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Dynamic signatures and their use in comparing the reliability of new and used systems. (English) Zbl 1182.90036

Summary: The signature of a system with independent and identically distributed (i.i.d.) component lifetimes is a vector whose \(i\)th element is the probability that the \(i\)th component failure is fatal to the system. System signatures have been found to be quite useful tools in the study and comparison of engineered systems. In this article, the theory of system signatures is extended to versions of signatures applicable in dynamic reliability settings. It is shown that, when a working used system is inspected at time \(t\) and it is noted that precisely \(k\) failures have occurred, the vector \(\mathbf s \in[0,1]^{n-k}\) whose \(j\)th element is the probability that the \((k + j)\)th component failure is fatal to the system, for \(j = 1,2,2026;,n - k\), is a distribution-free measure of the design of the residual system. Next, known representation and preservation theorems for system signatures are generalized to dynamic versions. Two additional applications of dynamic signatures are studied in detail. The well-known “new better than used” (NBU) property of aging systems is extended to a uniform (UNBU) version, which compares systems when new and when used, conditional on the known number of failures. Sufficient conditions are given for a system to have the UNBU property. The application of dynamic signatures to the engineering practice of “burn in” is also treated. Specifically, we consider the comparison of new systems with working used systems burned-in to a given ordered component failure time. In a reliability economics framework, we illustrate how one might compare a new system to one successfully burned-in to the \(k\)th component failure, and we identify circumstances in which burn-in is inferior (or is superior) to the fielding of a new system.

MSC:

90B25 Reliability, availability, maintenance, inspection in operations research
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[1] Bairamov, On the residual lifelengths of the remaining components in an n - k + 1 out of n system, Stat Probab Lett 78 pp 945– (2008) · Zbl 1141.62081
[2] Barlow, Statistical theory of reliability and life testing (1981)
[3] Block, Advances in distribution theory, order statistics and inference pp 279– (2006)
[4] Block, Burn-in, Stat Sci 12 pp 1– (1997)
[5] Block, Burn-in and mixed populations, J Appl Probab 30 pp 692– (1993) · Zbl 0781.60073
[6] Block, Some results on burn-in, Stat Sin 4 pp 525– (1994) · Zbl 0823.62080
[7] Boland, Mathematical reliability: An expository perspective pp 1– (2004)
[8] Dugas, On optimal system designs in reliability-economics frameworks, Naval Res Logist 54 pp 568– (2007) · Zbl 1143.90324
[9] Kochar, The ”signature” of a coherent system and its application to comparison among systems, Naval Res Logist 46 pp 507– (1999) · Zbl 0948.90067
[10] Harter, CRC handbook of tables for the use of order statistics in estimation (1996) · Zbl 0870.62002
[11] Navarro, Mean residual lifetimes of consecutive-k-out-of-n systems, J Appl Probab 44 pp 82– (2007) · Zbl 1135.62084
[12] Navarro, Properties of coherent systems with dependent components, Commun Stat Theory Methods 36 pp 175– (2007) · Zbl 1121.60015
[13] Navarro, Mixture representations of residual lifetimes of used systems, J Appl Probab 45 pp 1097– (2008) · Zbl 1155.60305
[14] Navarro, On the application and extension of system signatures to problems in engineering reliability, Naval Res Logist 55 pp 313– (2008)
[15] Navarro, Hazard rate ordering of order statistics and systems, J Appl Probab 43 pp 391– (2006) · Zbl 1111.62098
[16] Samaniego, On closure of the IFR class under formation of coherent systems, IEEE Trans Reliabil R-34 pp 69– (1985) · Zbl 0585.62169
[17] F. J. Samaniego, On the comparison of engineered systems of different sizes, in: Proceedings of the 12th Annual Army Conference on Applied Statistics, Aberdeen Proving Ground, Army Research Laboratory, 2006.
[18] Samaniego, International Series in Operations Research and Management Science 110 (2007)
[19] Shaked, Stochastic orders (2007)
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.