Abstract- An exposition is made of the exact method of moments which is based on the exact and finite Taylor expansion of the top-event probability in terms of ...
MIcr~/ectron. Rel/ab, Vol. 28, No. 6, pp. 945-965, 1988. Printed in Great Britain.
0026-2714/8853.00+ .00 © 1988Pergamon Press plc
UNCERTAINTY PROPAGATION IN FAULT-TREE ANALYSES USING AN EXACT METHOD OF MOMENTS ALl M. RUSHDI
|
and KAMEL F. KAFRAWY
2
Department of Electrlcal and Computer Englneerlng, and -Department of Engineering Mathematics, King Abdulaziz University, P.O. Box 9027, Jeddah-21413, Saudi Arabia (Received for publication 30 May 1988)
Abstract- An exposition is made of the exact method of moments which is based on the exact and finite Taylor expansion of the top-event probability in terms of the basic-event probabilities in a fault tree. This method allows calculation of the various moments with a readily quantifiable accuracy that can be arbitrarily improved. Typical approximations made in other versions of the method of moments are also discussed and their effects are empirically evaluated. The numerical results of the exact method of moments are in good agreement with those of the Monte Carlo method, and are superlor to those of other existing methods.
1. INTRODUCTION The study of uncertainty is a classical
propagation
problem in reliability
problem deals with the evaluation event probability probabilities.
arising
from uncertainties
[2,5,6,10-13,16-18].
the systematic
gate-by-gate
level
ression analysis
combination
in
top-
[1,5,6-8,10,15],
and the
Other existing methods on
and the approximate
a reg-
[9].
almost any type of uncertainty to common cause failures.
the benchmark values
that can simulate
[17] including uncertainties
To compare uncertainty
the Monte Carlo simulation
inaccurate
This
in basic-event
The Monte Carlo method is a powerful method
niques,
[1-19].
of random variables
[3,4,5,10,14,19],
analyses
for solving this problem
are the Monte Carlo simulation method
include
engineering
of the uncertainty
The most used methods
method of moments
in fault-tree
[9]. However,
if the number of samples
it can be limited by costs. 945
results
due
propagation
are usually
taken
the Monte Carlo method may is not sufficiently
techas be
large, and
946
A.M. RUSHDXand K. F. KAFRAWY
The method of moments
is an efficient
require the specification the basic-event riable Taylor second-order
truncated
independence
bility is a multiaffine
the for
terms is negligible.
method of moments
it has been observed
of statistical
at
it is valid only for systems
is difficult
apply to complex fault trees with many replicated
Recently,
of
It is usually based on a multiva-
of higher-derivative
the second-order
that does not
distributions
of the system function
term. Therefore,
which the importance Generally,
of the probabilistic
probabilities.
expansion
technique
events
to
[17].
[18] that under the assumption
of basic events,
the top-event
function of basic-event
proba-
probabilities,and
hence can be given by an exact and finite Taylor expansion. achieve a certain desirable ber of higher-order
degree of accuracy,
terms in this expansion
an adequate num-
is to be retained
such a way to ensure that the effect of the first truncated is really negligible.
Therefore,
the method of moments
made an "exact" one in the sense that all moments lated with a readily quantifiable improved.
Hence
accuracy
, the method of moments
in principle,
rarily increased by increasing cretizations
be
can be calcu-
can stand
on
combination
the accuracy
equal of random
can be arbit-
the number of simulations
paper is an exposition
or dis-
It also discusses
of the exact method
three typical approximations
usually made with the classical method of moments. extensive
term
[6].
The present moments.
can
in
that can be arbitrarily
footing with the Monte Carlo and systematic variables methods where,
To
set of ten test problems
zed via various versions
with the Monte Carlo method and other approximate
of the method of moments
the superiority
Finally,
an
in comparison methods.
This
of the exact version
and shows that the effects
mations made in its other versions
that are
is studied and computer analy-
of the method of moments
study clearly demonstrates
of
can be neglected
of approxiin many cases.
Uncertainty propagationin fault-tree analyses II. ASSUMPTIONS AND NOTATION
Assumptions 1. The system is modeled by a fault tree whose building blocks are logic gates [7,pp.48-54],[8,pp.157-162]. 2. Basic events of the fault tree are statistically independent.
None of these events represents a common cause
contribution. 3. Probabilities of the basic events of the fault tree are random variables characterized by their probabilistic distributions or moments.
Notation n
number of system components relevant to the fault tree
£i'xi
indicator variables for the occurrence and nonoccurrence of basic event i at time t. These are switching (Boolean) random variables;
Xi=l and Xi=0 if i occurs, and Xi=0
and X.=I if i does not occur l
indicator variable for the existence of the top event at time t T qi
implies the transpose of a vector probability of occurrence of basic event i (treated as a random variable);
qi = Pr{Xi=l} = E{Xi}= I-E{X i}
top-event probability;
also called system unavail-
ability (treated as a random function); Q = Pr{S=I} =
E{S}
n-dimensional vector of basic-event probabilities; = (ql q2 "'" qn )T mean value of ~ ; 51 = (~ii v21 "'" Vnl )T
"ij
central moment j of qi ; ~ij = E{(qi- ~il )J}' j = 2,3,4 mean value of Q
~j
central moment j of Q ; uj = E {(Q-~I)J}, j = 2,3,4
947
A.M. RusHDiand K . F . ~ w v
948
Q(~llki ) values of Q when all elements of ~ are set to their mean values,
except for qi which is set to k (where
k = 0 or I). Meanings of Q(~llki,£j) ..... etc. follow similarly Pr{.}
probability of event {.]
E{.}
expectation of random variable
m
median
(50th percentile)
{.]
of a lognormally distributed
variable F
error factor variable;
F = 95th percentile/50th
percentile/5th ~,~
(range factor)of a lognormally distributed percentile = 50th
percentile
mean and standard deviation of the natural logarithm of a lognormally distributed variable;
~=Ln(m)
;
= Ln(F)/I.645.
III. THE EXACT METHOD OF MOMENTS
The exact method of moments consists of the following steps: i. The logical relations of the gates of the fault tree are manipulated
to obtain a switching-domain
(Boolean-domain)
expre-
ssion for the top-event occurrence S(XI,X2 .... , Xn ). This expression usually takes a sum-of-products
(s-o-p) form [20], which
is equal to the union of all the minimal cutsets of the fault tree.
2. The switching expression S(XI,X2 .... ,Xn ) is converted to an equivalent disjoint,
form SM(XI,R2 ..... Xn ) in which any ORed terms are
and any ANDed alterms are statistically
independent.
Many techniques for the conversion from S to SM are available [18,20-28]. 3. The expression SM(XI,X2 .... ,Xn ) is converted directly, on a one-to-one basis, into the algebraic expression Q(ql,q2,...,qn ) by replacing each indicator variable Xi by its expectation qi and replacing the logical operators counterparts
(AND and OR) by their algebraic
(MULTIPLY and ADD), namely
Uncertainty propagation in fault-tree analyses
!Xi,Xi}
~'~{qi,(l-qi)} SM(XI'X2 ..... ~n) {n, u •}--~{, , + } ~
Q(ql,q2 ..... qn ).
949
(i)
Equation (I) results from the probability relations:
E{T I U T 2}
=
E{T I} + E{T 2} ; T I and T 2 are disjoint,
E{A I n A 2}
=
E{A I} * E{A 2} ; A I and A 2 are statistically independent.
(2)
(3)
4. The algebraic function Q(~) = Q(ql,q2 ..... qn ) is a multiaffine function, and hence it has a finite multivariable Taylor expansion. Consequently, the lower-order statistical moments of Q are obtained in terms of those of ~ as follows
[18]. ~i = Q ( V l )
n
(4)
'
C2
+
u2 " iffil z i vi2
+
+
E
z
l_
