Equivalency in joint signatures for binary/multi-state systems of different sizes

2021 ◽  
pp. 1-28
Author(s):  
He Yi ◽  
Narayanaswamy Balakrishnan ◽  
Xiang Li
Keyword(s):  
Order Statistics ◽  
Binary State ◽  
Equivalent Systems ◽  
Transformation Formulas ◽  
Mixed Systems ◽  
Theoretical Results ◽  
Independent And Identically Distributed

The joint signatures of binary-state and multi-state (semi-coherent or mixed) systems with i.i.d. (independent and identically distributed) binary-state components are considered in this work. For the comparison of pairs of binary-state systems of different sizes, transformation formulas of their joint signatures are derived by using the concept of equivalent systems and a generalized triangle rule for order statistics. Similarly, for facilitating the comparison of pairs of multi-state systems of different sizes, transformation formulas of their multi-state joint signatures are also derived. Some examples are finally presented to illustrate and to verify the theoretical results established here.

On the equivalence of systems of different sizes, with applications to system comparisons

2016 ◽  
Vol 48 (2) ◽  
pp. 332-348 ◽  
Author(s):  
Bo H. Lindqvist ◽  
Francisco J. Samaniego ◽  
Arne B. Huseby
Keyword(s):  
Optimization Problem ◽  
Coherent System ◽  
Sufficient Condition ◽  
Equivalent System ◽  
Coherent Systems ◽  
Two Systems ◽  
Equivalent Systems ◽  
Engineered Systems ◽  
Signature Vector ◽  
Mixed Systems

Abstract The signature of a coherent system is a useful tool in the study and comparison of lifetimes of engineered systems. In order to compare two systems of different sizes with respect to their signatures, the smaller system needs to be represented by an equivalent system of the same size as the larger system. In the paper we show how to construct equivalent systems by adding irrelevant components to the smaller system. This leads to simpler proofs of some current key results, and throws new light on the interpretation of mixed systems. We also present a sufficient condition for equivalence of systems of different sizes when restricting to coherent systems. In cases where for a given system there is no equivalent system of smaller size, we characterize the class of lower-sized systems with a signature vector which stochastically dominates the signature of the larger system. This setup is applied to an optimization problem in reliability economics.

scholarly journals Mixture Representations of Inactivity Times of Conditional Coherent Systems and their Applications

2010 ◽  
Vol 47 (03) ◽  
pp. 876-885 ◽  
Author(s):  
Zhengcheng Zhang
Keyword(s):  
Order Statistics ◽  
Reliability Function ◽  
Coherent System ◽  
Coherent Systems ◽  
Distributed Components ◽  
Inactivity Time ◽  
Independent And Identically Distributed

In this paper we obtain several mixture representations of the reliability function of the inactivity time of a coherent system under the condition that the system has failed at time t (> 0) in terms of the reliability functions of inactivity times of order statistics. Some ordering properties of the inactivity times of coherent systems with independent and identically distributed components are obtained, based on the stochastically ordered coefficient vectors between systems.

scholarly journals On the Skewness of Order Statistics in Multiple-Outlier Models

2011 ◽  
Vol 48 (01) ◽  
pp. 271-284 ◽  
Author(s):  
Subhash Kochar ◽  
Maochao Xu
Keyword(s):  
Order Statistics ◽  
Open Problem ◽  
Parallel System ◽  
Exponential Component ◽  
Convex Transform Order ◽  
Independent And Identically Distributed

Kochar and Xu (2009) proved that a parallel system with heterogeneous exponential component lifetimes is more skewed (according to the convex transform order) than the system with independent and identically distributed exponential components. In this paper we extend this study to the generalk-out-of-nsystems for the case when there are only two types of component in the system. An open problem proposed in Pǎltǎnea (2008) is partially solved.

A Note on the Mixture Representation of the Conditional Residual Lifetime of a Coherent System

2013 ◽  
Vol 50 (02) ◽  
pp. 475-485 ◽  
Author(s):  
Xiuying Feng ◽  
Shuhong Zhang ◽  
Xiaohu Li
Keyword(s):  
Order Statistics ◽  
Stochastic Ordering ◽  
Reliability Function ◽  
Residual Lifetime ◽  
Coherent System ◽  
Distributed Components ◽  
Residual Lifetimes ◽  
Independent And Identically Distributed

This paper builds a mixture representation of the reliability function of the conditional residual lifetime of a coherent system in terms of the reliability functions of conditional residual lifetimes of order statistics. Some stochastic ordering properties for the conditional residual lifetime of a coherent system with independent and identically distributed components are obtained, based on the stochastically ordered coefficient vectors.

scholarly journals Bounds on Variances of Lifetimes of Coherent and Mixed Systems

2009 ◽  
Vol 46 (3) ◽  
pp. 894-908 ◽  
Author(s):  
Krzysztof Jasiński ◽  
Jorge Navarro ◽  
Tomasz Rychlik
Keyword(s):  
Upper Bounds ◽  
Special Cases ◽  
Reliability Systems ◽  
Mixed Systems ◽  
Independent And Identically Distributed

We consider coherent and mixed reliability systems composed of elements with independent and identically distributed lifetimes. We present upper bounds on variances of system lifetimes, expressed in terms of variances of single components. We also discuss attainability conditions and some special cases and examples.

scholarly journals ON THE COMPARISON OF PERFORMANCE-PER-COST FOR COHERENT AND MIXED SYSTEMS

2020 ◽  
pp. 1-18
Author(s):  
Bo H. Lindqvist ◽  
Francisco J. Samaniego ◽  
Nana Wang
Keyword(s):  
Optimal System ◽  
Naval Research ◽  
Coherent Systems ◽  
Comparison Result ◽  
Stochastic Orderings ◽  
Cost Criterion ◽  
Naval Research Logistics ◽  
System Designs ◽  
Mixed Systems ◽  
Independent And Identically Distributed

The present paper is concerned with reliability economics, considering a certain performance-per-cost criterion for coherent and mixed systems, as introduced in [Dugas, M.R. & Samaniego, F.J. (2007). On optimal system designs in reliability-economics frameworks. Naval Research Logistics 54, 568–582]. We first present a new comparison result for performance-per-cost of systems with independent and identically distributed component lifetimes under certain stochastic orderings. We then consider optimization of the performance-per-cost criterion, first reconsidering and refining results from the above cited paper, and then considering mixtures of given subsets of coherent systems.

scholarly journals Mixture Representations of Inactivity Times of Conditional Coherent Systems and their Applications

2010 ◽  
Vol 47 (3) ◽  
pp. 876-885 ◽  
Author(s):  
Zhengcheng Zhang
Keyword(s):  
Order Statistics ◽  
Reliability Function ◽  
Coherent System ◽  
Coherent Systems ◽  
Distributed Components ◽  
Inactivity Time ◽  
Independent And Identically Distributed

In this paper we obtain several mixture representations of the reliability function of the inactivity time of a coherent system under the condition that the system has failed at time t (> 0) in terms of the reliability functions of inactivity times of order statistics. Some ordering properties of the inactivity times of coherent systems with independent and identically distributed components are obtained, based on the stochastically ordered coefficient vectors between systems.

scholarly journals Exact laws for sums of ratios of order statistics from the Pareto distribution

2006 ◽  
Vol 4 (1) ◽  
pp. 1-4 ◽  
Author(s):  
André Adler
Keyword(s):  
Order Statistics ◽  
Pareto Distribution ◽  
Random Variables ◽  
Weighted Sums ◽  
Laws Of Large Numbers ◽  
Large Numbers ◽  
Independent And Identically Distributed

AbstractConsider independent and identically distributed random variables {X nk, 1 ≤ k ≤ m, n ≤ 1} from the Pareto distribution. We select two order statistics from each row, X n(i) ≤ X n(j), for 1 ≤ i < j ≤ = m. Then we test to see whether or not Laws of Large Numbers with nonzero limits exist for weighted sums of the random variables R ij = X n(j)/X n(i).

Reliability analysis of a multi-state system with identical units having two dependent components

2020 ◽  
pp. 1748006X2095747
Author(s):  
Funda Iscioglu ◽  
Aysegul Erem
Keyword(s):  
Performance Evaluation ◽  
Reliability Analysis ◽  
Order Statistics ◽  
Performance Measures ◽  
Dynamic Performance ◽  
State System ◽  
Binary State ◽  
Degradation Rates ◽  
The Mean ◽  
Bivariate Order Statistics

The performance evaluation of a system having n identical units, each of which consists of two components has been successfully discussed in binary-state reliability analysis. In this paper, we study the performance evaluation of a multi-state system based on bivariate order statistics. The multi-state system consists of n independent and identical units, each having two components. The components of each unit are assumed to be s-dependent. However, the units work s-independently with each other. The system and each component of each unit having three performance levels “0 (failure), 1 (partially working) and 2 (completely working)” are considered. The degradation of the components follows Markov Process and also Farlie-Gumbel-Morgenstern distribution is used to model the s-dependence of the components. The reliability analysis of a multi-state k-out-of- n system are evaluated under the assumptions. Some dynamic performance measures for the system such as the mean residual and mean past lifetime functions based on bivariate order statistics are also evaluated. The performance of the system is especially examined for different values of s-dependence parameter, the degradation rates and different number of units for the system. The results are supported with some numerical examples and graphical representations.

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