Optimal allocation of relevations in coherent systems

2021 ◽  
Vol 58 (4) ◽  
pp. 1152-1169
Author(s):  
Rongfang Yan ◽  
Jiandong Zhang ◽  
Yiying Zhang
Keyword(s):  
Optimal Allocation ◽  
Necessary Conditions ◽  
Stochastic Order ◽  
Coherent Systems ◽  
Stochastic Comparisons ◽  
Hazard Rate Order ◽  
Usual Stochastic Order ◽  
Special Cases ◽  
Sufficient And Necessary Conditions ◽  
Allocation Strategies

AbstractIn this paper we study the allocation problem of relevations in coherent systems. The optimal allocation strategies are obtained by implementing stochastic comparisons of different policies according to the usual stochastic order and the hazard rate order. As special cases of relevations, the load-sharing and minimal repair policies are further investigated. Sufficient (and necessary) conditions are established for various stochastic orderings. Numerical examples are also presented as illustrations.

STOCHASTIC COMPARISONS OF LARGEST ORDER STATISTICS FROM MULTIPLE-OUTLIER EXPONENTIAL MODELS

2012 ◽  
Vol 26 (2) ◽  
pp. 159-182 ◽  
Author(s):  
Peng Zhao ◽  
N. Balakrishnan
Keyword(s):  
Order Statistics ◽  
Likelihood Ratio ◽  
Hazard Rate ◽  
Stochastic Order ◽  
Likelihood Ratio Order ◽  
Stochastic Comparisons ◽  
Hazard Rate Order ◽  
Reversed Hazard Rate ◽  
Exponential Models ◽  
Usual Stochastic Order

In this paper, we carry out stochastic comparisons of largest order statistics from multiple-outlier exponential models according to the likelihood ratio order (reversed hazard rate order) and the hazard rate order (usual stochastic order). It is proved, among others, that the weak majorization order between the two hazard rate vectors is equivalent to the likelihood ratio order (reversed hazard rate order) between largest order statistics, and that the p-larger order between the two hazard rate vectors is equivalent to the hazard rate order (usual stochastic order) between largest order statistics. We also extend these results to the proportional hazard rate models. The results established here strengthen and generalize some of the results known in the literature.

COMPARISONS OF SAMPLE RANGES ARISING FROM MULTIPLE-OUTLIER MODELS: IN MEMORY OF MOSHE SHAKED

2017 ◽  
Vol 33 (1) ◽  
pp. 28-49
Author(s):  
Narayanaswamy Balakrishnan ◽  
Jianbin Chen ◽  
Yiying Zhang ◽  
Peng Zhao
Keyword(s):  
Hazard Rate ◽  
Stochastic Order ◽  
Sufficient Conditions ◽  
Hazard Rates ◽  
Proportional Hazard ◽  
Stochastic Comparisons ◽  
Hazard Rate Order ◽  
Numerical Examples ◽  
Reversed Hazard Rate ◽  
Usual Stochastic Order

In this paper, we discuss the ordering properties of sample ranges arising from multiple-outlier exponential and proportional hazard rate (PHR) models. The purpose of this paper is twofold. First, sufficient conditions on the parameter vectors are provided for the reversed hazard rate order and the usual stochastic order between the sample ranges arising from multiple-outlier exponential models with common sample size. Next, stochastic comparisons are separately carried out for sample ranges arising from multiple-outlier exponential and PHR models with different sample sizes as well as different hazard rates. Some numerical examples are also presented to illustrate the results established here.

LIKELIHOOD RATIO AND HAZARD RATE ORDERINGS OF THE MAXIMA IN TWO MULTIPLE-OUTLIER GEOMETRIC SAMPLES

2012 ◽  
Vol 26 (3) ◽  
pp. 375-391 ◽  
Author(s):  
Baojun Du ◽  
Peng Zhao ◽  
N. Balakrishnan
Keyword(s):  
Likelihood Ratio ◽  
Hazard Rate ◽  
Stochastic Order ◽  
Likelihood Ratio Order ◽  
Stochastic Comparisons ◽  
Hazard Rate Order ◽  
Usual Stochastic Order ◽  
Likelihood Ratio Ordering ◽  
Special Case ◽  
Two Parameter

In this paper, we study some stochastic comparisons of the maxima in two multiple-outlier geometric samples based on the likelihood ratio order, hazard rate order, and usual stochastic order. We establish a sufficient condition on parameter vectors for the likelihood ratio ordering to hold. For the special case whenn= 2, it is proved that thep-larger order between the two parameter vectors is equivalent to the hazard rate order as well as usual stochastic order between the two maxima. Some numerical examples are presented for illustrating the established results.

scholarly journals Stochastic Comparisons of Parallel and Series Systems with Type II Half Logistic-Resilience Scale Components

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 470 ◽  
Author(s):  
Junrui Wang ◽  
Rongfang Yan ◽  
Bin Lu
Keyword(s):  
Stochastic Order ◽  
Distribution Functions ◽  
Series System ◽  
Type Ii ◽  
Stochastic Comparisons ◽  
Hazard Rate Order ◽  
Usual Stochastic Order ◽  
Resilience Scale ◽  
Series Systems ◽  
Parallel Series

This paper deals with stochastic comparisons of two parallel (series) systems with Type II half logistic-resilience scale (TIIHL-RS) distribution components with different baseline distribution functions. Under the conditions of interdependency and independency, the research shows that the system performance is better (worse) with the stronger component heterogeneity in the parallel (series) system under the usual stochastic order and the (reversed) hazard rate order.

Optimal allocation of a coherent system with statistical dependent subsystems

2021 ◽  
pp. 1-20
Author(s):  
Bin Lu ◽  
Jiandong Zhang ◽  
Rongfang Yan
Keyword(s):  
Hazard Rate ◽  
Optimal Allocation ◽  
Stochastic Order ◽  
Sufficient Conditions ◽  
Parallel Systems ◽  
Coherent System ◽  
Allocation Policy ◽  
Usual Stochastic Order ◽  
Series Systems ◽  
Parallel Series

Abstract This paper studies the optimal allocation policy of a coherent system with independent heterogeneous components and dependent subsystems, the systems are assumed to consist of two groups of components whose lifetimes follow proportional hazard (PH) or proportional reversed hazard (PRH) models. We investigate the optimal allocation strategy by finding out the number $k$ of components coming from Group A in the up-series system. First, some sufficient conditions are provided in the sense of the usual stochastic order to compare the lifetimes of two-parallel–series systems with dependent subsystems, and we obtain the hazard rate and reversed hazard rate orders when two subsystems have independent lifetimes. Second, similar results are also obtained for two-series–parallel systems under certain conditions. Finally, we generalize the corresponding results to parallel–series and series–parallel systems with multiple subsystems in the viewpoint of the minimal path and the minimal cut sets, respectively. Some numerical examples are presented to illustrate the theoretical findings.

Stochastic comparisons of interfailure times under a relevation replacement policy

2017 ◽  
Vol 54 (1) ◽  
pp. 134-145 ◽  
Author(s):  
Miguel A. Sordo ◽  
Georgios Psarrakos
Keyword(s):  
Stochastic Order ◽  
Replacement Policy ◽  
Convex Order ◽  
Stochastic Comparisons ◽  
Increasing Convex Order ◽  
Failure Times ◽  
Usual Stochastic Order ◽  
Comparison Results ◽  
Excess Wealth Order ◽  
Cumulative Residual

AbstractWe provide some results for the comparison of the failure times and interfailure times of two systems based on a replacement policy proposed by Kapodistria and Psarrakos (2012). In particular, we show that when the first failure times are ordered in terms of the dispersive order (or, the excess wealth order), then the successive interfailure times are ordered in terms of the usual stochastic order (respectively, the increasing convex order). As a consequence, we provide comparison results for the cumulative residual entropies of the systems and their dynamic versions.

scholarly journals Injectivity of linear combinations in $\mathcal{B}(\mathcal{H})$

2021 ◽  
Vol 37 ◽  
pp. 359-369
Author(s):  
Marko Kostadinov
Keyword(s):  
Linear Combination ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Combinations ◽  
Special Cases ◽  
Sufficient And Necessary Conditions ◽  
Alpha Beta ◽  
Necessary And Sufficient

The aim of this paper is to provide sufficient and necessary conditions under which the linear combination $\alpha A + \beta B$, for given operators $A,B \in {\cal B}({\cal H})$ and $\alpha, \beta \in \mathbb{C}\setminus \lbrace 0 \rbrace$, is injective. Using these results, necessary and sufficient conditions for left (right) invertibility are given. Some special cases will be studied as well.

Embedding properties of weighted Besov-type spaces

2015 ◽  
Vol 13 (05) ◽  
pp. 507-553 ◽  
Author(s):  
Wen Yuan ◽  
Dorothee D. Haroske ◽  
Leszek Skrzypczak ◽  
Dachun Yang
Keyword(s):  
Besov Spaces ◽  
Necessary Conditions ◽  
Muckenhoupt Weight ◽  
Exponential Weights ◽  
Special Cases ◽  
Type Spaces ◽  
Sufficient And Necessary Conditions ◽  
Weighted Besov Spaces ◽  
Dyadic Cubes ◽  
Besov Type Spaces

In this paper, we consider the embeddings of weighted Besov spaces [Formula: see text] into Besov-type spaces [Formula: see text] with w being a (local) Muckenhoupt weight, and give sufficient and necessary conditions on the continuity and the compactness of these embeddings. As special cases, we characterize the continuity and the compactness of embeddings in case of some polynomial or exponential weights. The proofs of these conclusions strongly depend on the geometric properties of dyadic cubes.

scholarly journals Erratum to: Mixture representations of residual lifetimes of used systems

2015 ◽  
Vol 52 (04) ◽  
pp. 1183-1186 ◽  
Author(s):  
Maria Kamińska-Zabierowska ◽  
Jorge Navarro
Keyword(s):  
Likelihood Ratio ◽  
Hazard Rate ◽  
Stochastic Order ◽  
Coherent Systems ◽  
Usual Stochastic Order ◽  
Residual Lifetimes

We have found a mistake in the proofs of Navarro (2008, Theorem 2.3(b) and 2.3(c)) due to misapplication of properties of hazard rate and likelihood ratio orders. In this paper we show with an example that the stated results do not hold. This example is interesting since it proves some unexpected properties for these orderings under the formation of coherent systems. The result stated in Navarro (2008, Theorem 2.3(a)) for the usual stochastic order is correct.

scholarly journals Optimal Allocation of Active Redundancies to k-out-of-n Systems with Heterogeneous Components

2010 ◽  
Vol 47 (1) ◽  
pp. 254-263 ◽  
Author(s):  
Xiaohu Li ◽  
Weiyong Ding
Keyword(s):  
Optimal Policy ◽  
Optimal Allocation ◽  
Stochastic Order ◽  
Independent Components ◽  
Usual Stochastic Order

In this note we deal with the allocation of independent and identical active redundancies to a k-out-of-n system with the usual stochastic order among its independent components. The optimal policy is proved both to assign more redundancies to the weaker component and to majorize all other policies. This improves the corresponding one in Hu and Wang (2009) and serves as a nice supplement to that in Misra, Dhariyal and Gupta (2009) as well.

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