zbMATH — the first resource for mathematics

On the application and extension of system signatures in engineering reliability. (English) Zbl 1153.90386
Summary: Following a review of the basic ideas in structural reliability, including signature-based representation and preservation theorems for systems whose components have independent and identically distributed (i.i.d.) lifetimes, extensions that apply to the comparison of coherent systems of different sizes, and stochastic mixtures of them, are obtained. It is then shown that these results may be extended to vectors of exchangeable random lifetimes. In particular, for arbitrary systems of sizes $$m < n$$ with exchangeable component lifetimes, it is shown that the distribution of an $$m$$-component system’s lifetime can be written as a mixture of the distributions of $$k$$-out-of-$$n$$ systems. When the system has $$n$$ components, the vector of coefficients in this mixture representation is precisely the signature of the system defined by the second author [IEEE Trans. Reliabil. R-34, 69–72 (1985)]. These mixture representations are then used to obtain new stochastic ordering properties for coherent or mixed systems of different sizes.

MSC:
 90B25 Reliability, availability, maintenance, inspection in operations research
Full Text:
References:
 [1] Balakrishnan, Rev Mat Complut 20 pp 7– (2007) · Zbl 1148.62031 · doi:10.5209/rev_REMA.2007.v20.n1.16528 [2] and , Statistical theory of reliability and life testing, Holt, Rinehart and Winston, New York, 1975 · Zbl 0379.62080 [3] and , ”The signature of a coherent system and its applications in reliability,” Mathematical reliability: an expository perspective, , and (Editors), Kluwer Publishers, Boston, 2004, pp. 1–29 [4] , and , ”Linking dominations and signatures in network reliability theory,” Mathematical and statistical methods in reliability, and (Editors), World Scientific, Singapore, 2003, pp. 89–103 · doi:10.1142/9789812795250_0007 [5] Cole, Ann Math Statist 22 pp 308– (1951) [6] David, Ann Math Statist 39 pp 272– (1968) [7] and , Order Statistics, 3rd ed., Wiley, Hoboken, New Jersey, 2003. · Zbl 1053.62060 · doi:10.1002/0471722162 [8] Dugas, Naval Res Logist 54 pp 568– (2007) [9] Esary, SIAM J Appl Math 18 pp 810– (1970) [10] ”The role of exchangeability in the theory of order statistics,” Exchangeability in probability and statistics, and (Editors), North-Holland, The Netherlands, 1982, pp. 75–86 [11] , and , ”On generalized orderings and ageing properties with their implications,” System and Bayesian Reliability, and (Editors), World Scientific, River Edge, NJ, 2001, pp. 199–228 [12] Kochar, Naval Res Logist 46 pp 507– (1999) [13] , and , Continuous multivariate distributions, vol. 1, 2nd ed., Wiley, New York, 2000. · doi:10.1002/0471722065 [14] Navarro, Naval Res Logist 58 pp 820– (2007) [15] Navarro, J Statist Plann Inference 138 pp 1242– (2008) [16] Navarro, Statist Probab Lett 72 pp 179– (2005) [17] Navarro, Commun Statist Theory Methods 36 pp 175– (2007) [18] Navarro, Statist Papers 49 pp 177– (2008) [19] Navarro, J Multivariate Anal 98 pp 102– (2007) [20] Navarro, Commun Statist Theory Methods 36 pp 1273– (2007) [21] Navarro, J Appl Probab 43 pp 391– (2006) [22] Samaniego, IEEE Trans Reliabil R-34 pp 69– (1985) [23] On the comparison of engineered systems of different sizes, Proc 12th ACAS, 2006. [24] System signatures and their applications in engineering reliability, Internat Ser Oper Res and Management Sci, vol. 110, Springer, New York, 2007. · Zbl 1154.62075 [25] Satyanarayana, IEEE Trans Reliabil R-27 pp 82– (1978) [26] Sillitto, Biometrika 51 pp 259– (1964) [27] and , Stochastic orders, Springer, New York, 2007. · Zbl 1111.62016 · doi:10.1007/978-0-387-34675-5 [28] Shaked, J Am Statist Assoc 98 pp 693– (2003)
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.