scholarly journals Increasing Mean Inactivity Time Ordering: A Quantile Approach

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
M. Kayid ◽  
S. Izadkhah ◽  
A. Alfifi
Keyword(s):  
Stochastic Order ◽  
Sufficient Conditions ◽  
Parallel Systems ◽  
The Other ◽  
Stochastic Orders ◽  
General Family ◽  
Order Relations ◽  
Inactivity Time ◽  
Time Order ◽  
Mean Inactivity Time

We study further the quantile mean inactivity time order. Relations between the proposed stochastic order and the other transform stochastic orders are obtained. Besides, sufficient conditions for the stochastic order are provided. Then, preservation of the order under monotone transformations, series, and parallel systems and mixtures of a general family of semiparametric distributions is studied. Examples are also given to illustrate the results.

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.

scholarly journals Stochastic Order Relations Among Parallel Systems from Weibull Distributions

2015 ◽  
Vol 52 (01) ◽  
pp. 102-116 ◽  
Author(s):  
Nuria Torrado ◽  
Subhash C. Kochar
Keyword(s):  
Likelihood Ratio ◽  
Hazard Rate ◽  
Stochastic Order ◽  
Random Variables ◽  
Parallel Systems ◽  
Parallel System ◽  
Weibull Distributions ◽  
Scale Parameters ◽  
Order Relations ◽  
Stochastic Order Relations

Let X λ1 , X λ2 , …, X λ n be independent Weibull random variables with X λ i ∼ W(α, λ i ), where λ i > 0 for i = 1, …, n. Let X n:n λ denote the lifetime of the parallel system formed from X λ1 , X λ2 , …, X λ n . We investigate the effect of the changes in the scale parameters (λ1, …, λ n ) on the magnitude of X n:n λ according to reverse hazard rate and likelihood ratio orderings.

CHARACTERIZATIONS OF THE RHR AND MIT ORDERINGS AND THE DRHR AND IMIT CLASSES OF LIFE DISTRIBUTIONS

2005 ◽  
Vol 19 (4) ◽  
pp. 447-461 ◽  
Author(s):  
I. A. Ahmad ◽  
M. Kayid
Keyword(s):  
Hazard Rate ◽  
Stochastic Comparison ◽  
Hazard Rate Order ◽  
Reversed Hazard Rate ◽  
Inactivity Time ◽  
Time Order ◽  
Mean Inactivity Time ◽  
Life Distributions ◽  
The Mean ◽  
Decreasing Reversed Hazard Rate

Two well-known orders that have been introduced and studied in reliability theory are defined via stochastic comparison of inactivity time: the reversed hazard rate order and the mean inactivity time order. In this article, some characterization results of those orders are given. We prove that, under suitable conditions, the reversed hazard rate order is equivalent to the mean inactivity time order. We also provide new characterizations of the decreasing reversed hazard rate (increasing mean inactivity time) classes based on variability orderings of the inactivity time of k-out-of-n system given that the time of the (n − k + 1)st failure occurs at or sometimes before time t ≥ 0. Similar conclusions based on the inactivity time of the component that fails first are presented as well. Finally, some useful inequalities and relations for weighted distributions related to reversed hazard rate (mean inactivity time) functions are obtained.

scholarly journals Reliability Properties of Residual Life Time and Inactivity Time of Series and Parallel System

2012 ◽  
Vol 8 (1) ◽  
pp. 5-16 ◽  
Author(s):  
Nitin Gupta ◽  
Neeraj Gandotra ◽  
Rakesh Bajaj
Keyword(s):  
Residual Life ◽  
Sufficient Conditions ◽  
Parallel Systems ◽  
Life Time ◽  
Parallel System ◽  
Time Data ◽  
Stochastic Comparisons ◽  
Aging Properties ◽  
Inactivity Time ◽  
In Series

Reliability Properties of Residual Life Time and Inactivity Time of Series and Parallel SystemThe concepts of residual life time and inactivity time are extensively used in reliability theory for modeling life time data. In this paper we prove some new results on stochastic comparisons of residual life time and inactivity time in series and parallel systems. These results are in addition to the existing results of Li & Zhang (2003) and Li & Lu (2003). We also present sufficient conditions for aging properties of the residual life time and inactivity life time of series and parallel systems. Some examples from Weibull and Gompertz distributions are provided to support the results as well.

Development on the mean inactivity time order with applications

2017 ◽  
Vol 45 (5) ◽  
pp. 525-529 ◽  
Author(s):  
M. Kayid ◽  
S. Izadkhah ◽  
S. Alshami
Keyword(s):  
Inactivity Time ◽  
Time Order ◽  
Mean Inactivity Time ◽  
The Mean

scholarly journals Stochastic Order Relations Among Parallel Systems from Weibull Distributions

2015 ◽  
Vol 52 (1) ◽  
pp. 102-116 ◽  
Author(s):  
Nuria Torrado ◽  
Subhash C. Kochar
Keyword(s):  
Likelihood Ratio ◽  
Hazard Rate ◽  
Stochastic Order ◽  
Random Variables ◽  
Parallel Systems ◽  
Parallel System ◽  
Weibull Distributions ◽  
Scale Parameters ◽  
Order Relations ◽  
Stochastic Order Relations

Let Xλ1, Xλ2, …, Xλn be independent Weibull random variables with Xλi ∼ W(α, λi), where λi > 0 for i = 1, …, n. Let Xn:nλ denote the lifetime of the parallel system formed from Xλ1, Xλ2, …, Xλn. We investigate the effect of the changes in the scale parameters (λ1, …, λn) on the magnitude of Xn:nλ according to reverse hazard rate and likelihood ratio orderings.

scholarly journals Stochastic Order Relations Among Parallel Systems from Weibull Distributions

2015 ◽  
Vol 52 (01) ◽  
pp. 102-116
Author(s):  
Nuria Torrado ◽  
Subhash C. Kochar
Keyword(s):  
Likelihood Ratio ◽  
Hazard Rate ◽  
Stochastic Order ◽  
Random Variables ◽  
Parallel Systems ◽  
Parallel System ◽  
Weibull Distributions ◽  
Scale Parameters ◽  
Order Relations ◽  
Stochastic Order Relations

Let X λ1 , X λ2 , …, X λ n be independent Weibull random variables with X λ i ∼ W(α, λ i ), where λ i > 0 for i = 1, …, n. Let X n:n λ denote the lifetime of the parallel system formed from X λ1 , X λ2 , …, X λ n . We investigate the effect of the changes in the scale parameters (λ1, …, λ n ) on the magnitude of X n:n λ according to reverse hazard rate and likelihood ratio orderings.

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