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The “signature” of a coherent system and its application to comparisons among systems. (English) Zbl 0948.90067
Summary: Various methods and criteria for comparing coherent systems are discussed. Theoretical results are derived for comparing systems of a given order when components are assumed to have independent and identically distributed lifetimes. All comparisons rely on the representation of a system’s lifetime distribution as a function of the system’s “signature,” that is, as a function of the vector \({\mathbf p}= (p_1,\dots,p)\), where \(p_i\) is the probability that the system fails upon the occurrence of the \(i\)th component failure. Sufficient conditions are provided for the lifetime of one system to be larger than that of another system in three different senses: stochastic ordering, hazard rate ordering, and likelihood ratio ordering. Further, a new preservation theorem for hazard rate ordering is established. In the final section, the notion of system signature is used to examine a recently published conjecture regarding componentwise and systemwise redundancy.

MSC:
90B35 Deterministic scheduling theory in operations research
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[1] and Statistical theory of reliability and life testing: Probability models, To begin with, Silver Springs, MD, 1981.
[2] and ?Comparing coherent systems,? Inequalities in statistics and probability, (Editor), IMS Lecture Notes, Monograph Series 5, 1984, pp. 187-192. · doi:10.1214/lnms/1215465646
[3] Boland, IEEE Trans Reliab 44 pp 614– (1995)
[4] Boland, J Appl Probab 31 pp 180– (1994)
[5] Joag-dev, Stat Probab Lett 22 pp 111– (1995)
[6] Total positivity, Stanford University Press, Stanford, CA, 1968.
[7] and Comparisons of series parallel (parallel-series) systems using Schur function theory, TR #M-778, Florida State University, Department of Statistics, Tallahassee, 1988.
[8] and Introduction to the theory of nonparametric statistics, Wiley, New York, 1991.
[9] Samaniego, IEEE Trans Reliab R-34 pp 69– (1985)
[10] and Stochastic orders and their applications, Academic, San Diego, CA, 1994.
[11] Singh, J Appl Probab 34 pp 283– (1997)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.