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.     

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     

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     

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.     

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。     

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.     

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.     

Reliability optimization of series-parallel systems using a genetic algorithm

A problem-specific genetic algorithm (GA) is developed and demonstrated to analyze series-parallel systems and to determine the optimal design configuration when there are multiple component choices available for each of several k-out-of-n:G subsystems. The problem is to select components and redundancy-levels to optimize some objective function, given system-level constraints on reliability, cost, and/or weight. Previous formulations of the problem have implicit restrictions concerning the type of redundancy allowed, the number of available component choices, and whether mixing of components is allowed. GA is a robust evolutionary optimization search technique with very few restrictions concerning the type or size of the design problem. The solution approach was to solve the dual of a nonlinear optimization problem by using a dynamic penalty function. GA performs very well on two types of problems: (1) redundancy allocation originally proposed by Fyffe, Hines, Lee, and (2) randomly generated problem with more complex k-out-of-n:G configurations.     

On the Evaluation of the Reliability of k-out-of-n Systems

The paper provides some correcting remarks on several methods to determine the k-out-of-n:G system reliability - a problem that has taken more than its share in the literature. A Jain-Gopal paper presents two algorithms JG-1 and JG-2 to evaluate the reliability of a k-out-of-n System. The comparison of the computational costs of JG-1 with those of JG-2 and with other existing algorithms, especially Barlow-Heidtmann, need to be restated.     

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.     

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.     

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.     

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.     

Optimal release time for software systems considering cost, testing-effort, and test efficiency

In this paper, we study the impact of software testing effort & efficiency on the modeling of software reliability, including the cost for optimal release time. This paper presents two important issues in software reliability modeling & software reliability economics: testing effort, and efficiency. First, we propose a generalized logistic testing-effort function that enjoys the advantage of relating work profile more directly to the natural flow of software development, and can be used to describe the possible testing-effort patterns. Furthermore, we incorporate the generalized logistic testing-effort function into software reliability modeling, and evaluate its fault-prediction capability through several numerical experiments based on real data. Secondly, we address the effects of new testing techniques or tools for increasing the efficiency of software testing. Based on the proposed software reliability model, we present a software cost model to reflect the effectiveness of introducing new technologies. Numerical examples & related data analyzes are presented in detail. From the experimental results, we obtain a software economic policy which provides a comprehensive analysis of software based on cost & test efficiency. Moreover, the policy can also help project managers determine when to stop testing for market release at the right time.     

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.     

Generalized multi-state k-out-of-n:G systems

In a binary k-out-of-n:G system, k is the minimum number of components that must work for the system to work. Let 1 represent the working state and 0 the failure state, k then indicates the minimum number of components that must be in state 1 for the system to be in state 1. This paper defines the multi-state k-out-of-n:G system: each component and the system can be in 1 of M+1 possible states: 0, 1, ..., M. In Case I, the system is in state ⩾j iff at least k_j components are in state ⩾j. The value of k_j I 1 can be different for different required minimum system-state level j. Examples illustrate applications of this definition. Algorithms for reliability evaluation of such systems are presented     

Stochastic ordering results for consecutive k-out-of-n:F systems

A linear (circular) consecutive k-out-of-n:F system is a system of n linearly (circularly) ordered components which fails if and only if at least k consecutive components fail. We use recursive relationships on the reliability of such systems with independent identically distributed components to show that for any fixed k, the lifetime of a (linear or circular) consecutive k-out-of-n:F system is stochastically decreasing in n. This result also holds for linear systems when the components are independent and not necessarily identically distributed, but not in general for circular systems.     

Optimal reliability allocation of redundant units and repair facilities by arbitrary failure and repair distributions

In this paper we solve three reliability optimization problems for repairable systems with arbitrary distributed times to unit failure and to unit repair. By solving the problems, we find the allocation of redundant units and repairmen. The optimized systems contain the k-out-of-n or cold standby seriesly connected subsystems. The numerical examples illustrate the presented approach for solving the problems.     

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.