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     

Markov model for k-out-of-n:G systems with built-in-test

This paper considers some new variations when the Built-In-Test technique is applied to k-out-of-n:G systems. A complete Markov model for the reliability and availability analysis of k-out-of-n:G systems with BIT is developed. Both fault coverage and false alarm are considered. A new inverse Laplace transform formula is derived to solve the differential simultaneous equations. Finally, generalized analytical functions for system reliability and availability are obtained. Our approach and solution are helpful and applicable for analyzing the impact of BIT design parameters on k-out-of-n:G system RAM and optimizing the redundancy management strategy.     

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     

A simple lower bound for reliability of k-out-of-n:G systems

Application of Boole's inequality results in a simple lower bound for system reliability in terms of reliabilities of subsystems for a k-out-of-n:G system. No assumptions regarding statistical dependence or independence of the outcomes (success or failure) for the subsystems are used. The no assumption lower bound for pure series or parallel systems has been previously published, but the more general case of the title has been overlooked. On the other hand, using the assumption of association of the outcomes, the lower bound in this paper can be improved upon slightly as shown in Barlow and Proschan (1981)     

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     

Comment on: Reliability of k-out-of-n:G systems with imperfectfault-coverage

This note comments on the paper “Reliability of k-out-of-n:G systems with imperfect fault-coverage” by S. Akhtar (1994). An alternative probability argument can be used to obtain the MTBF (mean time between failures) and MTTF (mean time to failure) for such systems. This has the advantage that higher moments of such failure times can also be determined     

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.     

Recursive Algorithm for Reliability Evaluation of k-out-of-n:G System

An algorithm for computing recursively the exact system reliability of k-out-of-n systems is proposed. It is simple, easy to implement, fast, and memory efficient. It gives a reliability expression with minimal number of terms, C(k, n) and involves only a few multiplications. The reduction in number of terms and multiplications is over 50 percent compared to some methods. The recursive nature of the algorithm enables one to design easily the number of units in the system to meet a reliability target. An alternative representation of the algorithm which is easy to remember and good for manual computation is given. However, it involves a few more multiplications compared to the original one but fewer than those required with existing methods.     

The Availability of a k-out-of-n:G Network

A method is described for calculating the analytic availability of a k-out-of-n:G network where the availability of each component may be different. An algorithm and a FORTRAN subroutine are provided to calculate this type of availability.     

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 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.     

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。     

Steady-State Availability of k-out-of-n:G System with Single Repair

This paper presents a solution and computer program for steady-state availability of a k-out-of-n:G system with single repair. Techniques and methodologies are commonly treated in text books to solve one and two element availabilities. This paper provides both the solution for k-out-of-n system availability and a FORTRAN source program for calculating the availability. The managers of mass transit and computer systems can put n elements on line with the assurance that, on the average, at least k elements will actually be available to complete the mission.     

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.     

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.     

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 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.     

An explicit closed-form formula for profit-maximizing k-out-of-n systems subject to two kinds of failures

This paper derives and analyzes an explicit closed-form formula for the optimal k in k-out-of-n systems consisting of i.i.d. components. The system can be in one of two possible modes with a pre-specified probability. The components are subject to failure in each of the two modes. The costs of the two kinds of system failures are generally not identical. Since the formula is explicit, it permits a calculation of the optimal k directly in terms of the parameters of the system. In addition, it yields many results concerning both the bounds of the optimal k and the effects of a change in parameters on the optimal k and on the optimized value of the system's expected profit.     

Efficient algorithms for k-out-of-n andconsecutive-weighted-k-out-of-n:F system

A new reliability model, consecutive-weighted-k-out-of-n:F system, is proposed and an O(n) algorithm is provided to evaluate its reliability. An O(n·min(n,k)) algorithm is also presented for the circular case of this model. The authors design an O(n) parallel algorithm using k processors to compute the reliability of k-out-of-n systems, that achieves linear speedup     

An O(n·(log2(n))2) algorithm forcomputing the reliability of k-out-of-n:G and k-to-l-out-of-n:G systems

This paper presents the RAFFT-GFP (Recursively Applied Fast Fourier Transform for Generator Function Products) algorithm as a computationally superior algorithm for expressing and computing the reliability of k-out-of-n:G and k-to-l-out-of-n:G systems using the fast Fourier transform. Originally suggested by Barlow and Heidtmann (1984), generating functions provide a clear, concise method for computing the reliabilities of such systems. By recursively applying the FFT to computing generator function products, the RAFFT-GFP achieves an overall asymptotic computational complexity of O(n·(log₂(n)) ²) for computing system reliability. Algebraic manipulations suggested by Upadhyaya and Pham (1993) are reformulated in the context of generator functions to reduce the number of computations. The number of computations and the CPU time are used to compare the performance of the RAFFT-GFP algorithm to the best found in the literature. Due to larger overheads required, the RAFFT-GFP algorithm is superior for problem sizes larger than about 4000 components, in terms of both computation and CPU time for the examples studied in this paper. Lastly, studies of very large systems with unequal reliabilities indicate that the binomial distribution gives a good approximation for generating function coefficients, allowing algebraic solutions for system reliability