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Article Excerpt
1. Introduction
Consider an n-component system, where each component either works or is failed. Let [X.sub.i] equal one if component i works and zero otherwise. For a non-decreasing function h([x.sub.1],...,[x.sub.n]), defined on n-dimensional binary vectors, we are interested in using simulation to efficiently estimate [alpha] = E[H], where H = h([X.sub.1],...,[X.sub.n]).
In the special case where h is a binary function, we can interpret it as a structure function for a component system (see Barlow and Proschan (1975)) and so E[H] would be the probability that the system functions. When the [X.sub.i] are independent (which we are not assuming), E[H] is the reliability function.
In Section 2 we present a new approach, based on an improved use of standard stratified sampling, for using simulation to estimate E[H] when the vector ([X.sub.i],...,[X.sub.n]) is exchangeable. In Section 3 we consider the case where there is a random environmental parameter [THETA] such that, conditional on [THETA] = [theta], the [X.sub.i] are independent with E[[X.sub.i]|[THETA] = [theta]] = [theta][p.sub.i]. In Section 3.1 we consider the case where the [X.sub.i] are independent, and show how to use this to improve the method of Section 3. In Section 4 we present other simulation approaches for estimating system reliability that have appeared in the literature.
2. The exchangeable Bernoulli case
Suppose that ([X.sub.1],...,[X.sub.n]) is an exchangeable binary random vector. Let S = [[SIGMA].sub.i=1.sup.n] [X.sub.i], and further suppose that the probability mass function:
[P.sub.j] = P{S = j} j = 0,...,n
is computable.
Before continuing let us mention some examples in which the preceding would hold.
Examples
1. The simplest model resulting in the [X.sub.i] being exchangeable and the distribution of S computable is when the components are independent, with each component working with probability p. In this case:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
2. A generalization of the preceding case is when the components are conditionally independent and identically distributed. That is, there is an environmental parameter [THETA], having distribution G, such that if [THETA] = [theta] then each component independently works with probability p([theta]). In this case:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The preceding probabilities, being one-dimensional integrals, are easily numerically computed.
3. Another possibility is the cascading failure model, introduced and studied in Dobson et al. (2005) and Lefevre (2006). In this model, each of n components has an initial load, with these loads being the values of n independent uniform (0, 1) random variables. Following a disturbance, an additional load d is added to each of these n loads. If the resulting load of a component exceeds one, that component fails. Each failure then adds a new load of fixed amount to those still unfailed components, possibly causing some of them to fail (if their total load now exceeds one) and thus adding new loads to the so-far unfailed components, and so on. The probability mass function of the total number of components that eventually fail is determined in Dobson et al. (2005) and Lefevre (2006).
To use simulation to estimate [alpha] = E[H] = E[h([X.sub.1],...,[X.sub.n])], note that:
[alpha] = [n.summation over (j = 0)]E[H|S = j][P.sub.j].
Our method will, on each simulation run, estimate all of the quantities E[H|S = j], j = 0,...,n. This is done by letting each simulation run be as follows.
Step 1. Generate...
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