- Email: [email protected]
Bounds on system reliability when components are dependent
Microelectron. Reliab., Vol. 26, No. 3, pp. 455-459,1986. Printed in Great Britain.
0026-2714/8653.00 + .00 Pergamon Journals Ltd.
BOUNDS ON SYSTEM RELIABILITY WHEN COblPONENTS ARE DEPENDENT C.D. LAI Department of Mathematics and Statistics, Massey University, New Zealand
(Received for publication 14 November 1985)
Abstract We present bounds on the reliability of a system which consists of several independent modules arranged in series and each module is a parallel redundant sub-system of dependent components. bounds are calculated.
An example is given and
These are compared with other bounds that are
obtained from an existing method.
i.
Introduction
In most reliability analyses, components are assumed to have independent life distributions.
However, as pointed out by many authors, in many
practical reliability situations,
it is more realistic to assume some form
of positive dependence among components.
This positive dependence among
component life lengths arises from common environmental stresses and shocks, from components depending on common sources of power, and so on.
An important task to a reliability analyst is to determine the system reliability.
However, it is well known that the exact computation of a
system reliability is usually a formidable task for complex systems. this reason one often prefers to find bounds on system reliability.
For If
components are statistically independent or statistically 'associated' (for a definition concerning association of random variables, see p29-30 of Barow and Proschan (1981)),
then upper and lower bounds are available
(see Barlow and Proschan (1981), p32-35, 114 & 123).
However, i n many practical applications,
it is difficult to establish i f
t h e components are associated even if we know that they are positively
correlated (positive correlation does not imply 'association'). Therefore those bounds presented in Barlow & Proschan (1981) may not be appropriate in such cases.
In this paper we consider a system which consists of one or several statistically independent modules (subsystems) such that they are arranged in series. parallel.
Within each module, the dependant components are arranged in We shall use an improved Bonferroni inequality ( Worsley
(1982)) to obtain upper and lower bounds on system reliabilities.
455
456
C.D. Lm
An i m p r o v e d Bonferroni inequality
2.
Many authors have used the Bonferroni
inequality to provide upper and
lower bounds for the probability of the union of a sequence of events AI,.,.,An: n
n
n
i=IEn-(A i ) - ~I
Ai
is the event
that
Xi ) c(i=1,2 . . . . . n)
a random v a r i a b l e
are interested in the distribution of X = max(Xi), and lower bounds for P r ( X )
c).
( 1) and we
then (I) gives upper
Worsley (1982) provides a few reference~
on applications of this inequality.
As was pointed out in Worsley (1982), the upper bound (I) is more important because it provides a conservative test, yet it is not so accurate as the lower bound.
He improved the upper bound in the
following way.
Represent vertices
events
vI
and
A1 . . . . ,A n vfiare
are not mutually
as vertices
joined
exclusive.
vl,...,vnof e i j i f and
by an e d g e Let
a graph Gwhere only if
T be a s u b g r a p h o f ~
Ai and Aj W o r s l e y showed
that n
PL.~=IAi) ~ Pr(Ai) E "= {i,j:eifT
}
Pr(A i N Aj)
(2)
if T is a tree.
In particular,
we h a v e n
n
n
Prj~__lAi) ~ E Pr(Ai) -1E1Pr(Ai fl Ai+I)" i=1 "= If
the sequence
A1,...,A
n c a n be r e l a b e l l e d
Pr(Aj. N A)) < Pr(A)¢ II A~+1)
(3)
such that
whenever J ~ k < j,
t h e n t h e u p p e r bound (3)
c a n be i m p r o v e d .
Consider now a reliability system which consists of several independent modules arranged in series such that each module is a parallel (hot) redtuJdant subsystem of dependent but similar (or identical) components. Such a system is often used as redundancy at the component level is more effective than redundancy at the system level.
This principle is
well-known among design engineers.
S i n c e t h e modules a r e assumed t o be i n d e p e n d e n t , consider
Let
only reliability
Tj(i=l, . . . , n )
be the lifetime
subsystem which has a lifetime T. n Pr(T>t) = --Pr~Jl(Ti i-) t)).
it
is sufficient
to
b o u n d s on e a c h m o d u l e .
of the
i th component in a hot r e d u n d a n t
Then clearly we have
It followed from (i) and (3) that
System reliability n
457
n
n
E Pr(Ti>t) - r r p r ( T i > t , rj>t)(pri(U__l(Ti>t) ) 41ElPr(Ti>t) i=l i j ": ....
(4)
n
- E P r ( T l > t , Ti+1>t) i=I
For a given mission time t, the upper bound can be improved further if we can relabel the components such that
Pr(ti>t, Tj>t)t,Tk+l>t)
whenever i ( k < j.
3.
An example and some comments
Consider a parallel redundant system with k exponential components jointly distributed according to the model proposed by Marshall and Olkin (1967) such that
F ( t l , t2, t3) = P r ( T l > t l , ~ > t 2, T3>t3) = e x p [ - X l t I - Xzt 2 - )'12 m a x ( t l , t2) - X I 3 m a x ( t l , t3) - X23 m~v(t2, t3) - X123 m~Y(t I , t2, t 3) . . . . . . Let
= 3"12 + x 1 3 + X23 + ~123, then f o r
(5)
t I = t 2 = t 3 = t,
we h a v e
(6a)
Pr(Tl>t, Tz>t ) = exp[-(~1+~ Z+~ )t] Pr(Tz>t, t3>t) = exp{-C~ Z ~ 3+~ )t]
(6b)
Pr(Tl>t, TR>t) = exp{-(~ l+~ 3+~ )t]
(6c)
It is easy to verify that corr(Ti, Tj)
=
i f ~123
)~+~ .+)~ i J and T3 are 'associated' (p129, 141 of Barlow and Proschan (1981)), .
As TI, ~
we may also use the upper and low bounds for associated components (see p38, 123 of Barlow and Proschan (1981)): 3
3
Pr(Ti>t) 4 Pr(T>t) 4 I - n Pr(Ti4t) i=l i=l We shall now compare the bounds given by (4) and (7).
(7)
To simplify
calculations, we let ~I = X Z = ~ 3 = l, X12 = X23 = XI3 = O, and then compute (4) and (7) for different values of ~ and t.
The results are now
tabulated below: = 0.i
t
Improved upper bound
O.l 0.5 1.O 1.5 2.0 2.5 3.0 3.5 4.0 5.0 6.0
1.0663 1.0310 0.7537 0.4904 0.3024 0.1813 0.1070 0.0626 0.0364 0.0122 0.0041
(Table i)
B o n f e r r o n i lower bound 0.2557 0.6810 0.6312 0.4476 0.2874 0.1760 O.1051 0.0619 0.0362 0.0122 0.0041
Upper bound o f (7) 0.9989 0.9243 0.7031 0.4724 0.2969 0.1798 0.1066 0.0625 0.0364 0.0122 0.0041
Lower bound o f (7) 0.7189 0.1920 0.0369 0.0071 0.0014 0.0003 0.0001 0.0000 0.0000 0.0000 0.0000
C.D. LAI
458
= 1
t 0.01 0.05 0.1 0.8 1.O 1.5 2.0 2.5 3.0 4.0
Improved upper bound
(Table
Bonferroni Lower b o u n d
0.9997 0.9935 0.9746 0.6574 0.3064 0.1271 0.0500 0.0191 0.0072 0.0010
0.01 0.05 0.1 0.5 1.0 1.5 2.0 2.5 3.0
Upper bound o f (7)
0.0293 0.11324 0.2337 0.4342 0.3064 0.1160 0.0475 0.0186 0.0071 0.0010
Improved Upper bound
Upper bound of (7)
0.0290 0.1259 0.2115 0.2634 0.0944 0.0259 0.0064 0.0015 0.0004
X = 4
t
Improved upper bound
0.01 0.05 0.1 0.5 1.O 1.5 2.0
0.9702 0.8548 0.7220 0.1467 0.0153 0.0014 0.0001
0.9418 0.7408 0.5488 0.0498 0.0025 0.0001 0.0000 0.0000 0.0000 0.0000
( T a b l e 3)
Bonferroni lower bound
0.9897 0.9447 0.8818 0.3987 0.1127 0.0283 0.0068 0.0016 0.0003
Lower b o u n d of (7)
0.9999 0.9991 0.9940 0.7474 0.3535 0.1420 0.0539 0.0201 0.0074 0.0010
= 2
t
2)
Lower bound o f (7)
0.9999 0.9973 0.9826 0.5311 0.1420 0.0330 0.0074 0.0017 0.0003
0.9[39 0.6376 0.4066 0.0111 0.0001 0.0000 0.0000 0.0000 0.0000
( T a b l e 4)
Bonferroni Lower b o u n d
Upper bound of (7)
0.0284 0.1139 0.1732 0.0969 0.0128 0.0013 0.0001
Lower b o u n d of (7)
0.9999 0.9892 0.9391 0.2266 0.0200 0.0016 0.0001
0.8607 0.4724 0.2231 0.0006 0.0000 0.0000 0.0000
Comparisons of lower bounds: A quick look over the tables reveals that the lower bounds obtained via (7) is not very accurate except for small mission time t.
This is not
surprising as the lower bound (7) is more appropriate for a series system of independent components. seem to do well when
On the other hand,
t ~ 0.5.
the Bonferroni
(Note that for t ) O . ~
bound is close to the improved upper bound).
lower bounds
the Bonferroni
lower
It appears from this example
that the maximum value of the two bounds for a fixed t would be a good lower bound on the system reliability.
Comparison of two upper bounds From the four tables we see the improved upper bound is more accurate than the upper bound of (7) except for X = 0. I.
For ~ = 0. I, we have
System reliability correlation coefficient p = O. 04Y6which
459
implies that the three components
are nearly independent.
By expanding the upper bound of (7), we note that n it is a truncated inclusion-exclusion formula for Pr~_l(Ti>t)) when T i are independent.
This explains why the upper bound of (7) is doing marginally better than the improved bound in Table I.
We also observe that for small t, a reasonable estimate of the system reliability can be obtained by averaging the improved upper bound and the lower bound of (7).
For t ) 6 5 , we would take the average of the
improved upper bound and the Bonferroni lower hound as our estimate.
References Barlow, R E and P r o s c h a n F (1981). and L i f e T e s t i n g .
Marshall, A W and Olkin, I (1967). Dikstribution.
Worsley, K J (1982).
Statistical
Silver Springs:
Theory o f R e l i a b i l i t y
To Begin With.
A Multivariate Exponential
J.Amer.Statlst,Assoc.
62, 30-44.
An improved Bonferroni inequality and Applications.
Biometrik8 69, 297-302.
0026-2714/8653.00 + .00 Pergamon Journals Ltd.
BOUNDS ON SYSTEM RELIABILITY WHEN COblPONENTS ARE DEPENDENT C.D. LAI Department of Mathematics and Statistics, Massey University, New Zealand
(Received for publication 14 November 1985)
Abstract We present bounds on the reliability of a system which consists of several independent modules arranged in series and each module is a parallel redundant sub-system of dependent components. bounds are calculated.
An example is given and
These are compared with other bounds that are
obtained from an existing method.
i.
Introduction
In most reliability analyses, components are assumed to have independent life distributions.
However, as pointed out by many authors, in many
practical reliability situations,
it is more realistic to assume some form
of positive dependence among components.
This positive dependence among
component life lengths arises from common environmental stresses and shocks, from components depending on common sources of power, and so on.
An important task to a reliability analyst is to determine the system reliability.
However, it is well known that the exact computation of a
system reliability is usually a formidable task for complex systems. this reason one often prefers to find bounds on system reliability.
For If
components are statistically independent or statistically 'associated' (for a definition concerning association of random variables, see p29-30 of Barow and Proschan (1981)),
then upper and lower bounds are available
(see Barlow and Proschan (1981), p32-35, 114 & 123).
However, i n many practical applications,
it is difficult to establish i f
t h e components are associated even if we know that they are positively
correlated (positive correlation does not imply 'association'). Therefore those bounds presented in Barlow & Proschan (1981) may not be appropriate in such cases.
In this paper we consider a system which consists of one or several statistically independent modules (subsystems) such that they are arranged in series. parallel.
Within each module, the dependant components are arranged in We shall use an improved Bonferroni inequality ( Worsley
(1982)) to obtain upper and lower bounds on system reliabilities.
455
456
C.D. Lm
An i m p r o v e d Bonferroni inequality
2.
Many authors have used the Bonferroni
inequality to provide upper and
lower bounds for the probability of the union of a sequence of events AI,.,.,An: n
n
n
i=IEn-(A i ) - ~I
Ai
is the event
that
Xi ) c(i=1,2 . . . . . n)
a random v a r i a b l e
are interested in the distribution of X = max(Xi), and lower bounds for P r ( X )
c).
( 1) and we
then (I) gives upper
Worsley (1982) provides a few reference~
on applications of this inequality.
As was pointed out in Worsley (1982), the upper bound (I) is more important because it provides a conservative test, yet it is not so accurate as the lower bound.
He improved the upper bound in the
following way.
Represent vertices
events
vI
and
A1 . . . . ,A n vfiare
are not mutually
as vertices
joined
exclusive.
vl,...,vnof e i j i f and
by an e d g e Let
a graph Gwhere only if
T be a s u b g r a p h o f ~
Ai and Aj W o r s l e y showed
that n
PL.~=IAi) ~ Pr(Ai) E "= {i,j:eifT
}
Pr(A i N Aj)
(2)
if T is a tree.
In particular,
we h a v e n
n
n
Prj~__lAi) ~ E Pr(Ai) -1E1Pr(Ai fl Ai+I)" i=1 "= If
the sequence
A1,...,A
n c a n be r e l a b e l l e d
Pr(Aj. N A)) < Pr(A)¢ II A~+1)
(3)
such that
whenever J ~ k < j,
t h e n t h e u p p e r bound (3)
c a n be i m p r o v e d .
Consider now a reliability system which consists of several independent modules arranged in series such that each module is a parallel (hot) redtuJdant subsystem of dependent but similar (or identical) components. Such a system is often used as redundancy at the component level is more effective than redundancy at the system level.
This principle is
well-known among design engineers.
S i n c e t h e modules a r e assumed t o be i n d e p e n d e n t , consider
Let
only reliability
Tj(i=l, . . . , n )
be the lifetime
subsystem which has a lifetime T. n Pr(T>t) = --Pr~Jl(Ti i-) t)).
it
is sufficient
to
b o u n d s on e a c h m o d u l e .
of the
i th component in a hot r e d u n d a n t
Then clearly we have
It followed from (i) and (3) that
System reliability n
457
n
n
E Pr(Ti>t) - r r p r ( T i > t , rj>t)(pri(U__l(Ti>t) ) 41ElPr(Ti>t) i=l i j ": ....
(4)
n
- E P r ( T l > t , Ti+1>t) i=I
For a given mission time t, the upper bound can be improved further if we can relabel the components such that
Pr(ti>t, Tj>t)
whenever i ( k < j.
3.
An example and some comments
Consider a parallel redundant system with k exponential components jointly distributed according to the model proposed by Marshall and Olkin (1967) such that
F ( t l , t2, t3) = P r ( T l > t l , ~ > t 2, T3>t3) = e x p [ - X l t I - Xzt 2 - )'12 m a x ( t l , t2) - X I 3 m a x ( t l , t3) - X23 m~v(t2, t3) - X123 m~Y(t I , t2, t 3) . . . . . . Let
= 3"12 + x 1 3 + X23 + ~123, then f o r
(5)
t I = t 2 = t 3 = t,
we h a v e
(6a)
Pr(Tl>t, Tz>t ) = exp[-(~1+~ Z+~ )t] Pr(Tz>t, t3>t) = exp{-C~ Z ~ 3+~ )t]
(6b)
Pr(Tl>t, TR>t) = exp{-(~ l+~ 3+~ )t]
(6c)
It is easy to verify that corr(Ti, Tj)
=
i f ~123
)~+~ .+)~ i J and T3 are 'associated' (p129, 141 of Barlow and Proschan (1981)), .
As TI, ~
we may also use the upper and low bounds for associated components (see p38, 123 of Barlow and Proschan (1981)): 3
3
Pr(Ti>t) 4 Pr(T>t) 4 I - n Pr(Ti4t) i=l i=l We shall now compare the bounds given by (4) and (7).
(7)
To simplify
calculations, we let ~I = X Z = ~ 3 = l, X12 = X23 = XI3 = O, and then compute (4) and (7) for different values of ~ and t.
The results are now
tabulated below: = 0.i
t
Improved upper bound
O.l 0.5 1.O 1.5 2.0 2.5 3.0 3.5 4.0 5.0 6.0
1.0663 1.0310 0.7537 0.4904 0.3024 0.1813 0.1070 0.0626 0.0364 0.0122 0.0041
(Table i)
B o n f e r r o n i lower bound 0.2557 0.6810 0.6312 0.4476 0.2874 0.1760 O.1051 0.0619 0.0362 0.0122 0.0041
Upper bound o f (7) 0.9989 0.9243 0.7031 0.4724 0.2969 0.1798 0.1066 0.0625 0.0364 0.0122 0.0041
Lower bound o f (7) 0.7189 0.1920 0.0369 0.0071 0.0014 0.0003 0.0001 0.0000 0.0000 0.0000 0.0000
C.D. LAI
458
= 1
t 0.01 0.05 0.1 0.8 1.O 1.5 2.0 2.5 3.0 4.0
Improved upper bound
(Table
Bonferroni Lower b o u n d
0.9997 0.9935 0.9746 0.6574 0.3064 0.1271 0.0500 0.0191 0.0072 0.0010
0.01 0.05 0.1 0.5 1.0 1.5 2.0 2.5 3.0
Upper bound o f (7)
0.0293 0.11324 0.2337 0.4342 0.3064 0.1160 0.0475 0.0186 0.0071 0.0010
Improved Upper bound
Upper bound of (7)
0.0290 0.1259 0.2115 0.2634 0.0944 0.0259 0.0064 0.0015 0.0004
X = 4
t
Improved upper bound
0.01 0.05 0.1 0.5 1.O 1.5 2.0
0.9702 0.8548 0.7220 0.1467 0.0153 0.0014 0.0001
0.9418 0.7408 0.5488 0.0498 0.0025 0.0001 0.0000 0.0000 0.0000 0.0000
( T a b l e 3)
Bonferroni lower bound
0.9897 0.9447 0.8818 0.3987 0.1127 0.0283 0.0068 0.0016 0.0003
Lower b o u n d of (7)
0.9999 0.9991 0.9940 0.7474 0.3535 0.1420 0.0539 0.0201 0.0074 0.0010
= 2
t
2)
Lower bound o f (7)
0.9999 0.9973 0.9826 0.5311 0.1420 0.0330 0.0074 0.0017 0.0003
0.9[39 0.6376 0.4066 0.0111 0.0001 0.0000 0.0000 0.0000 0.0000
( T a b l e 4)
Bonferroni Lower b o u n d
Upper bound of (7)
0.0284 0.1139 0.1732 0.0969 0.0128 0.0013 0.0001
Lower b o u n d of (7)
0.9999 0.9892 0.9391 0.2266 0.0200 0.0016 0.0001
0.8607 0.4724 0.2231 0.0006 0.0000 0.0000 0.0000
Comparisons of lower bounds: A quick look over the tables reveals that the lower bounds obtained via (7) is not very accurate except for small mission time t.
This is not
surprising as the lower bound (7) is more appropriate for a series system of independent components. seem to do well when
On the other hand,
t ~ 0.5.
the Bonferroni
(Note that for t ) O . ~
bound is close to the improved upper bound).
lower bounds
the Bonferroni
lower
It appears from this example
that the maximum value of the two bounds for a fixed t would be a good lower bound on the system reliability.
Comparison of two upper bounds From the four tables we see the improved upper bound is more accurate than the upper bound of (7) except for X = 0. I.
For ~ = 0. I, we have
System reliability correlation coefficient p = O. 04Y6which
459
implies that the three components
are nearly independent.
By expanding the upper bound of (7), we note that n it is a truncated inclusion-exclusion formula for Pr~_l(Ti>t)) when T i are independent.
This explains why the upper bound of (7) is doing marginally better than the improved bound in Table I.
We also observe that for small t, a reasonable estimate of the system reliability can be obtained by averaging the improved upper bound and the lower bound of (7).
For t ) 6 5 , we would take the average of the
improved upper bound and the Bonferroni lower hound as our estimate.
References Barlow, R E and P r o s c h a n F (1981). and L i f e T e s t i n g .
Marshall, A W and Olkin, I (1967). Dikstribution.
Worsley, K J (1982).
Statistical
Silver Springs:
Theory o f R e l i a b i l i t y
To Begin With.
A Multivariate Exponential
J.Amer.Statlst,Assoc.
62, 30-44.
An improved Bonferroni inequality and Applications.
Biometrik8 69, 297-302.