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CivilComp Proceedings
ISSN 17593433 CCP: 84
PROCEEDINGS OF THE FIFTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 210
A Mathematical Model to Evaluate the Unavailability of a Technical System L. González
Department of Mathematics, University of Las Palmas de Gran Canaria, Spain , "A Mathematical Model to Evaluate the Unavailability of a Technical System", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Fifth International Conference on Engineering Computational Technology", CivilComp Press, Stirlingshire, UK, Paper 210, 2006. doi:10.4203/ccp.84.210
Keywords: system unavailability, stochastic Boolean function, Hamming weight, intrinsic order, top binary ntuples, Pascal's triangle.
Summary
Many different technical systems in engineering depend on a large number n
of mutually independent basic components. Often, these components, , can be considered as random Boolean variables. That is, they only take two
possible values: if the ith component fails,
otherwise, and the (basic) probability
of failure of each component is assumed to be known a
priori [1]. So, each one of the possible situations is given by a binary
ntuple
of 0s and 1s, and it has its own occurrence probability
. Moreover, the technical system can be described by a stochastic
Boolean function
, depending on its n basic components. Then, assuming that
if the system fails, otherwise, the system unavailability
also called the top event probability is evaluated by computing the
probability
[1,2].
A wide class of different strategies has been proposed in reliability theory and risk analysis to estimate the top event probability (a central question in fault tree analysis) [2,3]. One of these methods is based on the canonical normal forms of the Boolean function , and it provides lower and upper bounds on the system unavailability, from any arbitrary subsets of binary ntuples for which , respectively [1,4]. The accuracy in the above mentioned estimation of improves at the same time as the total sum, , of the occurrence probabilities of all the selected binary ntuples increases. Consequently, the main (and the most difficult) question is how to select the minor number of binary strings with occurrence probabilities as large as possible, in order to minimize the computational cost. Note that the simplest answer to this question, namely, computing all the binary ntuple probabilities and then ordering them by their occurrence probabilities, is not feasible due to the exponential nature of the problem: There are ntuples of 0s and 1s. To avoid this obstacle, we have established a simple, positional criterion that allows to compare two given elementary state probabilities without computing them, simply looking at the positions of their 0s and 1s [4,5]. The socalled intrinsic order criterion (because it is independent of the basic probabilities and it is determined by the positions of the 0 and 1 bits) defines a partial order relation (intrinsic order), denoted by , on the set . Theoretical results and practical applications of the intrinsic order can be found in [5,6], and the graphical structure of the partially ordered set is described in [7]. One of the topics in the intrinsic order model is the set of binary ntuples whose occurrence probabilities are always (that is, for any set of parameters, , satisfying certain nonrestrictive assumptions) among the largest ones, the socalled top binary ntuples [8]. These binary strings can be characterized by several simple positional criteria related to their 0s and 1s. In this context, we present a new method for selecting different sets of binary ntuples, with large occurrence probabilities, in order to estimate the system unavailability with a low computational cost. Basically, the two main ideas underlying our new approach are: (i) we only select Top binary ntuples. (ii) we provide a simple, recursive formula for rapidly computing the sums of the occurrence probabilities of the binary ntuples with weight m whose 1s are placed among the k rightmost positions . This formula is tightly related to the famous Pascal's triangle. Moreover, this connection highlights, in an elegant way, the balance between accuracy and computational cost. In this way, we present an easily implementable algorithm which determines a priori different sets of top binary ntuples that assure us the estimation of the system unavailability via the above mentioned bounds with a prespecified maximum error. References
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