On allocating redundancies to k‐out‐of‐ n reliability systems

For two components in series and one redundancy with their lifetimes following the proportional hazard models, we build the likelihood ratio order and the hazard rate order for lifetimes of the redundant systems. Also, for k ‐out‐of‐ n system with components’ lifetimes having the arrangement increasing joint density and the redundancies having identically distributed lifetimes, allocating more redundancies to weaker components is shown to help improve the system's reliability. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

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.     

On active redundancy allocation for coherent systems—from the viewpoint of minimal cut decomposition

This paper deals with coherent systems with one active redundancy. For systems with the subclass of minimal cuts associated with one component covering that of another, assigning the redundancy to the former component is proved to bring forth a more reliable redundant system. As for symmetric systems with lower tail permutation decreasing component lifetimes, allocating the redundancy to the less reliable component results in a longer system lifetime. Several numerical examples are also presented to illustrate the new findings.     

Reliability and covariance estimation of weighted k-out-of-n multi-state systems

In the literature of reliability engineering, reliability of the weighted k-out-of-n system can be calculated using component reliability based on the structure function. The calculation usually assumes that the true component reliability is completely known. However, this is not the case in practical applications. Instead, component reliability has to be estimated using empirical sample data. Uncertainty arises during this estimation process and propagates to the system level. This paper studies the propagation mechanism of estimation uncertainty through the universal generating function method. Equations of the complete solution including the unbiased system reliability estimator and the corresponding unbiased covariance estimator are derived. This is a unified approach. It can be applied to weighted k-out-of-n systems with multi-state components, to weighted k-out-of-n systems with binary components, and to simple series and parallel systems. It may also serve as building blocks to derive estimators of system reliability and uncertainty measures for more complicated systems.     

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.     

A load share model for non-identical components of a k-out-of-m system

We consider a k-out-of-m load sharing system when the lifetimes of the components are not necessarily identically distributed random variables. For such systems, a model for the load sharing phenomenon through the exponentiated conditional survival functions of ordered failure times is proposed. This model is more general than the load sharing model with identically distributed component lifetimes and leads to a different family of distributions for ordered random variables. A general expression for the reliability of the system is given. The computations of the reliability for a two component parallel load sharing system corresponding to the exponential and Weibull distributions are discussed. For illustrative purpose, we discuss the inference procedures for a two component parallel load sharing system corresponding to the exponential distributions. A simulation study is carried out to assess the proposed estimation and testing procedures. The applicability of the proposed load sharing model is shown through two data sets.     

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.     

On relevation transform involved with statistical dependence between two component lifetimes

Krakowski (Rev Fr Autom Inform Rech Opèr. 1973;7:107–120.) introduced the relevation transform for component and active redundancy with independent lifetimes, and except for Johnson and Kotz (Am J Math Manag Sci. 1981;1:155–165; Nav Res Logist. 1983;30:163–169.) most subsequent researches were conducted under this framework. However, it is not uncommon that a component and its active redundancy bear some common stresses due to the environment and thus they have dependent lifetimes. In this note, we equip the involved lifetimes with a survival copula and then clarify the potential difference between the new and classical versions through making stochastic comparison. Moreover, by ordering the lifetime of system with relevation redundancy we also study the way of allocating a relevation redundancy at component level to ultimately improve the system reliability. The present results on series and parallel systems serve as a generalization of the corresponding ones of Belzunce et al. (Appl Stoch Models Bus Ind. 2019;35:492–503.). Several numerical examples are presented to illustrate these findings as well.     

Joint Reliability Function of Coherent Systems with Shared Heterogeneous Components

In this paper, we consider two coherent systems having shared components. We assume that in the two systems there are three different types of components; components of type one that just belong to the first system, components of type two that lie only in the second system and components of type three that are shared by the two systems. We use the concept of joint survival signature to assess the joint reliability function of the two systems. Using this concept, some representations for the joint reliability function of the system lifetimes are obtained under two different scenarios of component failures. In the first scenario, we assume that the components of the systems fail according to different counting processes such as non-homogeneous Poisson processes. In the second scenario, it is assumed that the component lifetimes of each type are exchangeable while the three types of component lifetimes can be independent or dependent. To illustrate the theoretical results, two systems with shared components are studied numerically and graphically.

     

Reliability analysis and optimization of weighted voting systems with continuous states input

Weighted voting systems are widely used in many practical fields such as target detection, human organization, pattern recognition, etc. In this paper, a new model for weighted voting systems with continuous state inputs is formulated. We derive the analytical expression for the reliability of the entire system under certain distribution assumptions. A more general Monte Carlo algorithm is also given to numerically analyze the model and evaluate the reliability. This paper further proposes a reliability optimization problem of weighted voting systems under cost constraints. A genetic algorithm is introduced and applied as the optimization technique for the model formulated. A numerical example is then presented to illustrate the ideas.     

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.     

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.     

Evaluating correct classification probability for weighted voting classifiers with plurality voting

Weighted voting classifiers (WVCs) consist of N units that each provide individual classification decisions. The entire system output is based on tallying the weighted votes for each decision and choosing the winning one (plurality voting) or one which has the total weight of supporting votes greater than some specified threshold (threshold voting). Each individual unit may abstain from voting. The entire system may also abstain from voting if no decision is ultimately winning. Existing methods of evaluating the correct classification probability (CCP) of WVCs can be applied to limited special cases of these systems (threshold voting) and impose some restrictions on their parameters. In this paper a method is suggested which allows the CCP of WVCs with both plurality and threshold voting to be exactly evaluated without imposing constraints on unit weights. The method is based on using the modified universal generating function technique.     

Interval uneffectiveness distribution for a k-out-of-n multistate reliability system with repair

This paper deals with a parallel load-sharing reliability system with cold standby redundancy and ample repair facilities. That is, we have n identical parallel units, of which at most k units are operating simultaneously. If less than k units are available, the system operates at a proportionally reduced level. For this system, an approximate method is given for the calculation of the probability distribution of that proportion of the system capacity that cannot be used in a given time period. The method is based on an approximation of the k-out-of-n multistate system by a two-state single component. Validation of the approximation using Monte-Carlo simulation shows satisfactory performance. Also, sensitivity results are given, showing in particular a decreasing sensitivity of the measures of performance to the distributional form of the unit lifetimes and repair times as the size of the system increases. Furthermore, it is found that the effect of the distributional form of the unit lifetimes dominates that of the unit repair times.     

Reliability equivalence factors of a system with mixture of n independent and non-identical lifetimes with delay time

Recently, [1], [2] generalized the reliability equivalence technique to a system with mixed of two non-identical lifetimes with delay time. The aim of this study is to generalize reliability equivalence technique to apply it to a system of mixture of n independent and non-identical lifetimes with delay time. We shall improve the system by using some reliability techniques: (i) reducing the failure for some lifetimes; (ii) add hot duplication components; (iii) add cold duplication components; and (iv) add cold duplication components with imperfect switches. We start by establishing two different types of reliability equivalence factors, the survival reliability equivalence (SRE) and mean reliability equivalence (MRE) factors. Also, we introduced some numerical results and conclusions.     

Stochastic ordering properties for systems with dependent identically distributed components

In this paper, we obtain ordering properties for coherent systems with possibly dependent identically distributed components. These results are based on a representation of the system reliability function as a distorted function of the common component reliability function. So, the results included in this paper can also be applied to general distorted distributions. The main advantage of these results is that they are distribution‐free with respect to the common component distribution. Moreover, they can be applied to systems with component lifetimes having a non‐exchangeable joint distribution. Copyright © 2012 John Wiley & Sons, Ltd.     

Complete voting systems with two classes of voters: weightedness and counting

We investigate voting systems with two classes of voters, for which there is a hierarchy giving each member of the stronger class more influence or important than each member of the weaker class. We deduce for voting systems one important counting fact that allows determining how many of them are for a given number of voters. In fact, the number of these systems follows a Fibonacci sequence with a smooth polynomial variation on the number of voters. On the other hand, we classify by means of some parameters which of these systems are weighted. This result allows us to state an asymptotic conjecture which is opposed to what occurs for symmetric games.     

The failure rate and likelihood ratio orderings of standby redundant systems

There are various notions of partial ordering between lifetimes of systems; stochastic ordering, failure rate ordering, and likelihood ratio ordering. In this paper we show that for series systems with noni.i.d. exponential lifetimes of components, standby redundancy at component level is better than that at system level in failure rate ordering and likelihood ratio ordering. We also demonstrate that for 2-component parallel systems withi.i.d. exponential lifetimes of components, standby system redundancy is better than standby component redundancy in failure rate ordering and likelihood ratio ordering.