A generalization of consecutive-k-out-of-n:Fsystems

A linear (m, n)-lattice system consists of m ·n elements arranged like the elements of a (m ,n)-matrix, i.e. each of the m rows includes m elements, and each of the n columns includes m elements. A circular (m,n)-lattice system consists of m circles (centered at the same point) and n rays. The intersections of the circle and the rays represent the elements, i.e. each of the circles includes n elements and each of the rays has m elements. A (linear or circular) (m, n)-lattice system is a (linear or circular) connected-X-out-of-(m,n):F lattice system if it fails whenever at least one subset of connected failed components occurs which includes failed components connected in the meaning of connected-X. The paper presents some practical examples and the reliability formulas of simple systems using results of consecutive-k-out-of-n:F systems     

m-consecutive-k-out-of-n:F systems

An m-consecutive-k-out-of-n:F system, consists of n components ordered on a line; the system fails if and only if there are at least m nonoverlapping runs of k consecutive failed components. Three theorems concerning such systems are stated and proved. Theorem one is a recursive formula to compute the failure probability of such a system. Theorem two is an exact formula for the failure probability. Theorem three is a limit theorem for the failure probability     

Bounds for reliability of consecutivek-within-m-out-of-n:F systems

Upper and lower bounds for the reliability of a (linear or circular) consecutive k-within-m-out-of-n:F system with unequal component-failure probabilities are provided. Numerical calculations indicate that, for systems with components of good enough reliability, these bounds quite adequately estimate system reliability. The estimate is easy to calculate, having computational complexity O(m²×n). For identically distributed components, a Weibull limit theorem for system time-to-failure is proved     

A coding scheme for m-out-of-n codes

A scheme for the construction of m-out-of-n codes based on the arithmetic coding technique is described. For appropriate values of n, k, and m, the scheme can be used to construct an (n,k) block code in which all the codewords are of weight m. Such codes are useful, for example, in providing perfect error detection capability in asymmetric channels such as optical communication links and laser disks. The encoding and decoding algorithms of the scheme perform simple arithmetic operations recursively, thereby facilitating the construction of codes with relatively long block sizes. The scheme also allows the construction of optimal or nearly optimal m-out-of-n codes for a wide range of block sizes limited only by the arithmetic precision used     

Reliability of a consecutivek-out-of-r-from-n:F system

The reliability of the consecutive k-out-of-r-from-n:F system is studied. For k=2 an explicit solution is given for n components in line or in cycle in the i.i.d. case. For k⩾3 sharp lower and upper bounds are given for the reliability of the system and demonstrated for different values of n, k, r, p. These bounds are exact for r=n, n-1, n-2, n-3, and for these values the exact analytic solution is also given     

Reliability of consecutive-k-out-of-n:F systemswith nonidentical component reliabilities

A system with n components in sequence is a consecutive- k-out-of-n:F system if it fails whenever k consecutive components are failed. Under the supposition that component failures need not be independent and that component failure probabilities need not be equal, a topological formula is presented for the exact system reliability of linear and circular consecutive-k -out-of-n:F networks. The number of terms in the reliability formula is O(n⁴) in the linear case and O(n⁵) in the circular case     

Index system and separability of constant weight Gray codes

A number system is developed for the conversion of natural numbers to the codewords of the Gray code G(n,k) of length n and weight k, and vice versa. The focus is on the subcode G(n,k) of G(n) consisting of those words of G(n) with precisely k 1-bits, 0<k<n. This code is called the constant weight Gray code of length n and weight k. As an application sharp lower and upper bounds are derived for the value of |i-j|, where i and j are indices of codewords g_i and g_j of G(n,k) such that they differ in precisely 2 m bits     

An approximation for largeconsecutive-k-out-of-n:F systems

The authors consider a consecutive-k-out-n:F system consisting of identically distributed and statistically independent components, where the life distribution of an individual component is Weibull distributed with scale parameter 1/λ and shape parameter B. Let T_n be the life length of the consecutive-k-out-of-n:F system. The authors prove that for large values of n, the distribution of the n ¹(ka)/T_n, is satisfactorily approximated by a Weibull distribution with the same scale parameter and shape parameter k times the original shape parameter     

A consecutive-k-out-of-n:G system: the mirrorimage of a consecutive-k-out-of-n:F system

A consecutive-k-out-of-n:F (consecutive-k -out-of-n:G) system consists of an ordered sequence of n components such that the system is failed (good) if and only if at least k consecutive components in the system are failed (good). In the present work, the relationship between the consecutive- k-out-of-n:F system and the consecutive-k-out-of-n:G system is studied, theorems for such systems are developed, and available results for one type of system are applied to the other. The topics include system reliability, reliability bounds, component reliability importance, and optimal system design. A case study illustrates reliability analysis and optimal design of a train operation system. An optimal configuration rule is suggested by use of the Birnbaum importance index     

Invariant permutations for consecutive k-out-of-ncycles

Consecutive-k-out-of-n cycles are proposed as topologies for k-loop computer networks and describe a circular system of n components where the system fails if and only if any k consecutive components all fail. Suppose that the components are interchangeable. The the question arises as to which permutation maximizes the system reliability, assuming that the components have unequal reliabilities. If there exists on optimal permutation which depends on the ordering, but not the values, of the component reliabilities, then the system (and the permutation) is called invariant. The circular system is found to be not invariant except for k=1, 2, n-2, n-1, and n     

Relayed consecutive-k-out-of-n:F lines

A consecutive-k-out-of-n:F line is a system of components in a sequence such that the system fails if and only if k consecutive components all fail. Relayed systems often quoted as examples of such systems differ from the definition by the fact that the first component must work to initiate the relay (in some cases the last component also must work). Such systems are differentiated from ordinary consecutive-k-out-of-n:F lines by adding the word `relayed'. It is shown that the main properties of the reliabilities of consecutive-k-out-of-n:F lines are preserved under this modification     

Preservation of partial orderings under the formation ofk-out-of-n:G systems of i.i.d. components

The authors discuss the preservation of certain partial orderings by a k-out-of-n:G system of i.i.d. components. If the lifetime of a component A is larger than that of a component B in the likelihood ratio, failure rate, or stochastic ordering, then a k-out-of-n:G system formed by n i.i.d. components of type A has a larger lifetime, in that ordering, than that of a similar system consisting of n i.i.d. components of type B. However, if the lifetime of a component A is larger than that of a component B in mean residual life, harmonic-average mean residual life, or variable orderings, it is not necessary that a k-out-of-n:G system formed by n i.i.d. components of type A has a larger lifetime, in that ordering, than that of a similar system consisting of n i.i.d. components of type B     

An O(kn) algorithm for a circularconsecutive-k-out-of-n:F system

An O(k×n) algorithm is described for evaluating the reliability of a circular consecutive-k-out- n:F system     

Distribution of time to failure of consecutive k-out-of n:F systems

It is shown that by modifying the path-evaluation technique due to R.C. Bollinger and A.A. Salvia (1985) it is possible to compute the cumulative distribution function (CDF) of the lifetime of any consecutive k-out-of-n:F system recursively, obtaining it as a mixture of the distributions of the failure times of the various paths. The distribution of the failure time given a path is a convolution of exponential distributions with the distributions of failure times of systems made up of disjoint modules in series, where each module is either a subsystem for which the recursive computation has already been done or an s-coherent system with nonoverlapping min-cut sets whose failure time CDF can be easily found     

Reliability of a general consecutive-k-out-of-n:Fsystem

A linear (circular) consecutive-k-out-of-n:F system consists of n components ordered on a line (circle). Each component and the system have two states: good or failed. The system fails if and only if at least k consecutive components fail. The reliability of such systems is computed. The most general case is examined without any restriction on the components     

Optimal designs of{k,n-k+1}-out-of-n:F systems(subject to 2 failure modes)

The problem of achieving optimal system size (n) for {k,n-k+1}-out-of-n systems, assuming that failure may take either of two forms, is studied. It is assumed that components are independently identically distributed (i.i.d.) and that the two kinds of system failures can have different costs. The optimal k or n that maximizes mean system-profit is determined, and the effect of system parameters on the optimal k or n is studied. It is shown that there does not exist a pair (k,n) maximizing the mean system-profit     

On the optimal design of k-out-of-n:G subsystems

For a k-out-of-n:G subsystem, the mathematical determination of the most economical number of components in the subsystem is sought. Optimal values of k (for fixed n) and n (for fixed k), which minimize the mean total cost of k-out-of-n:G subsystems, are given. A numerical example illustrates the results     

Exact reliability formula for consecutive-k-out-of-n:F systems with homogeneous Markov dependence

A direct, exact method for computing the reliability for a consecutive-k-out-of-n:F system with homogeneous Markov dependence is presented. This method calculates the reliability for a consecutive-k-out-n:F system where the probability that any component i fails depends upon, and only upon, the state of the component (i-1)     

Optimal-arrangement and importance of the components in aconsecutive-k-out-of-r-from-n:F system

The authors examine: the determination of an optimal consecutive k-out-of-r-from-n:F system, under permutations of the components, and the Birnbaum-importance of components in the i.i.d. case. The authors first study (theorem 1) the optimality of a general system, with not necessarily s-identical components, under permutation of the components. Then they study (theorem 2) the importance of components in the i.i.d. case. Theorem 2 is readily derived from theorem 1. The main results are given in theorems 1 and 2, and proofs are given. The assumptions are: the system and each component are either good or failed: all binary component states are mutually statistically independent, and all n can be arranged in any linear order; and the system fails if and only if within r consecutive components, there are at least k failed ones     

k-out-of-n:G systems: some cost considerations

The authors provide a tool that an engineer designing a subsystem can use to decide between one subsystem and a more reliable but more costly one. The authors provide methods for selecting redundancy levels in k-out-of-n:G systems in order to minimize particular cost considerations where the k-out-of-n:G system is a subsystem of a major system. The n and k are chosen to minimize the total cost of the subsystem plus the average loss due to subsystem failure. A BASIC program is available to determine the n and k which find this minimum. Five loss functions are considered, and illustrations are given