Combinatorial analysis for probabilistic assessment of dependent failures in systems and portfolios
Introduction
In structural reliability considerations, one is often interested in the number (or ratio) of buildings that exceed a particular damage limit state, and are thus defined as “failed”. This could be important in assessing the vulnerability of a community, wherein the number of damaged buildings of particular type (such as hospitals, residences, schools, etc.) may be crucial.
Let denote the event of failure of component , and the random variable describing the total number of component failures. To compute the probability of the union of all failure events, , corresponding to the failure probability of a series system, the inclusion/exclusion principle (e.g., [1]), can be invoked. It is:
Note that the to summation terms are over all possible subsets of failures, the number of which is given by the binomial coefficient .
In this paper we present a generalization of the inclusion–exclusion principle to compute the probability of more than events, and the probability of exactly events, where m is any particular value of between zero and n. This generalization is useful for computing the reliability of systems that fail as soon as components out of fail ( out of systems), but also for computing the distribution of the number of failures or damages in a portfolio. The proposed generalization is related to alternative system reliability approaches, such as Ditlevsen bounds [2], Daniels systems [3], system reliability based on linear programming concepts proposed by Song and Der Kiureghian [4], bounds by partial expansion [5], or other methods [6].
Section snippets
A generalization of the inclusion-exclusion principle for system reliability
The inclusion–exclusion principle can be extended to compute the probability of two or more failure events as follows: and to compute the probability of or more failure events as follows:
Note that the
Special Cases
A common special case occurs when all marginal failure probabilities are identical and all failure events have the same statistical dependence, i.e.: and so on, in which has been introduced for the probability of the intersection of r events. In this special case, Eq. (6) reduces to: In a more general setting, it is often possible to identify groups of components with identical marginal failure
Conclusions
Realistic safety assessment of systems is vitally important, including those with for which a large number of failure modes exhibit significant probabilities (i.e., those not dominated by a small number of modes). This is true for multiple modes of failure in a single structure, as well as performance criteria for a portfolio of systems within a community. For many such realistic systems those modes exhibit correlation and differing marginal probabilities. A straightforward understanding of the
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
The authors gratefully acknowledge the support of the National Science Foundation, USA , grant number CMMI 1063790. The opinions expressed in this paper are those of the authors, and do not necessarily reflect the views or policies of NSF.
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