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     

An O(kn)-time algorithm for computing thereliability of a circular consecutive-k-out-of-n:Fsystem

I. Antonopoulou and S. Papastavridis (1987) published an algorithm for computing the reliability of a circular consecutive-k-out-of-n:F system which claimed O (kn) time. J.S. Wu and R.J. Chen (1993) correctly pointed out that the algorithm achieved only O(kn²) time. The present study shows that the algorithm can be implemented for O(kn) time     

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     

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     

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     

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     

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     

Binary cyclic [n,k] codes used for simultaneoustEC and SBD

In a hybrid forward-error-correction-automatic-repeat-request system one may wish to use an [n,k] cyclic code because its decoding algorithm is well known. An analytic formula is given for determining the fraction of undetectable single bursts of different lengths when a cyclic code is used for simultaneous single-burst-error detection and t-random error correction     

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     

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     

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     

A geometric approach to root finding in GT(qm)

The problem of finding roots in F of polynomials in F [x] for F=GF(q^m), where q is a prime or prime power and m is a positive integer greater than 1 is considered. The problem is analyzed by making use of the finite affine geometry AG(m,q). A new method is proposed for finding roots of polynomials over finite extension fields. It is more efficient than previous algorithms when the degree of the polynomial whose roots are to be found is less than dimension m of the extension field. Implementation of the algorithm can be enhanced in cases in which optimal normal bases for the coefficient field are available     

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     

Robustness of mean E{X} and R charts

The effect of nonnormality on E{X} and R charts is reported. The effect of departure from normality can be examined by comparing the probabilities that E{X} and R lie outside their three-standard-deviation and two-standard-deviation control limits. Tukey's λ-family of symmetric distributions is used because it contains a wide spectrum of distributions with a variety of tail areas. The constants required to construct E{X} and R charts for the λ-family are computed. Control charts based on the assumption of normality give inaccurate results when the tails of the underlying distribution are thin or thick. The validity of the normality assumption is examined by using a numerical example     

k-out-of-m system availability with minimum-costallocation spares

An optimization method for determining the number of spare units that should be allocated to a k-out-of-m system to minimize the system-spares cost yet attain the specified system availability is presented. The objective function for optimization is a nonlinear integer type. The optimization method is a variation of the simplex search technique used for continuous functions. The optimization problem is cast in a form that minimizes the system-spares cost, with the required system availability as an inequality constraint. Results obtained by using the proposed optimization technique, as well as the computation time required for optimization, are compared to those for methods developed specifically for dealing with nonlinear integer problems. The method is simple, easy to implement, and yet very effective in dealing with the spare allocation problem for k-out-of-m:F systems     

Light-to-input nonlinearities in an R-LED series network

The light-to-current (L-I) and light-to-voltage (L-V) differential nonlinearities in the simple network of a customary LED and an external resistor R in series are analyzed and calculated theoretically and compared with experimental data. Particular emphasis is placed on the influence of the log-arithmetic slope ν of the L-I characteristic and the bias current I upon the ratio of the corresponding nonlinearity parameters. It is thus deduced that, for a given optical power P, over superlinear portions of the L-I curve (ν>1) the L-I linearity is typically better than its corresponding L-V linearity. On the contrary, when the L-I dependence is sublinear (ν<1) the voltage driving scheme may ensure for the R-LED network, or the LED alone, a local L-V response much more linear than the L-I response, provided that appropriate (optimum) I and/or R values are chosen     

Construction of t-EC/AUED codes

A t-EC/AUED code is constructed by appending a single check symbol from an alphabet S to each word of an n-bit binary t-EC code of even weight. Conditions are derived for the construction of S and a procedure is given which, for some values of t, n, leads to codes with fewer check bits than known codes with equivalent properties     

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