On redundant weighted voting systems with components having stochastic arrangement increasing lifetimes

As one generalization of the k-out-of-n structure, the weighted voting system has been paid much attention during the past two decades. This paper has a further study on active redundancies allocation to weighted voting reliability systems of components having LWSAI lifetimes. For redundancies with SAI lifetimes, allocating a more reliable redundancy to a weaker and more heavily weighted component is found to produce a more reliable system in the sense of having higher reliability. Also, in the context of redundancies with identically distributed lifetimes, we show that allocating more redundancies to a weaker and more heavily weighted component produces a more reliable system. Some numerical examples are presented to illustrate the main results as well.     

Optimal redundancy allocation problems in engineering systems with dependent component lifetimes

This paper deals with the problem of allocating spare components for improving the system reliability when lifetimes of components are dependent. Two policies, called active and standby redundancies, are investigated in details. The optimal allocations are given without specific assumptions on the dependency structure among the component lifetimes. When there exist the (positive/negative) quadratic dependence and (upper/lower) orthant order among the component lifetimes, the derived results are simplified. Various illustrative examples are also given. Copyright © 2015 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.     

Modeling and analysis of system reliability using phase‐type distribution closure properties

Phase‐type distribution closure properties are utilized to devise algorithms for generating reliability functions of systems with basic structures. These structures include series, parallel, K‐out‐of‐N, and standby structures with perfect/imperfect switch. The algorithms form a method for system reliability modeling and analysis based on the relationship between the system lifetime and component lifetimes for general structures. The proposed method is suitable for functional system reliability analysis, which can produce reliability functions of systems with independent components instead of only system reliability values. Once the system reliability function is obtained, other reliability measures such as the system's hazard function and mean time to failure can be obtained efficiently using only matrix algebra. Dimensional and numerical comparisons with computerized symbolic processing are also presented to show the superiority of the proposed method.     

Fisher-Z分布次序统计量的普通多元随机序

本文研究了Fisher-Z分布次序统计量的随机比较问题.利用Beta随机变量的性质以及比例风险率模型的次序统计量随机比较的结论,获得了Fisher-Z分布次序统计量向量的普通多元随机序的比较,推广了文献中的相关结果.     

On the hazard rate and reversed hazard rate orderings in two-component series systems with active redundancies

The lifetimes of two-component series systems with two active redundancies are compared using the hazard rate and the reversed hazard rate orders. We study the problem of where to allocate the spares in a system to obtain the best configuration. We compare redundancy at component level vs. system level using the likelihood ratio order. For this problem we find conditions under which there is no hazard rate ordering between the lifetimes of the systems.     

Reliability,signature, and relative quality functions of systems under time‐homogeneous load‐sharing models

For a coherent binary system made of binary components, we consider the assumption that the components' lifetimes are distributed according to a time‐homogeneous, load‐sharing model. Such models are characterized in terms of the so‐called multivariate conditional hazard rate functions. We aim to point out some related properties of the notions of signature, relative quality functions, and reliability functions. On this purpose, we preliminarily collect all the necessary background and review some related literature. This paper concludes with a discussion, also containing some hints for future work.     

Lifetime distribution and estimation problems of consecutive-k-out-of-n:F systems

Explicit formula is given for the lifetime distribution of a consecutive-k-out-of-n:F system. It is given as a linear combination of distributions of order statistics of the lifetimes of n components. We assume that the lifetimes are independent and identically distributed. The results should make it possible to treat the parametric estimation problems based on the observations of the lifetimes of the system. In fact, we take up, as some examples, the cases where the lifetimes of the components follow the exponential, the Weibull, and the Pareto distributions, and obtain feasible estimators by moment method. In particular, it is shown that the moment estimator is quite good for the exponential case in the sense that the asymptotic efficiency is close to one.This research was partially supported by the ISM Cooperative Research Program (94-ISM-CRP-5).     

两比例失效率元件组成串联系统元件冗余与系统冗余的随机比较

针对两个比例失效率元件组成的串联系统,在热冗余的情形下,讨论了串联系统的元件冗余与系统冗余两种方案,并基于随机序的方法,对普通随机序、失效率序、反失效率序建立了元件冗余优于系统冗余的随机比较理论.     

Allocations of cold standbys to series and parallel systems with dependent components

In the context of industrial engineering, cold‐standby redundancies allocation strategy is usually adopted to improve the reliability of coherent systems. This paper investigates optimal allocation strategies of cold standbys for series and parallel systems comprised of dependent components with left/right tail weakly stochastic arrangement increasing lifetimes. For the case of heterogeneous and independent matched cold standbys, it is proved that better redundancies should be put in the nodes having weaker [better] components for series [parallel] systems. For the case of homogeneous and independent cold standbys, it is shown that more redundancies should be put in standby with weaker [better] components to enhance the reliability of series [parallel] systems. The results developed here generalize and extend those corresponding ones in the literature to the case of series and parallel systems with dependent components. Numerical examples are also presented to provide guidance for the practical use of our theoretical findings.     

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.     

Two‐level Fourier analysis of multigrid for higher‐order finite‐element discretizations of the Laplacian

In this paper, we employ local Fourier analysis (LFA) to analyze the convergence properties of multigrid methods for higher‐order finite‐element approximations to the Laplacian problem. We find that the classical LFA smoothing factor, where the coarse‐grid correction is assumed to be an ideal operator that annihilates the low‐frequency error components and leaves the high‐frequency components unchanged, fails to accurately predict the observed multigrid performance and, consequently, cannot be a reliable analysis tool to give good performance estimates of the two‐grid convergence factor. While two‐grid LFA still offers a reliable prediction, it leads to more complex symbols that are cumbersome to use to optimize parameters of the relaxation scheme, as is often needed for complex problems. For the purposes of this analytical optimization as well as to have simple predictive analysis, we propose a modification that is “between” two‐grid LFA and smoothing analysis, which yields reasonable predictions to help choose correct damping parameters for relaxation. This exploration may help us better understand multigrid performance for higher‐order finite element discretizations, including for Q₂‐Q₁ (Taylor‐Hood) elements for the Stokes equations. Finally, we present two‐grid and multigrid experiments, where the corrected parameter choice is shown to yield significant improvements in the resulting two‐grid and multigrid convergence factors.     

Aging properties of the residual life length of k‐out‐of‐n systems with independent but non‐identical components

The k‐out‐of‐n structure is a very popular type of redundancy in fault‐tolerant systems, it has founded wide applications in industrial and military systems during the past several decades. This paper will investigate the residual life length of a k‐out‐of‐n system with independent (not necessarily identical) components, given that the (n?k)th failure has occurred at time t?0. Behaviour of PF₂, IFR, DRHR, DMRL, NBU(2) and NBUC classes of life distributions are derived in terms of the monotonicity of the residual life given the time of the (n?k)th failure. Copyright © 2004 John Wiley & Sons, Ltd.     

Probabilistic mean value theorems for conditioned random variables with applications

In this paper, we propose a probabilistic analogue of the mean value theorem for conditional nonnegative random variables ordered in the hazard rate and reversed hazard rate order, upon conditioning on intervals of the form (t,∞) and [0,t]. This result is then specialized within the proportional hazards model and the proportional reversed hazards model with applications to series systems in reliability theory and to absorption random times of linear birth‐death processes. We also study the comparison of residual entropies and discuss some connections to Wasserstein and stop‐loss distances of random variables. A treatment for the additive hazard rate model is finally provided, with an application to life annuities.     

Complex dynamics of a discrete fractional‐order Leslie‐Gower predator‐prey model

A proposed discretized form of fractional‐order prey‐predator model is investigated. A sufficient condition for the solution of the discrete system to exist and to be unique is determined. Jury stability test is applied for studying stability of equilibrium points of the discretized system. Then, the effects of varying fractional order and other parameters of the systems on its dynamics are examined. The system undergoes Neimark‐Sacker and flip bifurcation under certain conditions. We observe that the model exhibits chaotic dynamics following stable states as the memory parameter α decreases and step size h increases. Theoretical results illustrate the rich dynamics and complexity of the model. Numerical simulation validates theoretical results and demonstrates the presence of rich dynamical behaviors include S‐asymptotically bounded periodic orbits, quasi‐periodicity, and chaos. The system exhibits a wide range of dynamical behaviors for fractional‐order α key parameter.     

Stochastic comparisons of residual lifetimes and inactivity times of coherent systems with dependent identically distributed components

In this paper we compare the residual lifetime of a used coherent system of age t>0$t>0$ with the lifetime of the similar coherent system made up of used components of age t. Here ‘similar’ means that the system has the same structure and the component lifetimes have the same dependence (joint reliability copula). Some comparison results are obtained for the likelihood ratio order, failure rate order, reversed failure rate order and the usual stochastic order. Similar results are reported for comparing inactivity time of a coherent system with lifetime of similar coherent system having component lifetimes same as inactivity times of failed components.     

Learning formation control for fractional‐order multiagent systems

In this paper, we use 2 iterative learning control schemes (P‐type and PI‐type) with an initial learning rule to achieve the formation control of linear fractional‐order multiagent systems. To realize the finite‐time consensus, we assume repeatable operation environments as well as a fixed but directed communication topology for the fractional‐order multiagent systems. Both P‐type and PI‐type update laws are applied to generate the control commands for each agent. It is strictly proved that all agents are driven to achieve an asymptotical consensus as the iteration number increases. Two examples are simulated to verify the effectiveness of the proposed algorithms.     

An analysis for the compressible Stokes equations by first‐order system of least‐squares finite element method

This article applies the first‐order system least‐squares (fosls) finite element method developed by Cai, Manteuffel and McCormick to the compressible Stokes equations. By introducing a new dependent velocity flux variable, we recast the compressible Stokes equations as a first‐order system. Then it is shown that the ellipticity and continuity hold for the least‐squares functionals employing the mixture of H^?1 and L², so that the fosls finite element methods yield best approximations for the velocity flux and velocity. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:689–699, 2001