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     

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     

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     

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     

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     

Optimal replacement policies for k-out-of-nsystems

The authors study a discrete-time, infinite-horizon, dynamic programming model for the replacement of components in a binary k -out-of-n:F system. The goal is to trade off the component replacement and system failure costs. Under the criterion of minimizing the long-run average cost per period, it is optimal to follow a critical component policy (CCP), viz., a policy specified by a critical component set and the rule: replace a component if and only if it is failed and is in the critical component set. Computing an optimal CCP is a binary nonlinear programming problem, which can be solved by searching through a set with O(n^k-1) points. This approach to finding an optimal CCP is practical when k is small. In particular, assuming s-independent components, it requires O(n^2k-1) calculations. The authors analyze in detail the two most important cases with small k: the series (1-out-of-n:F) system and the 2-out-of-n:F system     

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     

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     

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     

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     

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     

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     

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     

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     

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     

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     

Two-dimensional consecutive-k-out-of-n:F models

A two-dimensional version of the consecutive-k-out-of-n:F model is considered. Bounds on system failure probabilities are determined by comparison with the usual one-dimensional model. Failure probabilities are determined by simulation for a variety of values of k and n     

Distribution of the lifetime of consecutive k-within-m-out-of-n:F systems

The distribution of the lifetime (MTTF) of any consecutive k -within-m-out-of-n:F system, with independent exponentially distributed component lifetimes, is shown to be a convex combination of the distributions (MTTFs) of several convolutions of independent random variables, where each convolution represents a distinct path in the evolution of the system's history, and where in each convolution all but the last random variable is exponential. The last random variable in each convolution is either a zero lifetime or the lifetime of several disjoint consecutive k_i within m_i-out-of-n:F systems in series with each k_i<k, each m_i<m, and each n_i<n. This enables the calculations to proceed recursively. Calculations are facilitated by the symmetric nature of the convex combination     

The number of failed components in aconsecutive-k-of-n:F system

For a consecutive-k-out-of-n:F system an exact formula and a recursive relation are presented for the distribution of the number of components, X, that fail at the moment the system fails. X estimates how many cold spares are needed to replace all failed components upon system failure. The exact formula expresses the dependence of the distribution of X upon parameters k , n. The recursive formula is suitable for efficient numerical computation of the distribution of X     

An efficient non-recursive algorithm for computing the reliabilityof k-out-of-n systems

An algorithm for computing the reliability of k-out-of- n systems is proposed. It is simple, easy to implement on a computer, time and memory efficient, and good for numerical computation. The memory complexity is O(n-k), and for a given value of n-k the computation time is proportional to n . Its FORTRAN implementation is presented