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June 2004 Monomial ideals and the Scarf complex for coherent systems in reliability theory
Beatrice Giglio, Henry P. Wynn
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Ann. Statist. 32(3): 1289-1311 (June 2004). DOI: 10.1214/009053604000000373


A certain type of integer grid, called here an echelon grid, is an object found both in coherent systems whose components have a finite or countable number of levels and in algebraic geometry. If α=(α1,…,αd) is an integer vector representing the state of a system, then the corresponding algebraic object is a monomial x1α1xdαd in the indeterminates x1,…,xd. The idea is to relate a coherent system to monomial ideals, so that the so-called Scarf complex of the monomial ideal yields an inclusion–exclusion identity for the probability of failure, which uses many fewer terms than the classical identity. Moreover in the “general position” case we obtain via the Scarf complex the tube bounds given by Naiman and Wynn [J. Inequal. Pure Appl. Math. (2001) 2 1–16]. Examples are given for the binary case but the full utility is for general multistate coherent systems and a comprehensive example is given.


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Beatrice Giglio. Henry P. Wynn. "Monomial ideals and the Scarf complex for coherent systems in reliability theory." Ann. Statist. 32 (3) 1289 - 1311, June 2004.


Published: June 2004
First available in Project Euclid: 24 May 2004

zbMATH: 1105.90312
MathSciNet: MR2065206
Digital Object Identifier: 10.1214/009053604000000373

Primary: 06A06 , 90B25

Keywords: coherent systems , inclusion–exclusion , monomial ideals , multistate systems , Network reliability , Scarf complex

Rights: Copyright © 2004 Institute of Mathematical Statistics


Vol.32 • No. 3 • June 2004
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