## 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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### References:

 [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. [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)
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