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Journal ArticleDOI

Relative behavior of a coherent system with respect to another coherent system

01 May 2015-Vol. 56, Iss: 2, pp 519-529
TL;DR: In this paper, the remaining lifetime and remaining number of working components of two independent coherent systems with different structures, and different types of components are considered, and signature-based expressions are obtained for the distribution of these conditional random variables.
Abstract: In this paper, two independent coherent systems with different structures, and different types of components are considered. The remaining lifetime and the remaining number of working components of system I after the failure of the system II when we know that the system II fails before the system I are studied. In particular, signature-based expressions are obtained for the distribution of these conditional random variables. Illustrative examples are provided.
Citations
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01 Jan 2016
TL;DR: In this paper, the authors defined the following types of pairs of variables: 0 ≤ p < ∞ [562]. 2 [616, 407]. $229.95 [2259] and 1 ≤ p = p <∞ [548].
Abstract: (k, l) [1885]. (n− k + 1) [1753, 1623]. 0 [772]. 0 < ρ < ρ0 [2026]. 1 [772, 548]. $109.95 [2218]. $125.00 [2161]. $129.95 [2259]. 1 ≤ p <∞ [562]. 2 [616, 407]. $229.00 [2019]. 24 [1398]. 24 [1476]. 2s−k [2039]. 3 [852, 2125]. $49.95 [2161]. $59.95 [2219]. $89.95 [2217]. (R) [2002, 1962]. 1 [1276]. 2 [1417]. k [798]. A [1577, 2312, 2346]. α [858, 1840, 2242, 1465, 1271, 1945, 2327, 1926]. AR(1) [1650, 790]. AR(p) [1346]. R̄ [652]. β [668, 856, 497]. c [2332]. χ [919]. D [2135, 2268, 836, 948, 1324, 2309, 2346, 2098, 2176]. δ [1959]. e [596]. E(x⊗ xx′) [1805]. E(xx′ ⊗ xx′) [1805]. F [1901, 915, 1361, 407, 681]. G [548]. Γ [722]. H [1664]. I(1) [845]. K [2308, 1551, 1223, 1837, 368, 479, 2177, 1322, 1198, 2224, 2309, 407, 1932]. k − 2 [1781]. L [439, 1974, 1584, 1399, 1322]. L1 [895]. L2 [2195, 2363, 2039]. LP [562]. LM [1343]. LR [1343]. M [1054, 1107, 2094, 2180]. Lp [1552]. N [816, 752, 48, 1753, 1623, 548]. p [1245, 1461, 2378, 1794, 1868, 1604, 1791, 1589, 2074, 2227]. P (X < Y ) [1867, 1492, 1955]. P (Y < X) [2222, 1380]. Φ [1345]. Pr(a < x < b) [263]. q [1651]. R [1238, 1848, 616, 1932]. r − k [2082, 1480]. R/S [1996]. R = Pr(X > Y ) [2007]. r [1446]. ρ∗ [1884]. S [2151, 1105]. S [848, 790, 909]. σ [1141, 684]. SUNn,2 [2046]. t [1054, 1777, 327, 1533, 1722, 982, 856, 1031, 1780, 1559, 1094, 1391, 1342, 1680]. T 2

45 citations

Journal ArticleDOI
TL;DR: In this paper, the convolution and the order statistics of k independent random lifetimes are considered as random times, and new characterizations of the exponential distribution are established based on the concept of residual life at such random times.

11 citations

Journal ArticleDOI
TL;DR: In this article, the authors consider stochastic comparisons in the transmuted-G family with different parametrized distributions, and show that the transmutation-G model is a useful technique to construct some new distributions by adding a parameter.
Abstract: The transmuted-G model is a useful technique to construct some new distributions by adding a parameter. This paper considers stochastic comparisons in the transmuted-G family with different paramet...

5 citations

Journal ArticleDOI
TL;DR: In this paper, the authors reveal several stochastic comparisons in the Harris family with different tilt parameters and different baseline distributions with respect to the usual baseline distributions, i.e., the shift-stochastic, shift-proportional, proportional, and shifted proportional orderings.
Abstract: Harris family of distributions models lifetime of a series system when the number of components is a positive random variable. In this paper, we reveal several stochastic comparisons in the Harris family with different tilt parameters and different baseline distributions with respect to the usual stochastic, shifted stochastic, proportional stochastic and shifted proportional stochastic orderings. Such comparisons are particularly useful in lifetime optimization of reliability systems. We shall also present two bounds for a Harris family survival function conditioned on its tilt parameter which are useful when aging properties are considered. Our results generalize several previous findings in this connection.

4 citations


Cites background from "Relative behavior of a coherent sys..."

  • ...For example, recently Eryilmaz and Tutuncu (2015) investigated relative behavior of a coherent system with respect to another coherent system using stochastic orderings and Torrado (2015) investigated stochastic properties of the smallest order statistics from Weibull distributions....

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Journal ArticleDOI
TL;DR: In this article , several characterizations for exponential distribution are derived from a new relative hazard rate measure, which is closely related to the concept of remaining lifetime at a random time, which considers the random times specified by the order statistics of a sample, the convolution of random variables, and the record values of a sequence of random variable.
Abstract: In this paper, several characterizations for exponential distribution are derived from a new relative hazard rate measure. This measure is closely related to the concept of remaining lifetime at a random time. It considers the random times specified by the order statistics of a sample, the convolution of random variables, and the record values of a sequence of random variables. The concept of completeness in functional analysis provides the technical background to obtain the main results.

1 citations

References
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BookDOI
28 Jan 2005

1,825 citations

Book
17 Oct 2007
TL;DR: This paper presents a meta-analysis of the application of Signature-Based Closure, Preservation and Characterization Theorems to Network Reliability and its applications in Reliability Economics and Signature-based Analysis of System Lifetimes.
Abstract: Background on Coherent Systems.- System Signatures.- Signature-Based Closure, Preservation and Characterization Theorems.- Further Signature-Based Analysis of System Lifetimes.- Applications of Signatures to Network Reliability.- Applications of Signatures in Reliability Economics.- Summary and Discussion.

387 citations

Journal ArticleDOI
TL;DR: In this paper, the authors derived the failure rate of an arbitrary s-coherent system when the lifetimes of its components are s-independently distributed according to a common absolutely continuous distribution.
Abstract: A representation is derived for the failure rate of an arbitrary s-coherent system when the lifetimes of its components are s-independently distributed according to a common absolutely continuous distribution F. The system failure rate is written explicitly as a function of F and its failure rate. The representation is used in several examples, including an example showing that the closure theorem for k-out-of-n systems in i.i.d. IFR components proven by Barlow & Proschan cannot be extended to all s-coherent systems. The class of s-coherent systems for which such closure obtains is characterized.

373 citations

Journal ArticleDOI
TL;DR: Theoretical results for comparing coherent systems are derived for comparing systems of a given order when components are assumed to have independent and identically distributed lifetimes as mentioned in this paper, and 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.
Abstract: 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 p= (p1, … , pn), where pi is the probability that the system fails upon the occurrence of the ith 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. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 507–523, 1999

268 citations

Journal ArticleDOI
TL;DR: In this paper, it was 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, and that the vector of coefficients in this mixture representation is precisely the signature of the system defined in Samaniego, IEEE Trans Reliabil R-34 (1985) 69−72.
Abstract: 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 in Samaniego, IEEE Trans Reliabil R–34 (1985) 69–72. These mixture representations are then used to obtain new stochastic ordering properties for coherent or mixed systems of different sizes. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008

203 citations