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     

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     

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     

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     

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     

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     

Reliability of a k-out-of-n warm-standby system

A general closed-form equation is developed for system reliability of a k-out-of-n warm-standby system (dormant failures). The equation reduces to the hot and cold standby cases under the appropriate restrictions     

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     

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     

Modeling a shared-load k-out-of-n:G system

A Markov model for analyzing the reliability and availability of an n-unit shared-load repairable k-out-of-n:G system with imperfect switching is presented. The equations for both time-dependent and steady-state system availability are given. An inverse Laplace transform is used to solve the simultaneous differential equations for the nonrepairable case. A generalized analytic function for system reliability is obtained. Examples are provided to demonstrate the model and the impact of a load-sharing strategy on the reliability. The load-sharing strategy can improve system reliability and availability, if the controller and switching parameters are adequate. The proposed approach and solution are helpful to system engineers and reliability analysts     

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     

Redundancy optimization of k-out-of-n systemswith common-cause failures

The problem of determining the optimal number of redundant units in k-out-of-n systems with common-cause failures (CCFs) is discussed. The mean cost rate is obtained, its behavior is examined considering both CCFs and random failures of the units, and the number of redundant units minimizing the mean cost rate is shown to be finite and unique. For the problem of determining the optimal number of redundant units and inspection period, the mean cost rate is obtained and a solution procedure is presented. Numerical studies are given which indicate that CCFs tend to reduce the optimal number of redundant units     

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     

Comment on `Modeling a shared-load k-out-of-n*Gsystem'

The commenter states that there is no reason to perform the steady-state analysis as attempted by J. Shao and L.R. Lamberson (see ibid., vol.40, no.2, p.205-9, June 1991) for a Markov chain that has an absorbing failure state. Corrected expressions are provided     

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     

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     

Lifetime distribution of circularconsecutive-k-out-of-n:F systems with exchangeablelifetimes

A recursive algorithm for computing the lifetime distribution of a circular consecutive-k-out-of-n:F system with statistically exchangeable component lifetimes is presented. The result applies to a load-sharing model     

On computing MTBF for a k-out-of-n:G repairablesystem

It is often necessary to calculate the MTBF (mean time between failures) quickly in order to make timely design decisions. An important system for which such calculations must be made is a k-out-of- n:G parallel system with unlimited repair and exponential interfailure and repair times at the unit level. Although a general formula is known, it is not easily remembered or derived. A method for deriving a formula for MTBF in this situation that is easily reproduced quickly by remembering a few simple concepts is presented     

An enhanced integer simplicial optimization method for minimum costspares for k-out-of-n systems

The authors report two enhancements to an integer simplicial optimization method developed for a spares allocation problem where it is necessary to minimize the spares cost of a k-out-of-n system configuration subject to an availability constraint. The first is an automated, simple, general method for generating an initial feasible good starting vector for optimization. This vector increases the likelihood of convergence to a global optimal solution and does not require homogenization of a suboptimum solution vector prior to restarting the optimization process of the penalty function for the lower values of the multipliers. The second is treatment of cases where the simplex strays into the feasible region. Results of testing the integer simplicial optimization procedure with the enhancements are compared to those obtained from methods developed specifically for dealing with this type of nonlinear integer problem. The tests were conducted for systems with various numbers of subsystems