Minimal-cost system reliability with discrete-choice sets for components

We focus on systems whose components come from discrete choice sets. In a choice set, the alternatives have increasing cost with increasing reliability. The objective is to ensure minimal cost for achieving a specified reliability for the systems under consideration. Earlier work restricted itself to series-parallel/parallel-series (S/P) systems and provided formulations and algorithms. However, these are not amenable for dealing with more general systems. In this paper, we develop alternative formulations and algorithms based on a dynamic programming approach, and these are generalized for S/P-reducible systems. The algorithms we obtain are pseudo-polynomial and possess fully polynomial approximation schemes. Moreover, the formulations & algorithms are amenable for further generalizations to k-out-of-n : G and k-out-of-n : G-reducible systems, though we cannot claim pseudo-polynomiality in these cases. The results of this paper are useful for developing reliable systems at minimum cost. As such, the formulation & algorithms are of vital interest for systems & reliability professionals & researchers.     

A unified algorithm of reliability estimation of k-out-of-n systems implied by the structure function

A new algebraic form of the structure function of a system will be proposed and its significant properties will be proved. The usefulness of these results in construction of algorithms of reliability estimation will be presented.The general approach to the desig of the algorithms of reliability estimation for both k-out-of-n and consecutive k-out-of-n systems will be presented. The method of the estimation of the lower and/or upper bound of system failure rate will be also discussed.The appropriate program in PASCAL will be given.     

Efficient computation of the sensitivity of k-out-of-n system reliability

The first-order sensitivity of system reliability is useful in evaluating several criticality measures, uncertainty measures, and the instantaneous failure rate of the system. Three new algorithms are described herein for the computation of the sensitivity of k-out-of-n system reliability. Generally, the numerical results of these algorithms check very well versus one another as well as versus those of known special cases. The computational complexities of these algorithms vary from almost double to slightly less than that of the best known algorithm for computing the k-out-of-n system reliability. Some observations are made on the important rankings of system components for different values of k, n and component reliabilities.     

顺序k/n(F)系统可靠性算法改进

在元件可靠性不相同的一般情况下,本文给出了线状/环状顺序k/n(F)系统可靠性的计算方法,算法完全排除了所有可能出现的相消项,使系统可靠性公式中项数降到最少,且具有极强的规律性,从而大大减少了计算复杂度。     

Optimal design of k-out-of-n:G subsystems subjected to imperfect fault-coverage

Systems subjected to imperfect fault-coverage may fail even prior to the exhaustion of spares due to uncovered component failures. This paper presents optimal cost-effective design policies for k-out-of-n:G subsystems subjected to imperfect fault-coverage. It is assumed that there exists a k-out-of-n:G subsystem in a nonseries-parallel system and, except for this subsystem, the redundancy configurations of all other subsystems are fixed. This paper also presents optimal design polices which maximize overall system reliability. As a special case, results are presented for k-out-of-n:G systems subjected to imperfect fault-coverage. Examples then demonstrate how to apply the main results of this paper to find the optimal configurations of all subsystems simultaneously. In this paper, we show that the optimal n which maximizes system reliability is always less than or equal to the n which maximizes the reliability of the subsystem itself. Similarly, if the failure cost is the same, then the optimal n which minimizes the average system cost is always less than or equal to the n which minimizes the average cost of the subsystem. It is also shown that if the subsystem being analyzed is in series with the rest of the system, then the optimal n which maximizes subsystem reliability can also maximize the system reliability. The computational procedure of the proposed algorithms is illustrated through the examples.     

Two formulas for computing the reliability of incompletek-out-of-n:G systems

Reliability computation of highly redundant systems most commonly uses approximate methods. Except for k-out-of-n:G systems or consecutive k-out-of-n:G systems, exact reliability formulas offering a broader range of applicability are rare. This paper gives two new formulas for this purpose: the first handles k-out-of-n:G systems of which some paths are not present; the second allows for the reliability calculation of a coherent binary system in general. Both formulas express system reliability in terms of the reliabilities of k-out-of-n:G systems. In practice, these new formulas cope with highly redundant systems with certain similarities to k-out-of-n:G systems. For example, a reliability of the control-rod system of a nuclear reactor is computed. Although the paper is directed to system reliability, the results can be used for computing the failure probability of a system which in practical applications is sometimes more convenient. In which case, the formulas are to be changed such that a system is given by its minimal cut-sets instead of minimal path-sets, and p should be a component unreliability instead of its reliability. The first proof of formula uses domination theory and, in thus contributes to the state of the art in this field     

Reliability evaluation of combined k-out-of-n:F,consecutive-k-out-of-n:F and linear connected-(r, s)-out-of-(m, n):Fsystem structures

Based on a real industrial application, three new system reliability models are proposed: combined k-out-of-n:F and consecutive-k _c-out-of-n:F system; combined k-out-of-m·n:F and linear connected-(r,s)-out-of-(m,n):F system; and combined k-out-of-m·n:F consecutive-k_c-out-of-n:F and linear connected-(r,s)-out-of-(m,n):F system. Reliability evaluation algorithms are provided for these models. The computation times of the algorithms for these models are, respectively: O(n·k), O(k·n·2 ^m·s^m-r+2), O(k·n·(2k_c )^m·s^m-r+1). The algorithms are used for system reliability evaluation of furnace systems. The concept of the combined k-out-of-n:F and 1-dimensional and 2-dimensional consecutive-k-out-of-n:F systems can be extended to other variations of the consecutive-k-out-of-n:F systems, e.g., the consecutive-k-out-of-n:G system and 1-dimensional and 2-dimensional r-within-k-out-of-n:F systems. The concept of Markov chain imbeddable (MIS) systems is another excellent tool that can be used for analysis of such combined system structures     

Reliability Analysis of the k-out-of-n:F System

A class of repairable systems known as k-out-of-n:F systems, 1 ? k ? n, consists of n units in parallel redundancy which are serviced by a single repairman; system failure occurs when k units are simultaneously inoperable for the first time. In this paper, assuming constant failure rates and general repair distributions, reliability characteristics of the k-out-of-n:F system are treated using two different methods. In Part I, a conditional transform approach is applied to the 2-out-of-n:F system. Transforms of distributions are obtained for T (the time to system failure), the time spent on repairs during (0, T) and the free time of the repairman during (0, T). In Part II, the supplementary variable technique is used to investigate time to failure characteristics of the k-out-of-n:F system for k = 2 and k = 3. A model of an airport limousine service illustrates the use of the results.     

An algorithm for computing the reliability of weighted-k-out-of-nsystems

This paper constructs a new k-out-of-n model, viz, a weighted-k-out-of-n system, which has n components, each with its own positive integer weight (total system weight=w), such that the system is good (failed) if the total weight of good (failed) components is at least k. The reliability of the weighted-k-out-of-n:G system is the complement of the unreliability of a weighted-(w-k+1)-out-of-n:F system. Without loss of generality, the authors discuss the weighted-k-out-of-n:G system only. The k-out-of-n:G system is a special case of the weighted-k-out-of-n:G system wherein the weight of each component is 1. An efficient algorithm is given to evaluate the reliability of the weighted-k-out-of-n:G system. The time complexity of this algorithm is O(n.k)     

Performance evaluation of generalized multi-state k-out-of-n systems

The generalized multi-state k-out-of-n:G system model defined by Huang provides more flexibilities for modeling of multi-state systems. However, the performance evaluation algorithm they proposed for such systems is not efficient, and it is applicable only when the k/sub i/ values follow a monotonic pattern. In this paper, we defined the concept of generalized multi-state k-out-of-n:F systems. There is an equivalent generalized multi-state k-out-of-n:G system with respect to each generalized multi-state k-out-of-n:F system, and vice versa. The form of minimal cut vector for generalized multi-state k-out-of-n:F systems is presented. An efficient recursive algorithm based on minimal cut vectors is developed to evaluate the state distributions of a generalized multi-state k-out-of-n:F system. Thus, a generalized multi-state k-out-of-n:G system can first be transformed to the equivalent generalized multi-state k-out-of-n:F system, and then be evaluated using the proposed recursive algorithm. Numerical examples are given to illustrate the effectiveness and efficiencies of the proposed recursive algorithms.     

The mean residual life function of a k-out-of-n structure at the system level

In the study of the reliability of technical systems, k-out-of-n systems play an important role. In the present paper, we consider a k-out-of-n system consisting of n identical components with independent lifetimes having a common distribution function F. Under the condition that, at time t, all the components of the system are working, we propose a new definition for the mean residual life (MRL) function of the system, and obtain several properties of that system.     

Reliability analysis of k-out-of-n systems with partially repairable multi-state components

In some environments the components might not fail fully, but can lead to degradation and the efficiency of the system may decreases. However, the degraded components can be restored back through a proper repair mechanism. In this paper, we present a model to perform reliability analysis of k-out-of-n systems assuming that components are subjected to three states such as good, degraded, and catastrophic failure. We also present expressions for reliability and mean time to failure (MTTF) of k-out-of-n systems. Simple reliability and MTTF expressions for the triple-modular redundant (TMR) system, and numerical examples are also presented in this study.     

Improved Method of Inclusion-Exclusion Applied to k-out-of-n Systems

The method of inclusion-exclusion is represented in general form for reliability analysis. Applying it to the reliability of k-out-of-n system causes many cancelling terms. The method is improved to use only noncancelling terms in evaluating bounds on the reliability of k-out-of-n systems. These bounds are appreciably better, and converge to the exact system reliability in at most n ? k + 1 steps. In conclusion some numerical considerations suggest the quality of the bounds. Similar results for the reliability analysis of networks were derived by Satyanaraynna & Prabhakar.     

Reliability analysis of a k-out-of-n:G vehicle fleet

This paper presents a reliability analysis of a k-out-of-n:G on-surface vehicle fleet. The transit system is in a failed state when (n − k + 1) vehicles failed. Laplace transforms of state probabilities and reliability of the transit system are derived. The transit system steady-state probabilities and availability formulas are also developed.     

A unified way of comparing the reliability of coherent systems

Consider a coherent system S hat has n nodes and a structure function /spl phi/. The state of S is determined by the "states of the components allocated at the n nodes" and /spl phi/. If the reliabilities of these it components are expressed by an n-dimensional vector p, then the reliability of S is a function of p. Using the concepts of node-criticality introduced by Boland et al. (1989), the main theorem in this paper proposes a unified approach by which the reliabilities of S corresponding to 2 different values of p can be compared. The conditions in the main theorem are both necessary and sufficient for k-out-of-n systems. Application of this theorem to various situations yields a unified approach for obtaining previous results in the literature. Also, the application of this theorem immediately extends the results established for k-out-of-n systems to coherent systems.     

A method to evaluate reliability of systems with critical and non-critical human errors

This paper presents a newly developed method to perform reliability analysis of redundant systems with critical and non-critical human errors. The method is demonstrated using k-out-of-n units, parallel and series-parallel configurations. Time dependent analyses are developed for exponentially and Rayleigh distributed failure times. System reliability and mean time to failure formulas are presented.     

Optimal reliability of systems subject to imperfect fault-coverage

This paper maximizes the reliability of systems subjected to imperfect fault-coverage. The results include the effect of common-cause failures and `maximum allowable spare limit'. The generalized results are presented and then the policies for some specific systems are given. The systems considered include parallel, parallel-series, series parallel, k-out-of-n, and NMR (k-out-of-(2k-1)) systems. The results are generalized for the non s-identical component case     

Effects of Weibull hazard rate on common cause failure analysis of reliability networks

This paper presents formulae for system mean life, variance of time to failure, hazard rate and reliability of parallel, k-out-of-n, parallel-series and bridge networks with common cause failures. The components in every configuration are assumed to be identical and characterized by a Weibull time to failure density function. The graphical plots of the system reliability and mean life gain are shown.     

一类多状态连续n中取k(G)系统的可靠性计算

为了计算多状态连续厅中取后（G）系统的可靠性,引入4个定理,将满足引理的多状态系统转换为二元状态系统。分别推导了多状态线形连续k／n（G）系统和环形连续k／n（G）系统的可靠性计算公式。证明了固定k值增加一个新部件,若部件可靠性独立同分布,线形和环形系统可靠性均增加;若部件可靠性独立但不同分布,环形系统存在一个极值,新增加部件可靠性大于这个极值时得到的新系统可靠性增加,反之系统可靠性下降。     

Reliability Bounds for Multi-State $k$-out-of- $n$ Systems

Algorithms have been available for exact performance evaluation of multi-state k-out-of-n systems. However, especially for complex systems with a large number of components, and a large number of possible states, obtaining "reliability bounds" would be an interesting, significant issue. Reliability bounds will give us a range of the system reliability in a much shorter computation time, which allow us to make decisions more efficiently. The systems under consideration are multi-state k-out-of-n systems with i.i.d. components. We will focus on the probability of the system in states below a certain state d, denoted by Q_sd. Based on the recursive algorithm proposed by Zuo & Tian [14] for performance evaluation of multi-state k-out-of-n systems with i.i.d. components, a reliability bounding approach is developed in this paper. The upper, and lower bounds of Q_sd are calculated by reducing the length of the k vector when using the recursive algorithm. Using the bounding approach, we can obtain a good estimate of the exact Q_sd value while significantly reducing the computation time. This approach is attractive, especially to complex systems with a large number of components, and a large number of possible states. A numerical example is used to illustrate the significance of the proposed bounding approach.