# A simple method of determining moments of a top event.

Introduction

The study of uncertainty propagation on fault tree analyses is a classical problem in reliability engineering This problem deals with the evaluation of uncertainty in the top event probability arising from uncertainties in basic event probabilities. The methods used for solving this problem are either based on Monte Carlo simulation[1-4] or use the method of moments[1,3,5-9]. Other methods include systematic gate-by-gate level combination of random variables[1,3,10-13], and an approximate regression analysis[14].

The Monte Carlo simulation provides an approximate method and its application is restricted by the cost of computation. The method of moments is an efficient technique and in fact does not require the distributions of the basic event probabilities to be specified. Rushdi[7] and Rushdi and Kafrawy[15] have demonstrated that under the assumption of the statistical independence of basic events, the top event probability is a multiaffine function of basic event probabilities and hence can be obtained through an exact and finite Taylor series expansion. These expressions have been provided in the Appendix for the sake of completeness.

The present paper approaches the problem in a more direct way and obtains the exact moments from the basic principles of probability theory. At first, we would explain the concept involved in the computation of moments and then we describe an algorithm which computes these for systems with two state components. Finally, the method is extended to a system with multi state components.

Notation

The following notation is used:

n Number of relevant components, failure of which contributes to the

occurrence of top event of the fault tree; [q.sub.i] Failure probability of ith component (for system with two state elements);

[q.sub.ij] Failure probability of ith component in jth failure mode, j = 1, 2.. ., L - 1.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Q Top event probability; m Number of terms in the expression of Q; E Expectation operator; Var Variance operator; G(.) Gamma function.

Assumptions

For systems consisting of two state components, it is assumed that

* the building blocks of the system fault tree are logic gates;

* basic events of the fault tree are statistically independent (subsequently, it is shown how this assumption can be relaxed) and none of the events represents a common cause contribution;

* probabilities of the basic events of the fault tree are random variables characterized by their distributions or moments.

Basic steps

The algorithm developed in this paper is based on the fact that once each indicator variable [X.sub.i] in the disjointed structure function of top event of a fault tree has been replaced by [q.sub.i] and [??.sub.i] is replaced by 1- [q.sub.i], the resulting function is a multivariate Taylor series expansion. Thus Rushdi[7] and Rushdi and Kafrawy[15] have obtained moments of top event probability (Q) using Taylor series expansion in terms of moments of [q.sub.i] which have been treated as random variables (see Appendix). A finite Taylor series expansion implies that exact moments of Q (treated as a random variable) can be computed by direct application of an expectation operator Eon Q, [Q.sup.2], [Q.sup.3], etc., assuming that all [q.sup.S] [sub.i] are independent random variables.

Example 1

Let us consider a fault tree shown in Figure 1. Here,

[Figure 1 ILLUSTRATION OMITTED]

T = [X.sub.1][X.sub.2] + [X.sub.1][X.sub.3].

On disjointing the terms, we obtain

T = [X.sub.1][X.sub.2] + [X.sub.1][X.sub.2][X.sub.3].

Thus top event probability Q is given by

(1) Q = [q.sub.1][q.sub.2] + [q.sub.1] (1 - [q.sub.2])[q.sub.3]

= [q.sub.1][q.sub.1] + [q.sub.1][q.sub.3] - [q.sub.1][q.sub.2][q.sub.3] and therefore,

[Q.sup.2] = ([q.sub.1][q.sub.2] + [q.sub.3] - [[q.sub.1][q.sub.2][q.sub.3]).sup.2]

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similarly,

[Q.sup.3] = [([q.sup.1][q.sub.2] + [q.sub.2][q.sup.3] - [q.sub.1][q.sub.2][q.sub.3]).sup.3]

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

On using expectation operator on (1), we obtain

(4) (E(Q) = E([q.sub.1]) E([q.sub.2])+E([q.sub.1]) E([q.sub.3]-E([q.sub.1]) E([q.sub.2]) E([q.sub.3]).

Similarly, using the E-operator on (2) and (3),we obtain

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To evaluate numerically, let us assume that

E([q.sub.1])= 0.2, E([q.sub.2])= 0.13333,E([q.sub.3]) = 0.1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

On substituting these values in (4), (5) and (6), we obtain

E(Q) = 0.044,E([Q.sup.2] = 0.003297521, E([Q.sup.3]) = 0.000355.

Any other moments of higher order can be computed in the same way. However, the first three or four moments are enough to characterize a distribution.

We can now summarize the steps to compute the moments of top event probability of any order in an algorithmic form, from the moments of basic events' probabilities. In this algorithm, the terms involving various powers of Q are obtained by a simple addition of rows of a matrix A which represent the terms involved in the expression of Q. For a system with multistate elements, the same algorithm can be used with some modification as will explained in a later section.

Algorithm

The steps involved in the computation are:

(1) Find the structure function of the top event in the sum of product (sop)

form (which actually is the union of cut sets).

(2) Disjoint the terms of the expression of the structure function using any

of the procedures discussed in (1).

(3) Replace each indicator variable [X.sub.i] by [q.sub.i] and ?? by 1 -

[q.sub.i]. If it is in the form

of complemented subproduct form like ??, then it is replaced by

1 - [q.sub.i]

(4) Simplify the expression by removing all parentheses. For illustration, in

case of Example 1, we would obtain

Q = [q.sub.][q.sub.2]+[q.sub.1][q.sub.3]-[q.sub.1][q.sub.2][q.sub.3].

(5) Disregarding the sign of a term, form a matrix A whose each row

represents a term namely, [S.sub.1], [S.sub.2], [S.sub.3], ..., [S.sub.m]

for an expression of Q

involving m terms. The presence of [q.sub.i] in the ith column of a term is

represented by 1 and its absence is indicated by recording a 0 in the

corresponding position. Thus for expression (1), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(6) Form a coefficient vector for the terms (which includes sign of the term)

in the expression for Q. For Example 1, we have C' = [1, 1-1] where, C'

stands for the transpose of matrix C.

(7) Form vectors of kth moments (about origin) of basic event probabilities.

For the fault tree of Example 1, we have vector for kth moment as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, we obtain

[M'.sub.0] = [1, , 1]

[M'.sub.1] = [E[q.sub.1]. E[q.sub.2], E[q.sub.3]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(8) Compute the kth moment of Q, i.e., E([Q.sup.k]). This is done as follows:

8(i) Take k rows (rows need not be different, i.e., a row can be

considered as many as k times) say, [S.sub.il], l = 1,2,. . ., k and add

to form a vector [V.sup.k] (an n-tuple). The elements of [V.sup.k] will

be positive integers ranging form O to k. Replace each element in

[V.sub.k] by the corresponding moments, (i.e., if jth element of

[V.sub.k] is l', replace it by the jth entry of [M'.sub.1]) and form a

product by multiplying the elements. Finally, multiply the product

obtained with coefficients from C' corresponding to the k rows considered.

8(ii) Repeat 8(i) for [i.sub.1] = 1, 2, ..., m; ..., [i.sub.k] - 1, 2,

..., m, and add all the products obtained to get E([Q.sup.k]).

Uncertainty modelling and propagation in multi-state system

In the previous section, we have considered systems with two state elements. In multi-state systems, though the assumption of independent element is a reasonable assumption, the probability of failure of a unit in different modes of failure in general is negatively correlated. Thus we may have to model uncertainty in each state-probability (marginal) as well as in the association among them. One of the distributions that is suitable for representing our state knowledge is Dirichlet's distribution[16]. It is a multivariate generalization of the beta distribution and its pdf is given by

(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The distribution is of L-1 dimension since [q.sub.cL] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The moment of order ([k.sub.1], [k.sub.2], [k.sub.3], ..., [k.sub.L-1]) is given by

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where, Z = [summation] [V.sub,i].

The mean and variance of [q.sub.ci] are:

(9) E([q.sub.ci]) = [V.sub.i]/Z, i = 1, ..., L - 1

(10) Var([q.sub.ci]) = [V.sub.i][Z.sub.i]/[[Z.sup.2].(Z+1)], i = 1, ...,

L - 1

where, [Z.sub.i] = Z - [V.sub.i]

The covariance between [q.sub.ci] and [q.sub.cj] is:

(11) Cov([q.sub.ci], [q.sub.ci]) = [-V.sub.i].[V.sub.j]/[Z.sup.2].(Z +

1)], [is not equal to] j = 1, ..., L - 1

Example 2

Consider an example given in Misra[17] of an electronic filter shown in Figure 2, which consists of a diode with three states, a resistor with two states and a capacitor with three states The output can be normal or one which is false but not dangerous or false and dangerous (if an a.c. appears an output it may destroy certain components connected to the output of the filter). Thus three outputs are distinguished which are taken as system states. The component and its states are distinguished as shown in Table I.

[Figure 2 ILLUSTRATION OMITTED]
```Table I. The component and its states

Component

Capacitor: [X.sub.1]   [X.sub.11] short circuit fault
[X.sub.12] open circuit fault
[X.sub.13] working normally
Diode: [X.sub.2]       [X.sub.21] short circuit fault
[X.sub.22] open circuit fault
[X.sub.23] working normally
Resistor: [X.sub.3]    [X.sub.31] open circuit fault
[X.sub.32] working normally
```

System states are distinguished as follows:

* [T.sub.1] normal output;

* [T.sub.2] false output signal but safe;

* [T.sub.3] false and dangerous output signal.

The fault trees leading to system states [T.sub.2] and [T.sub.3] are shown in Figure 3. We have., [T.sub.2] = [X.sub.22] + [X.sub.31] + [X.sub.12][X.sub.23] + [X.sub.11][X.sub.23] + [X.sub.11][X.sub.21]. On disjointing the term we obtain,

[T.sub.2] = [X.sub.22] + [X.sub.33][X.sub.22] + [X.sub.12][X.sub.23][X.sub.31] + [X.sub.11][X.sub.23][X.sub.31] + [X.sub.11][X.sub.21][X.sub.31];

[T.sub.3] = [X.sub.12][X.sub.21][X.sub.32] + [X.sub.13][X.sub.21][X.sub.32].

[Figure 3 ILLUSTRATION OMITTED]

Let the top event probabilities associated with [T.sub.2] and [T.sub.3] be represented by [Q.sub.2] and [Q.sub.3], respectively.

(12) [Q.sub.2] = [q.sub.22] + [q.sub.31](1 - [q.sub.22]) + [q.sub.12][q.sub.23](1 - [q.sub.31]) + [q.sub.11][q.sub.23](1 - [q.sub.31]) + [q.sub.11][q.sub.21](1 - [q.sub.31])

and

(13) [Q.sub.3] = [q.sub.12][q.sub.21][q.sub.32] + [q.sub.13][q.sub.21][q.sub.32].

Since success probability of a unit is functionally dependent on the sum of its failure mode probabilities (failure mode probabilities in turn are stochastically dependent ,i.e. there exists correlation between them) we must express top event probabilities in terms of each unit's failure mode probabilities, to find out higher order moments of top event probability. Therefore, [Q.sub.2] and [Q.sub.3] are expressed as

(14) [Q.sub.2] = [q.sub.11] + [q.sub.12] + [q.sub.22] + [q.sub.31] - [q.sub.1][q.sub.22] - [q.sub.1][q.sub.31] - [q.sub.12][q.sub.21] - [q.sub.12][q.sub.22] - [q.sub.22][q.sub.31] - [q.sub.12][q.sub.31] + [q.sub.11][q.sub.22][q.sub.31] + [q.sub.12][q.sub.21][q.sub.31] + [q.sub.12][q.sub.22][q.sub.31].

(15) [Q.sub.3] = [q.sub.21] - [q.sub.11][q.sub.21] - [q.sub.21][q.sub.31] + [q.sub.11][q.sub.21][q.sub.31].

We can now describe an algorithm to compute the moments of top event probability from the expression of [Q.sup.S] [sub.i].

Algorithm

This consists of the following steps.

(1) Express the top event probability variable in terms of the basic event

probability variables. Also express success probability of units appearing

in the expression in terms of its failure mode probabilities.

(2) Form a matrix A with each row representing a term in the expression for

top event probability. Columns are designated such that first (L - 1)

columns represent (L -1) failure states of the first component, next (L - 1)

column represents that of second component, etc. The presence of a

variable [q.sub.jk] in ith term is represented by a 1 in position

corresponding to ith row and kth column in the jth group of columns

representing the failure states of the jth element, and the absence is

indicated by a zero in the corresponding position. Thus for expression (14)

we have,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note that components [X.sub.1] and [X.sub.2] have two failures states and hence two

columns to represent their states.

(3) Form a coefficient vector for the terms in the expression for Q. For [Q.sub.2] in

example 2, we have, C' = [1, 1, 1, 1, -1, -1, -1, -1, -1, -1, 1, 1, 1].

(4) Computation of E([Q.sup.k]):

* Take k rows (rows need not be different, i.e. a row can be considered as many as k times) say, [S.sub.il], l = 1, 2, ..., k and add to form a vector [V.sub.k].

From this, form an n-vector, the jth element of which is the moment

that can be obtained by taking the entries in the jth group of columns

in [V.sub.k] as the order of the moment of distribution of the jth component.

Finally, the product obtained by multiplying the elements in the

e-vector formed is multiplied by coefficients (from C') corresponding

to the k rows considered.

* Repeat (i) for [i.sub.1] = 1, 2... m;..., [i.sub.k] = 1, 2, ..., m and add all the products obtained to get E([Q.sup.k]).

As an illustration let us assume that [X.sub.1] follows D(1, 1, 16), [X.sub.2]

follows D(2, 1, 15) and [X.sub.3] follows D(1, 9), where D([V.sub.1], [V.sub.2], ... [V.sub.L]) is the

Dirichlet distribution with parameters [V.sub.1], [V.sub.2],... [V.sub.L]. Table II gives the

required moments of components' failure probabilities to compute

moments of top event probability up to third order.

Table II. Moments about origin of component failure mode probabilities
```Order of                                Components
moments
[K.sub.1]   [K.sub.2]   [X.sub.1]    [X.sub.2]   [X.sub.3]
0           0       1            1           1
1           0       0.0555555    0.1111111   0.1
0           1       0.0555555    0.0555555   -
1           1       0.0029239    0.0058479   -
2           0       0.0058479    0.0175439   0.0181818
0           2       0.0058479    0.0058479   -
2           1       0.0002924    0.0008772   -
1           2       0.0002924    0.0005848   -
3           0       0.0008772    0.0035087   0.00454545
0           3       0.0008772    0.0008772   -
```

Using the algorithm described earlier in this section, we obtain moments of Q about origin as follows:

* E([Q.sub.2]) = 0239;

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Finally, we would like to mention that the method adopted in this algorithm to treat the dependent variables, i.e. grouping those variables in adjacent columns in matrix A, can be applied to systems with two state components in which failure probability of some of the components are dependent.

Conclusion

A simple algorithm is developed for finding exact moments of top event failure probability from the moments of basic events in a fault tree. This procedure does not require Taylor series expansion of a top event failure probability function and its subsequent partial derivatives. The method has been extended to systems with multistate components and a general algorithm has been developed for the purpose. The method is basically very simple and easy to program.

References

[1.] Ahmed, S., Metcalf, DR. and Pegram, J.W., "Uncertainty analysis of system reliability and availability assessment", Nuclear Engineering and Design, Vol. 68, 1981, pp. 1-3.

[2.] Jackson, P.S., Hockenbury, R.W. and Yeater, M.L., "Uncertainty analysis of system reliability and availability assessment", Nuclear Engineering and Design, Vol. 68, 1981, pp. 5-29.

[3.] Ahmed, S., Clark, R.E. and Metcalf, D.R., "A method for propagating probability in probabilistic risk assessment", Nuclear Technology, Vol. 59,1982, pp. 238-45.

[4.] Chang, S.H., Park, J.Y. and Kim, M.K., "The Monte-Carlo method without sorting for uncertainty propagation analysis in PRA", Reliability Engineering, Vol. 10, 1985, pp. 233-43.

[5.] Jackson, P.S., "A second order moments method for uncertainty analysis", IEEE Transactions on Reliability, R-31, 1982, pp. 382-4.

[6.] Dezfuli, H. and Modaress, M., "Uncertainty analysis of reactor safety systems with statistically correlated failure data", Reliability Engineering, Vol. 11, 1985, pp. 47-64.

[7.] Rushdi, A.M., "Uncertainty analysis of fault-tree outputs", IEEE Transaction on Reliability, R-34, 1985, pp. 458-62.

[8.] Misra, K.B. and Gadani, J.P., "Confidence limits assessment using sensitivity calculations", Reliability Engineering, Vol. 3,1982, pp. 63-77.

[9.] Apostolakis, G. and Lee, Y.T., "Method for estimation of confidence bounds for the top-event unavailability of fault trees", Nuclear Engineering and Design, Vol. 41, 1977, pp. 411-19.

[10.] Colombo, A.G. and Jaarsma, R.J., "A powerful numerical method to combine random variables", IEEE Transactions on Reliability, R-29, 1980, pp. 126-9.

[11.] Colombo, A.G., "Uncertainty propagation in fault tree analysis", in Apostolakis, G. et al. (Eds), Synthesis and Analysis Methods for Safety and Reliability Studies, Plenum Press, New York, NY, 1980.

[12.] Keey, R.B. and Smith, C.H., "The propagation of uncertainties in failure events", Reliability Engineering, Vol. 10,1982, pp. 105-27.

[13.] Masera, M., "Uncertainty propagation in fault tree analysis using lognormal distribution", IEEE Transaction on Reliability, R-36, 1987, pp. 145-9.

[14.] Mazumdar, M., "An approximate method for computation of probability intervals for the top-event probability of fault trees", Nuclear Engineering and Design, Vol. 71, 1982, pp. 45-50.

[15.] Rushdi, A.M. and Kafrawy, K.F., "Uncertainty analysis in fault tree analyses using an exact method of moments", Microelectronics and Reliability, Vol. 28,1988, pp. 945-65.

[16.] Haim, M. and Porat, Z., "Bayes reliability modeling of a multistate consecutive k-out-of-n:F system", in Proceedings of Annual Reliability and Maintainability Symposium, 1991, pp. 582-6.

[17.] Misra, K.B., Reliability Analysis and Prediction (A Methodology Oriented Treatment), Elsevier Science Publishers, Amsterdam, 1992.

Appendix

Taylor series expansion method of finding moments of top event probability.

Notation

n number of system component relevant to the fault tree;

[[micro].sub.1] mean value of Q;

[[micro.sub.j] jth central moment of Q, j [is not equal to] 1;

[m.sub.ij] jth central moment of [q.sub.i], j [is not equal to] 1;

q n-dimensional vector of basic event probabilities (treated as random variables);

[M'.sub.1] n-dimensional vector representing mean value of q;

Q top event probability (treated as a random variable).

Rushdi[7] and Rushdi and Kafrawy[15] derived the following expressions for moments using Taylor series expansion method:

[[micro].sub.1] = Q([M'.sub.1])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
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