On allocation of redundancies in two-component series systems

Allocation of active [standby] redundancies in a system is a topic of great interest in reliability engineering and system safety because optimal configurations can significantly increase the reliability of a system. In this paper, we study the problem of allocating two exponentially distributed active [standby] redundancies in a two-component series system using the tools of stochastic ordering. We establish two interesting results on likelihood ratio ordering which have no restriction on the parameters.     

Stochastic comparison of residual lifetime mixture models

In this paper, we consider stochastic comparisons of mixtures ($F^1$ and $G^1$) of residual life distributions ($F_{\theta }$ and $G_{\theta }$, $\theta >0$) arising out of different baseline distributions ($F$ and $G$) and different mixing distributions ($H_1$ and $H_2$). Under certain conditions on $F$, $G$, $H_1$ and $H_2$, we establish that the distributions $F^1$ and $G^1$ are stochastically ordered. Further, we make stochastic comparison of mixture distribution $F^1$ with the distribution of residual lifetime at random time ${\Theta }_1$.     

Some stochastic comparisons of conditional coherent systems

This paper investigates coherent systems with independent and identical components. Stochastic comparison on the residual life and the inactivity time of two systems with stochastically ordered signatures is conducted. Copyright © 2008 John Wiley & Sons, Ltd.     

Optimal allocation of redundancies in series systems

It is of great interest for the problem of how to allocate redundancies in a system so as to optimize the system performance in reliability engineering and system security. In this paper, we consider the problems of optimal allocation of both active and standby redundancies in series systems in the sense of various stochastic orderings. For the case of allocating one redundancy to a series system with two exponential components, we establish two likelihood ratio order results for active redundancy case and standby redundancy case, respectively. We also discuss the case of allocating K active redundancies to a series system and establish some new results. The results developed here strengthen and generalize some of the existing results in the literature. Specifically, we give an answer to an open problem mentioned in Hu and Wang [T. Hu, Y. Wang, Optimal allocation of active redundancies in r-out-of-n systems, Journal of Statistical Planning and Inference 139 (2009) 3733–3737]. Numerical examples are provided to illustrate the theoretic results established here.     

A note on likelihood ratio ordering of order statistics from two samples

The purpose of this paper is to establish the stochastic comparisons of order statistics from two samples in the sense of likelihood ratio order. We strengthen and complement some results in Zhao and Balakrishnan (2012) and Ding et al. (2013).     

On stochastic comparisons of series systems with heterogeneous dependent and independent location-scale family distributed components

This paper carries out comparisons of heterogeneous series systems with location-scale family distributed components It is shown that the systems with dependent components in series sharing Archimedean copula with more dispersion in the location or scale parameters result in better performance in the sense of the usual stochastic order. Moreover, if the components are independently distributed, it is possible to obtain more generalized results as compared to the dependent set-up.     

Reliability and non-reliability studies of Poisson variables in series and parallel systems

In this paper, we have derived the distribution of the minimum and maximum of two independent Poisson random variables. A useful procedure for computing the probabilities is given and a total of four numerical examples are presented. Of these four examples, the first two are on the generated data and the other two are on the Champion League Soccer data in order to illustrate the model which is considered here. The hazard rate and the reversed hazard rate, of the minimum and maximum of two independent discrete random variables, are also obtained and their monotonicity is investigated. The results for the Poisson-distributed variables are obtained as special cases.     

Stochastic comparisons of multivariate mixture models

In this paper we consider sufficient conditions in order to stochastically compare random vectors of multivariate mixture models. In particular we consider stochastic and convex orders, the likelihood ratio order, and the hazard rate and mean residual life dynamic orders. Applications to proportional hazard models and mixture models in risk theory are also given.     

Some characterization results on generalized cumulative residual entropy measure

The cumulative residual entropy (CRE) has been found to be a new measure of information that parallels Shannon entropy, refer to Rao et al. (2004). In this paper we study a generalized cumulative residual information measure based on Verma’s entropy function and a dynamic version of it. The exponential, Pareto and finite range distributions, which are commonly used in reliability modeling, have been characterized using this generalized measure.     

Stochastic comparisons of series and parallel systems with generalized linear failure rate components

In this paper, we discuss stochastic comparisons of series and parallel systems comprising independent and heterogeneous generalized linear failure rate components. When a system with possibly different shape and scale parameters has its matrix of parameters changing to another matrix of parameters in a certain mathematical sense, we examine the usual stochastic order of the smallest and largest order statistics under such a setup. Copyright © 2017 John Wiley & Sons, Ltd.     

Mean residual life order of convolutions of heterogeneous exponential random variables

In this paper, we study convolutions of heterogeneous exponential random variables with respect to the mean residual life order. By introducing a new partial order (reciprocal majorization order), we prove that this order between two parameter vectors implies the mean residual life order between convolutions of two heterogeneous exponential samples. For the 2-dimensional case, it is shown that there exists a stronger equivalence. We discuss, in particular, the case when one convolution involves identically distributed variables, and show in this case that the mean residual life order is actually associated with the harmonic mean of parameters. Finally, we derive the “best gamma bounds” for the mean residual life function of any convolution of exponential distributions under this framework.     

Some new results involving general standby systems

This article presents a stochastic comparison on the total lifetime of the general standby system and a discussion of the optimal allocation of a general standby component in a series system with two independent components. Several examples are also presented to justify the main results, which provide nice generalizations of some existing conclusions in the literature. Copyright © 2009 John Wiley & Sons, Ltd.     

Stochastic comparisons of lifetimes of series and parallel systems with dependent and heterogeneous components

This work considers stochastic comparisons of lifetimes of series and parallel systems with dependent and heterogeneous components having lifetimes following the proportional odds (PO) model. The joint distribution of component lifetimes is modeled by Archimedean survival copula. We discuss some potential applications of our findings in system reliability and actuarial science.     

New results on comparisons of parallel systems with heterogeneous gamma components

This paper discusses ordering properties of lifetimes of parallel systems with two independent heterogeneous gamma components in terms of dispersive and star orders. It is proved, among others, that the p-larger order between the two scale vectors implies the dispersive order and the star order between lifetimes of two parallel systems. Another sufficient condition is also provided for the star order between the lifetimes of two parallel systems. The results established here extend some of the known results in the literature.     

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??In this paper, we compare the smallest order statistics arising from multiple-outlier models when the numbers of independent and identically distributed random variables are different. Let and denote the smallest order statistics among, and, respectively, whereand. We then prove that $ and are ordered in terms of the usual stochastic order, hazard rate order and likelihood ratio order under the majorization relationship between and.     

Comparisons and bounds for expected lifetimes of reliability systems

Sharp bounds on expectations of lifetimes of coherent and mixed systems composed of elements with independent and either identically or non-identically distributed lifetimes are expressed in terms of expected lifetimes of components. Similar evaluations are concluded for the respective mean residual lifetimes. In the IID case, improved inequalities dependent on a concentration parameter connected to the Gini dispersion index are obtained. The results can be used to compare systems with component lifetimes ordered in the convex ordering. In the INID case, some refined bounds are derived in terms of the expected lifetimes of series systems of smaller sizes, and the expected lifetime of single unit for the equivalent systems with IID components. The latter can be further simplified in the case of weak Schur-concavity and Schur-convexity of the system generalized domination polynomial.     

Stochastic comparisons of series and parallel systems with randomized independent components

Consider a series or parallel system of independent components and assume that the components are randomly chosen from two different batches, with the components of the first batch being more reliable than those of the second. In this note it is shown that the reliability of the system increases, in usual stochastic order sense, as the random number of components chosen from the first batch increases in increasing convex order. As a consequence, we establish a result analogous to the Parrondo’s paradox, which shows that randomness in the number of components extracted from the two batches improves the reliability of the series system.