A16(s) = 3 .~, (A14(s) + Alh(S))
&ff s) = ,Lo &3(s) + ;~1 A14(s) + .~2 &s AF(s) = (s + C3(0)) A13(s) - #1 A14(s) - #~ A,5(s) The system reliability, R3/3(t ) is given by: 2
R3/3(t) =
EPj(t) j=0
eS~°t[B42 + B4s + B4s] + e~t[B43 + B46 + B49] +
=
eS'2t[B44 + B47 + Bso]
(53)
Where symbols sl0 to s12 are real and unique roots of the polynomial function Av(s). Symbols B42 to B5o are defined in Appendix A3. A plot of (53) and corresponding tabular values for specified values of system parameters are shown in Figure 2 and Table 1 respectively. The system mean time to failure is given by: 2
MTTF3/3
=
lim~-'Pj(8) s---*0 j=9
Y3/z Z3/3
(54)
where,
Y3/3 = C3(1)63(2) + 3 .~1 C3(2) + .~ 63(1) Z3/3 ~- 6 3 ( 0 ) [ 6 3 ( 1 )
6 3 ( 2 ) ] - / 1 1 [3 ,'~1 C 3 ( 2 ) ] - ]Is [.~s C 3 ( 1 ) ]
Z3/3 = AF(S : O) The variance of time to failure is:
0"32/3 :
--2 lira R~/3(s ) -- (MTTF3/3) 2 S~0
= ~ [2(xm~,l - ~ 9 ~ 1 - x~0]
(~5)
Symbols x9 to x12 are defined in Appendix A3. Using Equations (26), (41) and (54) as well as analyses performed on several more special cases, the following generalized system mean time to failure expression is obtained:
M T T F k / . = Yk/~ Zk/.
(56)
Where Yk/n and Zk/~ can easily be obtained via the following recursive relationships3:
For k = n, n - 1, n - 2,. ....... ,1 we have the following set of recursive equations which can easily be solved:
Yn/. = 6.(1) c.(2) + . ~, 6.(2) + ~. G(1) Yn-1/. = Cn(3)Y./. - n.~l #3 Cn(1) - n.~1#3 C.(2) + O . 3For the specific case of k = 1, a non-recursive expression for the system mean time to failure is presented in Ref. [5].
B.S. DHILLON and O. C. ANUDE
438
Y,~-2/. = C . ( 4) Yn-~/n - (n - 1) )h b4 Y./n + (n - 1)~10. Y.-3/,~ = C . ( 5 ) Y ~ - 2 / .
- (n - 2 ) A l # s Y n _ l / n
+ (n - 1)(n- 2)A~ 0,~
Y a / . = C . ( n + 1) Y 2 / . - 2 A~ #.+~ Y31. + ( n - 1)! A~-~ O.
where On = (n A1)2 Cn(2) + hA1 A.Cn(1) and,
= ~./,,
z,,/,,
-
~1 [,¢,/.]
Z . _ ~ l . = a._~l,,
- #1
-
u, [¢./,,]
[~,,-~/,.,] - ~,, [¢,.-1/-]
Z,,_ w , = a . _ ~ l . - #1 [ / ~ . - 2 / . ]
z,/,,
= ~,/,,
- ~,, [,o,/,,] -
- m [¢.-~/.]
~, [¢,/,,]
Where, a./,, = C,,(O) Cn(1) Cn(2) a,~_a/n = Cn(3)an/,~ - n A , # 3 C n ( O ) C n ( 1 ) -
nAll~3Cn(O)C.(2)
a,,-21n = C,,( 4 ) a n - l l n - (n - 1) Al P,4 anl,~ a . - 3 / ~ = C . ( 5 ) e . - 2 / . - (n - 2) ~ #5 ~ . - 1 / .
aal,, = C,,(n + 1)as/,, - 2 Aa/~.+a a 3 / . / 3 ~ / . = n ~1 C . ( 2 )
/3._,/~ = C . ( 3 ) / 3 . / . + n A , ~ 3 ( ~ . - n ~ , )
,~n-2/n = C,.,(4) t3n_11,~ - (n - 1) Alp,4 ,On/. ~.-3/,,
= c.(5)/~,,-2/,,
- (- - 2) ,h~,6,,-~/,,
¢,,./,-, = As Cn(1) ¢,,_,/.
= c,,(3)
¢,,/,, - ..xl
m (~. -
,-, .~)
¢ n - 2 / n = Cn(4) ¢,~-1/,~ - (n - 1) Aa #4 Cn/n ¢.-3/.
= C . ( 5 ) ¢ . - 5 / . - (n - 2) A~ ~ ¢ . - 1 / .
¢ ~ / . = C,,(n + 1)¢2/~ - 2 A1#~+1 ¢ 3 / .
Cn(1) = n ~1 -~ ~cl -F f$1 C . ( 2 ) = n A, + At2 + / ~ .
Cn(3) = (n - 1) Aa + At3 -t- 2#3 c . ( 4 ) = (n - 2) ~1 + ~o, + u,
C . ( n + 1) = A, + Acn+, + #n+,
For specified values of system parameters, Equation (56) was used to generate the values given in Table 2 and the plots shown in Figure 3.
Consecutive-k-out-of-n:G system T a b l e 2: S y s t e m M T T F
439
v a l u e s (in h o u r s ) f o r t h r e e
d i f f e r e n t v a l u e s o f k (n = 3)
Critical Common-Cause Failure Rate, Ac0 (per Hr.)
k= 1
k= 2
k= 3
0.00000
6499.34
3393.38
1841.04
0.00001
6422.42
3355.43
1822.31
0.00002
6347.30
3318.32
1803.95
0.00003
6273.93
3282.02
1785.96
0.00004
6202.22
3246.51
1768.32
0.00005
6132.14
3211.75
1751.03
7000.0-
k-1
/ 6000.0-
5000.0O c-
.c 4000.0-
/k=2 2 E
5000.0-
a)
/
2000.0-
k=3
n----3 1000.0 -
)`1 --- O.O004-/Hr
)`s = O.O001/Hr
)`Cl -- Xc2 -- )`c3 -- )`c4- = O.O0003/Hr /1,s --= O.O006/Hr 0.0
o.o
/z4_ = O.O003/Hr
//'3 = O.O0015/Hr /1,1 ---- O.O004-/Hr
1'.o
~.o
~.o
,;.o
Crit.ical c o m m o n - c a u s e failure rot,e,
i.o .10 -5
}'co
Fig. 3. M q T F ~ plots for three different values of k.
REFERENCES
1. Apostolakis, G., "On the Reliability of Redundant Systems", Transactions of the American Nuclear Society on Safety Systems Reliability Methods and Applications, Vol. 22, 1975, pp. 477-478. 2. Moody, J, It., Follen, S, M., "Common-Cause Modeling of Reactor Trip Breaker Configuration", Proc. of the International Topical Meeting on Probabilistic Methods and Applications, EPRI NP-3912-SR, Vol. 3, Sessions 17-23, Palo Alto, CA., 1985, Feb., pp. 180/1-180/10.
440
B.S. DHILLON and O. C. ANUDE 3. Gangloff, W, C., "Common-Mode Failure Analysis", I E E E Transactions on Power Apparatus and Systems, Vol. 94, 1975, Jan-Feb., pp. 27-30.
4. Watson, I, A., "Common Mode/Cause Failures - An Overall Review", in Mechanical Reliability, Science and Technology Press Ltd., London, U.K., 1980, pp. 136-162. 5. Dhillon, B, S., Anude, 0, C., "Common-Cause Failure Analysis of a Redundant System with Repairable Units", International Journal of Systems Scienoe (to appear).
Appendix
A1
B1 = Ao(s = s , ) / D ~ I B2 = Ao(s = s2)/Ds2 B3 = Ao(s = s3)/Ds3 B4 = Ao(s = s 4 ) / D s , B5 = A d s = s s ) / D , ~ Bs = Al(S = s l ) / D ~ B7 = A , ( s = s 2 ) / D , 2 B8 = A I ( s = s3)/Dsa B9 = A i ( s = s 4 ) / D , ~ B,o = A , ( s = ss)/D~ 5 B n = A2(s = s , ) / D ~ , B12 = A : ( s = s2)/Ds2 B,3 = A2(s = s3)/D~ 3
B14
=
A2(s = s 4 ) / D s ,
B15 = A2(s = ss)/Ds5 B , 6 = A3(s = s l ) / D ~ ,
BI7 = A3(s = s 2 ) / D s , B , s = n 3 ( s = s3)/Dss
B19 = A3(s = s4)/Ds4 B2o = A3(s = ss)/D~ 5 B21 = A4(s = s l ) / D ~ 1 B22 = A4(s = s 2 ) / D ~
B23 = A4(s = s3)/D~s B24 -- A4(s = s 4 ) / D s , /3*25 = A4(s = s s ) / D , 5 where, Ds, = (s, D, 2 = (s2
-
s~)(sl
-
s3)(s,
- s4)(sl
-
ss)
-
s,)(s2
-
s3)(s~
- s4)(s~
-
ss)
D,~
=
(s3 -
sl)(s3
-
s~)(83
- s4)(~3
- ss)
D~,
=
(S4
-
Sl)(S4
-
s~)(s4
- 83)(s4
- ~s)
Ds5 =
X1
=
C3(0) C3(1) C3(2)C3(3) C3(4) - 3 A1 #3 C3(0)C3(1) C3(4) -
3 A, #3 C3(O)C3(2) C3(4) - 2 AI#4 C3(I) C3(2) C3(3) 3 AI #, C3(2) C3(3) C3(4) + 6 A~#1 #4 C3(2) 3 A, A, #1 #3 C3(4) + 9 A~#, #3 C3(4) #s As C3(1) C3(3) C3(4) + 2 A1 As #s ~4 C3(1)
--
9 A~#3 #s C3(4) + 3 A1 As #s #3 C3(4) 2:2
=
C3(I) C3(2) C3(3) C3(4) - 3 A1#3 C3(I) C3(4) - 3 Al #3 C3(2) C3(4) 2 A, #4 C3(I) C3(2) + 3 A, C3(2) C3(3) 6",3(4) - 6 A~#4 C3(2) +
Consecutive-k-out-of-n: G system
441
3 Al Ad~3C3(4) - 9 A~#363(4) + A, 63(i) C3(3) C3(4) 2 A, A, #4 6'3(I) + 9 A~#3 63(4) - 3 A, A, #3 63(4) + 9 A~C3(2) C3(4) + 3 A, As63(1) 63(4) + 18 A3 63(2) + 6
~ ~. c ~ o )
----- C3(I) C3(2) 63(3) 63(4) - 3 A, #3 63(I) 63(4) - 3 A, #3 C3(2) 63(4) -
23
2 At #4 6'3(1) 6'3(2) + C3(0) 6'3(1) 63(2) 6'3(3) - 3 A, #3 63(0) 63(1) + 6'3(0) C3(2)63(3)63(4) - 3 A, #3 C3(0) C3(4) + C3(0) C3(I)63(2) C3(4) + 63(0) 63(1) C3(3) 63(4) 2 A; #4 63(0) C3(1)
-
3 Al #3 63(0) C3(2)
--
2 A, #4 C3(0) 63(2)
-
--
3 A1#3 63(0) 63(4)
-
3 '~1 #1 C3(3) C3(4) +
6 )%2#1 #4 - 3~1 #1 63(2) 63(3) - 3 ~, #1 63(2)63(4) 3 At )%,#I #3 + 9 )%~#I #3
-
-
#s AsC3(3)63(4) +
2 )%1)%,,, m - #, ~ c3(1) 63(3) - #, ~ c3(1) c~(4) 9 A12#o #3 + 3 )%z)%,#3 #~ 24
=
63(1) 63(2) 63(3) -- 3 )%lm 63(1) + 63(2) 63(3) C3(4) -3 )%1#3 6'3(4) + 6'3(1)C3(2) 63(4) + C3(1) (73(3) (73(4) 3 )q #3 C3(2) -- 3 )q #3 63(4) - 2 )%1#4 63(i) 2 )%1#4 63(2) q- 3 )%163(3) 63(4)
--
3 A1 63(2) 63(4) + ~. 63(3) 63(4)
6 )%12#4 q- 3 )~1 63(2) 63(3) + --
2 A1 A. #4 -[- )% 63(1) 6'3(3) +
)%.63(1) C3(4) + 9 A1263(2) + 9 A2 63(4) + 3 AlAs 63(1) +
3)%1)%.c3(4) + is ~ + 6 ~12~o
Appendix
A2
B26 = A7(8 = 86)/D86 B~T = Az(8 = s~)lD87
B28 = AT(s = ss)/D~ 9
B2* =
AT(s = S9)/D,9
B3o = A s ( s = s6)/Ds6 B31
=
A 8 ( 8
:
s7)/D,,
B32
=
As(s
=
ss)/D~ a
B33 = As(8 = 89)/Ds.
934 ----A9(S = s 6 ) / D , . B3s = Ag(s = s T ) / D , , B36 = A9(s ----s s ) / D , 8 B37 = A9(s = s9)/D.9 B3s = A,o(s = s 6 ) / D , 6
B39 = AlO(S -'- sT)/Ds~ B4o = Alo(s = s s ) / D , 8
B4, =
A,o(s
=
s9)/D,9
where, D,. = (86 - 8~)(s6 - s8)(86 - 89) D,, = (sT - ~6)(~ - ~8)(s~ - ~9)
zs = C3(0)C3(1) 63(2) 63(3) - 3 A, #3 6'3(0) 63(1) - 3 AI #3 63(0) C3(2) 3 A, #i C3(2) 63(3) + 9 A~#1 ~3 - 3 A1As#I #3 #s As63(I) C3(3) + 3 A, Aa#3 #a - 9 A~#3#a x6 = 6'3(1) 63(2) C3(3) - 3 A1#3 C3(I) - 3 A,p3 63(2) + 3 A, 63(2) 63(3) - 9 A~ #3 + 3 A, A, #3 +
442
B.S. DH1LLON and O. C. ANUDE A. C3(1) C3(3) - 3A1 As#3 + 9 A~/~3 + 3 A1 As C3(1) + 9 A~ C3(2) C3(1) C3(2) C3(3) - 3 A1/~3 C3(1) - 3 A1 #3 C3(2) +
~7
C3(0) C 3 ( 2 ) C 3 ( 3 ) - 3 A, #3 C3(0) + C3(0)C3(1)C3(2) + C3(0) (73(1) C3(3) - 3 A1 #3C3(0) - 3 A] g, (73(2) -
3 A1/.tl C3(3) -/~s As C3(1) - ,as As C3(3) C3(2) C3(3) - 3 A,#3 + C3(1) C3(2) + C3(1)C3(3) -
x8
3 A1/~3 + 3 A, C3(2) + 3 A, C3(3) + A, C3(1) +
As C3(3) + 3A, As
Appendix
A3
B42 --- A13(s = slo)/Ds,o ~43 = A , 3 ( s = s,1)/Dsl, B44 = A , 3 ( s : s , 2 ) / D , , , B45 = A14(s -- 31o)/Dslo B4s = A14(s = s l l ) / D s , , B47 = A14(s = s12)/D.,2 B4s = A15(s = s , o ) / D , , o
B49 = A , s ( s = s l , ) / D s , ,
Bso = AIS(S = Sl2)/D,I= where, D,,o = (~o - ~,,)(~,o - ~,~) D~,~ = (s11 - ~xo)(~,, - ~2)
x9
=
C3(0) C3(1) C3(2) - 3 al ~i C3(~)
- -
As ~s C3(i)
z,o =
C3(1)c3(2) + 3A~ C3(2) + A, C3(1)
xll
C3(1)C3(2) + C3(0) C3(1) + C3(0) C3(2) -
=
3 A I # I - #~A~
x12 =
C3(I)+C3(2)+3A,+As
Vol. 34, No. 3, pp. 429-442, 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0026-2714/94 $6.00+.00
Pergamon
COMMON-CAUSE
FAILURE
ANALYSIS REPAIRABLE
B. S. DHILLON Department
of Mechanical
Engineering,
OF A k-OUT-OF-n UNITS
:G SYSTEM
WITH
and 0. C. ANUDE
University Canada
of Ottawa,
Ottawa,
Ontario
KIN
6N5,
(Received for publication 23 February 1993)
ABSTRACT A newly developed of-n
: G system
is presented.
generalized composed
Also, reported
time to failure
expression
for the mean time to failure
of repairable
units and subject
are some special
of a k - out -
to common-cause
case system
reliability
failures
and variance
of
formulas.
INTRODUCTION
The importance
of incorporating of complex
engineering
In fact, as equipment
redundancy
increases,
ures potentially example,
may become
in the reliability
uration,
Moody
contributors
Broadly
to system
speaking,
clude multiple emergency
assessment
This paper
standby.
study
failure
jet engines
in redundant
is concerned
This system
examples
active
is a more general
units
diagram start
mechanism
failure
version
functioning
of operable technique
mode.
The system
condition
for system
mean
k, needed
success.
429
failures
one warm reported
system Laplace
in
1. At time
with the warm
normally
time to failure,
end results.
with
system
The system
The standby
of parallel
for system
have been used to obtain
in-
is shown in Figure
standby
is considered
to be
(k 5 n). The standby
and/or
as well as due to non-common-cause
reliability,
fail
of a k - out - of -
its switch
can fail due to a common-cause
in terms of the number units,
analysis
components
instantaneously condition.
unit fails.
items failures
loss of power station
of the parallel
for this system
in perfect
when multiple
of valves and multiple
identical
as soon as one active
Expressions
config-
were the main
[4].
is activated
are developed
fail-
[I]. For
trip breaker
of common-cause
maintenance
as long as at least k units are operating
working
not chance
failures
due to bird ingestion,
operational
normal
in reliability
be over-looked.
unavailability
reactor
is said to occur
structures
of repairable
Ref. [S]. The state-transition
in its standby
failures,
common-cause
with the common-cause
composed
and its switching
failures simply
to system
of a nuclear
that
due to inadequate
corrosion
t = 0, all n active
cannot
common-cause
contributor
cause [3]. Typical
loss of aircraft
n : G system
common-cause systems
unavailability.
a common-cause
feedwater
due to stress
the major
and Follen [2] concluded
due to a single underlying
MR Y/I--D
and quantifying
and safety assessment
failure
can fail from the
failures.
variance
of time to failure
units n and the least number transforms
and the Markov
430
B.S. DHILLON and O. C. ANUDE
r=n-k+2 m=n-k+l
Fig. 1. State-space diagram of the system.
ASSUMPTIONS The following assumptions are associated with this system model:
i)
The system has n + 1 identical and repairable units (n active and one on standby).
ii)
The common-cause and hardware failures are statistically independent. The system is operable as long as at least k units are operating normally (k < n)
iii)
A common-cause failure can occur from any of the operable states of the system. All failure and repair rates are constant.
iv)
v) vi) vii) viii) ix)
x)
The switching mechanism for the standby is considered automatic and instantaneous. The standby may fail in its standby mode, in addition, to the failure of its switching mechanism. The system, once in a failed state, is non-repairable A repaired unit is as good as new All common-cause failures are completely coupled
NOTATION The following symbols are used in this article: t n
k
Ai A°
time Number of parallel system units Least number of active units needed for system success Laplace transform variable Constant failure rate of a unit Constant failure rate of switching mechanism and/or standby itself
Consecutive-k-out-of-n:G system
431
Constant critical common-cause failure rate
~e0
from system state O. he/
Constant common-cause failure rate from system
J
jth state of the system; for j = O, 1,2,3, ....,n - k + 4
up-state i; for i = 1,2,3, ....,n - k + 2 Constant repair rate of switching mechanism
#s
and/or standby itself Constant repair rate from system up-state i; f o r i = l , 3 , 4 , ...... n - k + 2 The probability that the system is in state j at time t;
#i
Pj(t)
f o r j = 0,1,2,3, ..... , n - k + 4 Common-cause failure Derivative of the Laplace transform of the reliability function, Rn(s), with respect to s Inverse Laplaxe Transform
C.C.F R'(s)
In Figure 1, the system state is denoted by a numeral and/or a letter at the bottom of each box, ANALYSIS The set of differential equations a~sociated with this system model shown in Figure 1 is:
dPo(t) d----'-~+(nAl + $, + A~o)Po(t) = pl P1(t) + p, P2(t) dPl(t)
d-""~ 31- (n/~l "[- ~cl + ]~1) Pl(t) = n~l Po(t) "~/23 e3(t)
de2(t)
d-----t--+ (n)h + At2 + #.) P2(t) = Ao Po(t) + #3 P3(t)
dP3(t) d~
(1)
(2) (3)
+ [(n - 1)A1 + Ac3 + 2p3] P3(t) = nA1 Pl(t) + hA1 Pz(t) + ~4 P4(t) (4)
dPr(t)
d----i--+ [k~ + ~ . + ~,~]P~(t) = (k + 1);~ P~_~(t)
(5) (6)
,~P~+l(t) _ k~l P~(t) dt r
dPr+2(t) - ~ Aej Pj(t) dt j=0
(7)
At time t = 0, P0(0) = 1 and all other initial state probabilities are equal to zero. The system reliability, Rk/n(t), is given by:
r
r
Rk/.(t) = ~ P~(t) = L - ~ [ ~ Pj(,)] j=O
(S)
j=O
The mean time to failure, MTTFk/., of the system is:
r
MTTFk/. = lim S'~ P3(s)
(9)
432
B.S. DHILLONand O. C. ANUDE The variance of the time to failure, a~ln, is expressed by the following relationship:
a~/,~ = - 2 lim R'k..~(s ) - ( M T T Fk/n) 2
(10)
Special Cases i) For k = 1, and n = 3 in Figure 1, the set of differential equations describing the system becomes:
dPo(t)
d---~- + (3~, + A, + Aco)Po(t) = #1 Pl(t) + #, P2(t)
dPx(t)
d----~+ (3A1 + Acl + .1) P,(t) = 3A, Po(t) + P3 P3(t)
(11) (12)
dP2(t) dt t-(3Al+Xc2+ps)P2(t)=XsPo(t)+p3P3(t)
(13)
dP3(!) =(2Al+A~3+21~3)P3(t)=3A1Pl(t)+3;hP2(t)+~qP4(t)
(14)
dP,(t) dt
(15)
dt
(A, + At4 + I~4)P4(t) = 2)h Pz(t)
-
dPs(t) d-----~- = ~1 P4(t)
dP6(*) dt
= ~ Acj Pj(t) j=o
(16) (17)
At t i m e r = 0 , P0(0) = 1 and all other initial state probabilities are equal to zero. Using Laplace transforms, Equations (11)-(17) yield:
Ao(~) P0(~) - Ao(s)
(18)
A,(~)
(19)
A2(s)
(20)
A3(s)
(21)
A4(s) P4(~) = :4D(~)
(22)
As(s) P s ( s ) - s "AD(~)
(23)
A6(s) P6(s)- s*AD(s)
(24)
P,(s) - AD(S) P2(~) - AD(S)
P3(s)
-
AD(S)
where,
A0(s) -- (s + C3(4))(s + C3(1))[(s + C3(2))(s + C3(3)) - 3 AI~3] 3 AI #3 (s + C3(2))(s + C3(4)) - 2 A, #4 (s + C3(1))(s + C3(2))
Al(s)
=
3A, (s + C3(2))[(s + C3(3))(s + C3(4)) - 2)q/x4] + 3 AI A~~/3(s ~- C3(4)) - 9 X~#3 (s + C3(4))
Consecutive-k-out-of-n:G system
A~(,)
433
As (s + C3(1))[(s + C3(3))(s + C3(4)) - 2A, P4] + 9A~p3(s + C3(4)) - 3Ai A,#3(s + C3(4))
A3(s) = 9 A~ (s + C3(2))(8 + C3(4)) + 3 A, A, (s + C3(1))(s + C3(4)) A4(s) =
18 ~13( , -}- C3(2)) Jr 6 )~]2"~s (8 + C3(1))
As(~) =
~, A4(s) 4
As(s)
=
j=O
Ao(s)
=
(s + C3(0)) Ao(s) - #, A I ( s ) - It, A2(s)
and, 6'3(0) = 3 A, + A,o + A, C3(I) = 3AI + Ae] +/q C3(2) = 3 A] + At2 + #, C3(3) = 2AI + Ae3 + 2#3 C3(4) = Az + At4 + #4 From Equation (8), the system reliability is: 4
R,/~(t) = ~PAt)
j=o =
e °at [B1 Jr Bs + Bll -k B16 + B21] +
e s==[B2 + Br + B12 + B l r + B22] + e szt
[B3 + 138 + BI3 "1-BlS + B23] +
e s ' t [134 e"t
+ B9 Jr B14 -}- B19 + B24] -r-
[Bs -{- BlO -k B,s + B2o + B25]
(25)
where symbols sl to ss are real and unique roots of the polynomial function A D ( S ) . The symbols B1 to B2s are defined in Appendix A1. A plot of (25) and corresponding tabular values for specified values of system parameters are shown in Figure 2 and Table 1 respectively.
1.00.90.8"
\
0.7"
\,
\ N
0.6-
k=2
~'~.
0
E
0.50.40.3-
~.
=
0.20.1-~
~ 0.0
0.0
~
-
3-- ~ 4 =O.O0003/~r i
1500.0
i
3000.0 Time in hours, t
i
4500.0
Fig. 2. System reliability plots for three different values of k.
i
6000.0
434
B. S, DHILLON and O. C. ANUDE T a b l e 1: S y s t e m values
Reliability
for three
different
o f k ( n = 3)
Time (in hours)
k = 1
k = 2
k = 3
0.00
1.0000
1.0000
1.0000
1500
0.9264
0.7800
0.4708
3000
0.7589
0.4456
0.1607
4500
0.5682
0.2310
0.0534
6000
0.4067
0.1164
0.0177
oo
0.0000
0.0000
0.0000
Using Equation (9), the system mean time to failure is: 4
MTTF1/3
= lim ~ Pi(s) Y~/3 Z1/3
(26)
where,
Y1/3
Ca(l) C3(4)[63(2) C3(3) - 3 h, P3] - 2 h, P4 C3(1) C3(2) 3 hi/z3 63(2) C3(4) + 3 hi C3(2) [63(3) C3(4) - 2 hi P4] + h, 63(1)[63(3)63(4) - 2 hi P4] + 9 h~ 63(2)63(4) + 3 h, h. 63(1) 63(4) + 18 h~ C3(2) + 6 hl h. C3(1)
Z1/3 = C3(0) [63(1) C3(4)(63(2) C3(3) - 3 hi #3) - 3 h, P3 C3(2) C3(4) 2 hi P4 Ca(l) C3(2)] - pl [3 A1 C3(2) (63(3) C3(4) - 2 hi p4) + 3 hi P3 63(4) (As - 3 hi)] - p, [h, 63(1) (63(3) C3(4) - 2 A1P4) -
3 hi ~,3 63(4)
(hs -
3 hi)]
Z1/3 = An(s = O) From Equation (10), the variance of the time to failure is:
a~/3 -- - 2 1 i m E 1 / 3 ( s ) - ( M T T F1/3) 2 $~0
=
"
1 2 ~12 [2(x2x3 - xlx4) - x2]
(27)
Symbols zl to x4 are defined in Appendix A1. ii) For k = 2 and n = 3 in Figure 1, the set of system differential equations becomes:
dPo(t) q- C3(0) Po(t) = #1 Pl(t) + #, P2(t) dt
(28)
dPl(t) 4_ 63(1 ) PI(Q -- 3hi P0(~) -{-/-L3P3(t) dt
(29)
dP2(t) dt [- 63(2)P2(t) = )% Po(t) + P3 P3(t)
(30)
Consccutive-k-out-of-n:G system
435
dP3(t) + C3(3)P3(t) = 3 )il P, (t) ~- 3 )il P2(t)
(31)
dP4(t) = 2A1 P3(t) dt
(32)
dPs(t) 3 dt - ~ )icj Pj(t)
(33)
dt
j=o
At time t = 0, Po(0) = 1 and all other initial state probabilities are equal to zero. Using Laplace transforms, from Equations (28)-(33), the following relationships result: AT(s)
(34)
As(s) P,(s) = A S ( s )
(35)
Ag(s)
(36)
P3(s) = AIo(s) AS(S)
(37)
Po(s)- As(s)
P2(s) = As(s)
P4(s) - s Ps(s) -
All(S) •
(38)
AS(s)
A,2(~)
(39)
s * AE(S)
where, AT(s)
=
(s + Ca(l)) [(s + C3(2))(s + C3(3)) - 3 )il ]-/31-- 3 )il ~-/3(8 "4"63(2))
As(s) = 3)ii [(s + 63(2))(s -I- 63(3)) - 3 )ii It3] + 3 )il )is Its A9(s) =
)is (s + C3(1))(s + C3(3)) - 3 )il ]/,3()is - 3 At)
Aio(s)
=
3AiAs(s+C3(1))+9A2(s+63(2))
All(s)
=
2A1Am(s)
A12(8)
=
)ic0 Av(s) "4-)ic, As(s) "4-)ic2 A g ( s ) "~- )ic3 A,o(s)
AE(S)
=
( s + C 3 ( O ) ) n T ( s ) - It, A s ( s ) - ItsA9(s)
The system reliability is given by: 3
R2/3(t) = ~ Pj(t) j=0 :
e s6t [B26 + B3o + B34 + B3a] + e sTt [B27 + B31 + B3s + B39] + e sst [B2s + B32 + B36 + B40] + e sgt [B29 + B33 + B37 + B41] (40)
where symbols s6 to s9 are real and unique roots of the polynomial function A s ( s ) . Symbols B~s to B41 are defined in Appendix A2. A plot of (40) and corresponding tabular values for specified values of system parameters are shown in Figure 2 and Table 1 respectively. The mean time to failure of this system is: 3
MTTF2/3
=
lira ~ ' P j ( s )
s ~ 0 j=O
Y213 Z2/3
(41)
436
B.S. DHILLON and O. C. ANUDE where,
Y2/3
C3(1) [(73(2) C3(3) - 3 A1 ,u3] - 3 A1 #3 C3(2) + 3 A1 [C3(2) 6'3(3) - 3 ~1,0,3] + 3 A1 As#3 + As C3(1) C3(3) - 3 A 1 ,a3 [As - 3 A,] + 3 A, A, C3(i) + 9 A]2 C3(2)
Z2/3
=
C3(0) [C3(1) (C3(2) C3(3) - 3 A1 ju3) - 3 ~1/z3 C3(2)] #1 [3 ,kl C3(2) C3(3) + 3 A1 #3 (A, - 3 A1)] #, [),, C3(1) C3(3) - 3 Ax #z(A, - 3 A1)]
Z2/3 = AE(S = O) Using Equation (10), the formular for variance of the time to failure is:
a~/3
=
-21imR~t3(s ) - (MTTF2/3) 2
=
1 x-~ [2(=6x7 - x5=8) - ~ ]
$40
/
(42)
Symbols z5 to xs are defined in Appendix A2. iii) For k = 3, and n = 3 in Figure 1, we obtain the following set of system differential equations:
dPo(t) + C3(0) Po(t) -= #1 Pl(t) + #, P2(t) dt
(43)
dPl(t) + C3(1 ) Pl(t) = 3A1 Po(t) dt
(44)
dP2(t) + C3(2)P~(t) = A, Po(t) dt
(45)
dP3(t) dt - 3A1 Pl(t) + 3A1 P2(t)
(46)
2
dP,(t) _
dt
Z ~J P~(t)
(47)
j=o
Equations (43)-(47) using Laplace transforms lead to: A13(s) AF(S)
(48)
AI,(S) P I ( S ) - AF(S)
(49)
P°(s)-
Als(s)
P 2 ( s ) - AF(S) Alo(s) P3(s) = s • AF(S ) A17(s) P4(s) = s • AF(s) where, A,3(s) = (s + C3(1))(s + C3(2)) Al~(s) = 3 ~1 (s + C3(2)) A15(s) : A,(s + c3(1))
(~o) (51)
(52)
437
Consecutive-k-out-ofon: G system
A16(s) = 3 .~, (A14(s) + Alh(S))
&ff s) = ,Lo &3(s) + ;~1 A14(s) + .~2 &s AF(s) = (s + C3(0)) A13(s) - #1 A14(s) - #~ A,5(s) The system reliability, R3/3(t ) is given by: 2
R3/3(t) =
EPj(t) j=0
eS~°t[B42 + B4s + B4s] + e~t[B43 + B46 + B49] +
=
eS'2t[B44 + B47 + Bso]
(53)
Where symbols sl0 to s12 are real and unique roots of the polynomial function Av(s). Symbols B42 to B5o are defined in Appendix A3. A plot of (53) and corresponding tabular values for specified values of system parameters are shown in Figure 2 and Table 1 respectively. The system mean time to failure is given by: 2
MTTF3/3
=
lim~-'Pj(8) s---*0 j=9
Y3/z Z3/3
(54)
where,
Y3/3 = C3(1)63(2) + 3 .~1 C3(2) + .~ 63(1) Z3/3 ~- 6 3 ( 0 ) [ 6 3 ( 1 )
6 3 ( 2 ) ] - / 1 1 [3 ,'~1 C 3 ( 2 ) ] - ]Is [.~s C 3 ( 1 ) ]
Z3/3 = AF(S : O) The variance of time to failure is:
0"32/3 :
--2 lira R~/3(s ) -- (MTTF3/3) 2 S~0
= ~ [2(xm~,l - ~ 9 ~ 1 - x~0]
(~5)
Symbols x9 to x12 are defined in Appendix A3. Using Equations (26), (41) and (54) as well as analyses performed on several more special cases, the following generalized system mean time to failure expression is obtained:
M T T F k / . = Yk/~ Zk/.
(56)
Where Yk/n and Zk/~ can easily be obtained via the following recursive relationships3:
For k = n, n - 1, n - 2,. ....... ,1 we have the following set of recursive equations which can easily be solved:
Yn/. = 6.(1) c.(2) + . ~, 6.(2) + ~. G(1) Yn-1/. = Cn(3)Y./. - n.~l #3 Cn(1) - n.~1#3 C.(2) + O . 3For the specific case of k = 1, a non-recursive expression for the system mean time to failure is presented in Ref. [5].
B.S. DHILLON and O. C. ANUDE
438
Y,~-2/. = C . ( 4) Yn-~/n - (n - 1) )h b4 Y./n + (n - 1)~10. Y.-3/,~ = C . ( 5 ) Y ~ - 2 / .
- (n - 2 ) A l # s Y n _ l / n
+ (n - 1)(n- 2)A~ 0,~
Y a / . = C . ( n + 1) Y 2 / . - 2 A~ #.+~ Y31. + ( n - 1)! A~-~ O.
where On = (n A1)2 Cn(2) + hA1 A.Cn(1) and,
= ~./,,
z,,/,,
-
~1 [,¢,/.]
Z . _ ~ l . = a._~l,,
- #1
-
u, [¢./,,]
[~,,-~/,.,] - ~,, [¢,.-1/-]
Z,,_ w , = a . _ ~ l . - #1 [ / ~ . - 2 / . ]
z,/,,
= ~,/,,
- ~,, [,o,/,,] -
- m [¢.-~/.]
~, [¢,/,,]
Where, a./,, = C,,(O) Cn(1) Cn(2) a,~_a/n = Cn(3)an/,~ - n A , # 3 C n ( O ) C n ( 1 ) -
nAll~3Cn(O)C.(2)
a,,-21n = C,,( 4 ) a n - l l n - (n - 1) Al P,4 anl,~ a . - 3 / ~ = C . ( 5 ) e . - 2 / . - (n - 2) ~ #5 ~ . - 1 / .
aal,, = C,,(n + 1)as/,, - 2 Aa/~.+a a 3 / . / 3 ~ / . = n ~1 C . ( 2 )
/3._,/~ = C . ( 3 ) / 3 . / . + n A , ~ 3 ( ~ . - n ~ , )
,~n-2/n = C,.,(4) t3n_11,~ - (n - 1) Alp,4 ,On/. ~.-3/,,
= c.(5)/~,,-2/,,
- (- - 2) ,h~,6,,-~/,,
¢,,./,-, = As Cn(1) ¢,,_,/.
= c,,(3)
¢,,/,, - ..xl
m (~. -
,-, .~)
¢ n - 2 / n = Cn(4) ¢,~-1/,~ - (n - 1) Aa #4 Cn/n ¢.-3/.
= C . ( 5 ) ¢ . - 5 / . - (n - 2) A~ ~ ¢ . - 1 / .
¢ ~ / . = C,,(n + 1)¢2/~ - 2 A1#~+1 ¢ 3 / .
Cn(1) = n ~1 -~ ~cl -F f$1 C . ( 2 ) = n A, + At2 + / ~ .
Cn(3) = (n - 1) Aa + At3 -t- 2#3 c . ( 4 ) = (n - 2) ~1 + ~o, + u,
C . ( n + 1) = A, + Acn+, + #n+,
For specified values of system parameters, Equation (56) was used to generate the values given in Table 2 and the plots shown in Figure 3.
Consecutive-k-out-of-n:G system T a b l e 2: S y s t e m M T T F
439
v a l u e s (in h o u r s ) f o r t h r e e
d i f f e r e n t v a l u e s o f k (n = 3)
Critical Common-Cause Failure Rate, Ac0 (per Hr.)
k= 1
k= 2
k= 3
0.00000
6499.34
3393.38
1841.04
0.00001
6422.42
3355.43
1822.31
0.00002
6347.30
3318.32
1803.95
0.00003
6273.93
3282.02
1785.96
0.00004
6202.22
3246.51
1768.32
0.00005
6132.14
3211.75
1751.03
7000.0-
k-1
/ 6000.0-
5000.0O c-
.c 4000.0-
/k=2 2 E
5000.0-
a)
/
2000.0-
k=3
n----3 1000.0 -
)`1 --- O.O004-/Hr
)`s = O.O001/Hr
)`Cl -- Xc2 -- )`c3 -- )`c4- = O.O0003/Hr /1,s --= O.O006/Hr 0.0
o.o
/z4_ = O.O003/Hr
//'3 = O.O0015/Hr /1,1 ---- O.O004-/Hr
1'.o
~.o
~.o
,;.o
Crit.ical c o m m o n - c a u s e failure rot,e,
i.o .10 -5
}'co
Fig. 3. M q T F ~ plots for three different values of k.
REFERENCES
1. Apostolakis, G., "On the Reliability of Redundant Systems", Transactions of the American Nuclear Society on Safety Systems Reliability Methods and Applications, Vol. 22, 1975, pp. 477-478. 2. Moody, J, It., Follen, S, M., "Common-Cause Modeling of Reactor Trip Breaker Configuration", Proc. of the International Topical Meeting on Probabilistic Methods and Applications, EPRI NP-3912-SR, Vol. 3, Sessions 17-23, Palo Alto, CA., 1985, Feb., pp. 180/1-180/10.
440
B.S. DHILLON and O. C. ANUDE 3. Gangloff, W, C., "Common-Mode Failure Analysis", I E E E Transactions on Power Apparatus and Systems, Vol. 94, 1975, Jan-Feb., pp. 27-30.
4. Watson, I, A., "Common Mode/Cause Failures - An Overall Review", in Mechanical Reliability, Science and Technology Press Ltd., London, U.K., 1980, pp. 136-162. 5. Dhillon, B, S., Anude, 0, C., "Common-Cause Failure Analysis of a Redundant System with Repairable Units", International Journal of Systems Scienoe (to appear).
Appendix
A1
B1 = Ao(s = s , ) / D ~ I B2 = Ao(s = s2)/Ds2 B3 = Ao(s = s3)/Ds3 B4 = Ao(s = s 4 ) / D s , B5 = A d s = s s ) / D , ~ Bs = Al(S = s l ) / D ~ B7 = A , ( s = s 2 ) / D , 2 B8 = A I ( s = s3)/Dsa B9 = A i ( s = s 4 ) / D , ~ B,o = A , ( s = ss)/D~ 5 B n = A2(s = s , ) / D ~ , B12 = A : ( s = s2)/Ds2 B,3 = A2(s = s3)/D~ 3
B14
=
A2(s = s 4 ) / D s ,
B15 = A2(s = ss)/Ds5 B , 6 = A3(s = s l ) / D ~ ,
BI7 = A3(s = s 2 ) / D s , B , s = n 3 ( s = s3)/Dss
B19 = A3(s = s4)/Ds4 B2o = A3(s = ss)/D~ 5 B21 = A4(s = s l ) / D ~ 1 B22 = A4(s = s 2 ) / D ~
B23 = A4(s = s3)/D~s B24 -- A4(s = s 4 ) / D s , /3*25 = A4(s = s s ) / D , 5 where, Ds, = (s, D, 2 = (s2
-
s~)(sl
-
s3)(s,
- s4)(sl
-
ss)
-
s,)(s2
-
s3)(s~
- s4)(s~
-
ss)
D,~
=
(s3 -
sl)(s3
-
s~)(83
- s4)(~3
- ss)
D~,
=
(S4
-
Sl)(S4
-
s~)(s4
- 83)(s4
- ~s)
Ds5 =
X1
=
C3(0) C3(1) C3(2)C3(3) C3(4) - 3 A1 #3 C3(0)C3(1) C3(4) -
3 A, #3 C3(O)C3(2) C3(4) - 2 AI#4 C3(I) C3(2) C3(3) 3 AI #, C3(2) C3(3) C3(4) + 6 A~#1 #4 C3(2) 3 A, A, #1 #3 C3(4) + 9 A~#, #3 C3(4) #s As C3(1) C3(3) C3(4) + 2 A1 As #s ~4 C3(1)
--
9 A~#3 #s C3(4) + 3 A1 As #s #3 C3(4) 2:2
=
C3(I) C3(2) C3(3) C3(4) - 3 A1#3 C3(I) C3(4) - 3 Al #3 C3(2) C3(4) 2 A, #4 C3(I) C3(2) + 3 A, C3(2) C3(3) 6",3(4) - 6 A~#4 C3(2) +
Consecutive-k-out-of-n: G system
441
3 Al Ad~3C3(4) - 9 A~#363(4) + A, 63(i) C3(3) C3(4) 2 A, A, #4 6'3(I) + 9 A~#3 63(4) - 3 A, A, #3 63(4) + 9 A~C3(2) C3(4) + 3 A, As63(1) 63(4) + 18 A3 63(2) + 6
~ ~. c ~ o )
----- C3(I) C3(2) 63(3) 63(4) - 3 A, #3 63(I) 63(4) - 3 A, #3 C3(2) 63(4) -
23
2 At #4 6'3(1) 6'3(2) + C3(0) 6'3(1) 63(2) 6'3(3) - 3 A, #3 63(0) 63(1) + 6'3(0) C3(2)63(3)63(4) - 3 A, #3 C3(0) C3(4) + C3(0) C3(I)63(2) C3(4) + 63(0) 63(1) C3(3) 63(4) 2 A; #4 63(0) C3(1)
-
3 Al #3 63(0) C3(2)
--
2 A, #4 C3(0) 63(2)
-
--
3 A1#3 63(0) 63(4)
-
3 '~1 #1 C3(3) C3(4) +
6 )%2#1 #4 - 3~1 #1 63(2) 63(3) - 3 ~, #1 63(2)63(4) 3 At )%,#I #3 + 9 )%~#I #3
-
-
#s AsC3(3)63(4) +
2 )%1)%,,, m - #, ~ c3(1) 63(3) - #, ~ c3(1) c~(4) 9 A12#o #3 + 3 )%z)%,#3 #~ 24
=
63(1) 63(2) 63(3) -- 3 )%lm 63(1) + 63(2) 63(3) C3(4) -3 )%1#3 6'3(4) + 6'3(1)C3(2) 63(4) + C3(1) (73(3) (73(4) 3 )q #3 C3(2) -- 3 )q #3 63(4) - 2 )%1#4 63(i) 2 )%1#4 63(2) q- 3 )%163(3) 63(4)
--
3 A1 63(2) 63(4) + ~. 63(3) 63(4)
6 )%12#4 q- 3 )~1 63(2) 63(3) + --
2 A1 A. #4 -[- )% 63(1) 6'3(3) +
)%.63(1) C3(4) + 9 A1263(2) + 9 A2 63(4) + 3 AlAs 63(1) +
3)%1)%.c3(4) + is ~ + 6 ~12~o
Appendix
A2
B26 = A7(8 = 86)/D86 B~T = Az(8 = s~)lD87
B28 = AT(s = ss)/D~ 9
B2* =
AT(s = S9)/D,9
B3o = A s ( s = s6)/Ds6 B31
=
A 8 ( 8
:
s7)/D,,
B32
=
As(s
=
ss)/D~ a
B33 = As(8 = 89)/Ds.
934 ----A9(S = s 6 ) / D , . B3s = Ag(s = s T ) / D , , B36 = A9(s ----s s ) / D , 8 B37 = A9(s = s9)/D.9 B3s = A,o(s = s 6 ) / D , 6
B39 = AlO(S -'- sT)/Ds~ B4o = Alo(s = s s ) / D , 8
B4, =
A,o(s
=
s9)/D,9
where, D,. = (86 - 8~)(s6 - s8)(86 - 89) D,, = (sT - ~6)(~ - ~8)(s~ - ~9)
zs = C3(0)C3(1) 63(2) 63(3) - 3 A, #3 6'3(0) 63(1) - 3 AI #3 63(0) C3(2) 3 A, #i C3(2) 63(3) + 9 A~#1 ~3 - 3 A1As#I #3 #s As63(I) C3(3) + 3 A, Aa#3 #a - 9 A~#3#a x6 = 6'3(1) 63(2) C3(3) - 3 A1#3 C3(I) - 3 A,p3 63(2) + 3 A, 63(2) 63(3) - 9 A~ #3 + 3 A, A, #3 +
442
B.S. DH1LLON and O. C. ANUDE A. C3(1) C3(3) - 3A1 As#3 + 9 A~/~3 + 3 A1 As C3(1) + 9 A~ C3(2) C3(1) C3(2) C3(3) - 3 A1/~3 C3(1) - 3 A1 #3 C3(2) +
~7
C3(0) C 3 ( 2 ) C 3 ( 3 ) - 3 A, #3 C3(0) + C3(0)C3(1)C3(2) + C3(0) (73(1) C3(3) - 3 A1 #3C3(0) - 3 A] g, (73(2) -
3 A1/.tl C3(3) -/~s As C3(1) - ,as As C3(3) C3(2) C3(3) - 3 A,#3 + C3(1) C3(2) + C3(1)C3(3) -
x8
3 A1/~3 + 3 A, C3(2) + 3 A, C3(3) + A, C3(1) +
As C3(3) + 3A, As
Appendix
A3
B42 --- A13(s = slo)/Ds,o ~43 = A , 3 ( s = s,1)/Dsl, B44 = A , 3 ( s : s , 2 ) / D , , , B45 = A14(s -- 31o)/Dslo B4s = A14(s = s l l ) / D s , , B47 = A14(s = s12)/D.,2 B4s = A15(s = s , o ) / D , , o
B49 = A , s ( s = s l , ) / D s , ,
Bso = AIS(S = Sl2)/D,I= where, D,,o = (~o - ~,,)(~,o - ~,~) D~,~ = (s11 - ~xo)(~,, - ~2)
x9
=
C3(0) C3(1) C3(2) - 3 al ~i C3(~)
- -
As ~s C3(i)
z,o =
C3(1)c3(2) + 3A~ C3(2) + A, C3(1)
xll
C3(1)C3(2) + C3(0) C3(1) + C3(0) C3(2) -
=
3 A I # I - #~A~
x12 =
C3(I)+C3(2)+3A,+As