A Priori Error Estimates for the Method of Inclusion-Exclusion with Applications
Pages 443 - 450
Abstract
The method of inclusion-exclusion can be used to obtain successive upper and lower bounds on the probability of the occurrence of complex events.
Usually only the first upper and the first lower bound are considered. But as they can be crude, further bounds must not be excluded from methodical studies and practical computations. In this paper the errors of all inclusion-exclusion approximations are estimated. As a priori error estimates are independent of the actual computation of the bounds, they can precede. So the order of the approximation, being at least as close to the exact value as demanded, can be determined in advance.
As a second point, the paper shows how to apply the method of inclusion-exclusion and the discussed a priori error bounds skillfully to approximate the reliability and the probability of connectivity for random graphs representing various structures.
References
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Richard E. Barlow, Frank Proschan, Statistical theory of reliability and life testing, Holt, Rinehart and Winston, Inc., New York, 1975xiii+290
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H. B. Baskin, B. R. Borgerson, R. Roberts, PRIME—A modular architecture for terminal-oriented systems, AFIPS Conf. Proc., 40 (1972), 431–437
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Th. Brand, K. D. Heidtmann, The inversion of sets being decisive of system reliability, Report on Computer Science, 24, Fernuniversität, Hagen, 1982
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Kai Lai Chung, On the probability of the occurrence of at least m events among n arbitrary events, Ann. Math. Statistics, 12 (1941), 328–338
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Howard Frank, Ivan T. Frisch, Communication, transmission, and transportation networks, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1971xiii+479
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Maurice Fréchet, Les probabilités associées à un système d'événements compatibles et dépendants. I. Événements en nombre fini fixe, Actual. Sci. Ind., no. 859, Hermann et Cie., Paris, 1940viii+80
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K. D. Heidtmann, Masters Thesis, Zur Approximation der Verfügbarkeit technischer Systeme (On the approximation of the reliability of technical systems), Ph.D. Dissertation, Dept. Mathematics & Computer Science, Fernuniversität, Hagen, 1981
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K. D. Heidtmann, Improved method of inclusion-exclusion applied to k-out-of-n systems, IEEE Trans. Rel., 31 (1982), 36–40
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C. Jordan, Le théorème de probabilité de Poincaré, generalisé au cas du plusieurs variables indépendentes, Acta Sci. Math., 7 (1934–35), 103–111, Szeged
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Copyright © 1984 Society for Industrial and Applied Mathematics.
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Society for Industrial and Applied Mathematics
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Published: 01 April 1984
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