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 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 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.     

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.     

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.     

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.     

Fast optimal diagnosis procedures for k-out-of-n:G systems

A k-out-of-n:G system consists of a set of components, where each component is either faulty or fault-free. The system is working if at least k components are fault-free. The problem of finding an optimal diagnosis procedure for a given k-out-of-n:G system has been considered in several research fields including medical diagnosis, redundant-system testing, and searching data-files. A polynomial-time algorithm for this problem was presented first by Salloum, and later by Salloum and Breuer, and independently by Ben-Dov. This paper implements the Salloum-Breuer-Ben-Dov algorithm, leading to an optimal diagnosis procedure that can determine the state of any given system in O(n·log(n)) time complexity and O(n) space complexity. The efficiency is achieved by using a generalized radix sorting procedure that uses a heap data structure. For some k-out-of-n:G systems, including those with equal testing costs for all components, the components along the leftmost and rightmost paths in the optimal diagnostic tree uniquely determine the other components in the tree. This property is used to devise a faster optimal diagnosis procedure than the one for the general k-out-of-n:G system. With regard to complexity, these procedures are the best solutions for the problem under consideration. This conjecture is supported by the fact that all these procedures require a sorting operation which has O(n·log(n)) as a lower bound on its time complexity     

k-out-of-n:G System Reliability With Imperfect Fault Coverage

Systems requiring very high levels of reliability, such as aircraft controls or spacecraft, often use redundancy to achieve their requirements. Reliability models for such redundant systems have been widely treated in the literature. These models describe k-out-of-n:G systems, where n is the number of components in the system, and k is the minimum number of components that must work if the overall system is to work. Most of this literature treats the perfect fault coverage case, meaning that the system is perfectly capable of detecting, isolating, and accommodating failures of the redundant elements. However, the probability of accomplishing these tasks, termed fault coverage, is frequently less than unity. Correct modeling of imperfect coverage is critical to the design of highly reliable systems. Even very high values of coverage, only slightly less than unity, will have a major impact on the overall system reliability when compared to the ideal system with perfect coverage. The appropriate coverage modeling approach depends on the system design architecture, particularly the technique(s) used to select among the redundant elements. This paper demonstrates how coverage effects can be computed, using both combinatorial, and recursive techniques, for four different coverage models: perfect fault coverage (PFC), element level coverage (ELC), fault level coverage (FLC), and one-on-one level coverage (OLC). The designation of PFC, ELC, FLC, and OLC to distinguish types of coverage modeling is suggested in this paper.     

On Optimal Redundancy of Multivalue-Output Systems

This paper considers improving the reliability of multivalue-output systems by the use of n-redundant systems in which n copies of systems are used redundantly and the output is determined from the outputs of those copies by the voter. A k-out-of-n redundant system minimizes the mean loss caused by the occurrence of output errors under the condition that the voter can be composed of only two kinds of operators, logical sum and logical product. The optimal k depends on the probability and loss matrices, but it can be specified in some special cases. The mean loss of multivalue-output systems with multichannels can be minimized by adopting k-out-of-n redundancy for each channel. The results provide a powerful guide to the improvement of fail-safe characteristics of many systems and the design of fault-tolerant systems.     

Comparative reliability analysis of simplex and redundant systems

This paper presents the comparative reliability analysis of simplex and selective redundant configurations. System configurations such as k-out-of-n units, bridge, series-parallel and parallel-series are compared with the simplex system when the unit failure rate is constant and time dependent. Comparative reliability plots are shown for different reliability networks and failure time distributions. As expected, the detailed investigation shows that the system mean-time-to-failure may be a misleading indicator of performance for mission-oriented systems.     

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     

A Heuristic Method for Determining Optimal Reliability Allocation

The paper presents a method for obtaining an optimal reliability allocation of an n-stage series system. In each stage, redundant comnponents can be added (in parallel, stand-by, or k-out-of-n:G, etc.), or a more reliable component can be used in order to improve the system reliability. The solution is obtained by repeatedly using a more reliable candidate at each stage that has the greatest value of a `weighted sensitivity function'. The balance between the objective unction and the constraints is controlled by a `balancing coefficient'. The overall computational procedure is given and an example is presented. The computations are given for a set of randomly generated test problems in which the optimal parallel redundancy under linear onstraints is determined. The proposed method is then compared with other methods.     

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.     

Reliability of k-out-of-n:G systems with imperfect fault-coverage

k-out-of-n:G systems are modeled to determine their reliability and availability. Markov models are obtained to examine the fault-tolerant operation of the system. From the Markov chains, reliability and availability measures are found as state probabilities. Recursive expressions for mean time-between-failures and mean time-to-failure are obtained for repairable systems, considering perfect and imperfect fault-coverage     

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.     

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     

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。     

Combined k-out-of-n:G, and consecutive k/sub c/-out-of-n:G systems

Qualification tests for a system are normally carried out according to either a k-out-of-n:G scheme, or a consecutive k/sub c/-out-of-n:G structure. The reliability of a combination of the two systems is evaluated, showing its benefit over each of the individual structures. As expected, the mean time to failure of the combined system is larger than any of them. Generalizations of the analysis are presented for tests with multi-state results, and for dependent tests. Illustrative numerical results are presented to substantiate the theory.     

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)     

Lifetime of Combined $k$-out-of-$n$ , and Consecutive $k_{c}$ -out-of-$n$ Systems

A combined k-out-of-n:F(G) & consecutive k_c -out-of-n :F(G) system fails (functions) iff at least k components fail (function), or at least fcc consecutive components fail (function). Explicit formulas are given for the lifetime distribution of these combined systems whenever the lifetimes of components are exchangeable, and have an absolutely continuous joint distribution. The lifetime distributions of the aforementioned systems are represented as a linear combination of distributions of order statistics by using the concept of Samaniego's signature. Formulas for the mean lifetimes are given. Some numerical results are also presented.