In this paper, we study the number of failed components in a consecutive-k-out-of-n:G system. The distributions and expected values of the number of failed components when system is failed or working at a particular time t are evaluated. We also apply them to the optimization problems concerned with the optimal number of components and the optimal replacement time. Finally, we present the illustrative examples for the expected number of failed components and give the numerical results for the optimization problems.

A consecutive-k-out-of-n:G system consists of n components which are arranged in a line and the system works if and only if at least k consecutive components work. This paper discusses the optimization problems for a consecutive-k-out-of-n:G system. We first focus on the optimal number of components at the system design phase. Then, we focus on the optimal replacement time at the system operation phase by considering a preventive replacement, which the system is replaced at the planned time or the time of system failure which occurs first. The expected cost rates of two optimization problems are considered as objective functions to be minimized. Finally, we give study cases for the proposed optimization problems and evaluate the feasibility of the policies.

We propose a reality-based model of a two-echelon repair system with “priority resupply” and present a method for analyzing the availability of the system operated in each base. The two echelon repair system considered in our model consists of one repair station, called depot, and several bases. In each base, n items which constitute a k-out-of-n: G system, called k/n system, are operated. Each item has two failure modes, failures repaired at a base (level 1) and failures repaired at the depot (level 2). When a level 2 failure occurs in a base, either a normal order or an emergency order of a spare item is issued depending on the number of operating items in the base. The spare item in the depot is sent preferentially to the base where the emergency order is placed. We propose two models, both including priority resupply. Firstly, we propose an approximation method for analyzing the basic model where a k/n system is operated in a base. Using a simulation method, we verify the accuracy of our approximation method. Secondly, we expand the basic model to a dual k/n system where the items of the system are interchangeable between two k/n systems in the case of an emergency, which is called “cannibalization”. Then, we show a numerical example and discuss the optimal timing for placing an emergency order.

In this letter, the k-out-of-n rule for cooperative sensing is considered. For a given n, we derive the optimal value of k that minimizes the total sensing error probability subject to the sensing accuracy, considering both the effective of sensing errors and the primary activities. According to the optimal k, we analyze the performance and compare with other schemes, which illustrate the effectiveness of the proposed scheme.

A linear consecutive-k-out-of-n: F system is an ordered sequence of n components. This system fails if, and only if, k or more consecutive components fail. Optimal arrangement is one of the main problems for such kind of system. In this problem, we want to obtain an optimal arrangement of components to maximize system reliability, when all components of the system need not have equal component failure probability and all components are mutually statistically independent. As n becomes large, however, the amount of calculation would be too much to solve within a reasonable computing time even by using a high-performance computer. Hanafusa and Yamamoto proposed applying Genetic Algorithm (GA) to obtain quasi optimal arrangement in a linear consecutive-k-out-of-n: F system. GA is known as a powerful tool for solving many optimization problems. They also proposed ordinal representation, which produces only arrangements satisfying the necessary conditions for optimal arrangements and eliminates redundant arrangements with same system reliabilities produced by reversal of certain arrangements. In this paper, we propose an efficient GA. We have modified the previous work mentioned above to allocate components with low failure probabilities, that is to say reliable components, at equal intervals, because such arrangements seem to have relatively high system reliabilities. Through the numerical experiments, we observed that our proposed GA with interval k provides better solutions than the previous work for the most cases.

In this paper, first, we propose a new recursive algorithm for evaluating generalized multi-state k-out-of-n:F systems. This recursive algorithm can be applied to the systems even though the states of all components in the system are assumed to be non-i.i.d. random variables. Our algorithm is useful for any multi-state k-out-of-n:F system, including the decreasing, increasing and constant multi-state k-out-of-n:F system. Furthermore, our algorithm can evaluate the state distributions of the other non-monotonic multi-state k-out-of-n:F systems. Next, we calculate the order of computing time and memory capacity of the proposed algorithm. We perform numerical experiments in the non-i.i.d. case. The results show that the proposed algorithm is efficient for evaluating the system state distribution of multi-state k-out-of-n:F system when n is large and kl are small.

In this paper the problem of determining optimal workload for a load sharing system is considered. The system is composed of total n components and it functions until (n-k+1) components are failed. The works that should be performed by the system arrive at the system according to a homogeneous Poisson process and it is assumed that the system can perform sufficiently large number of works simultaneously. The system is subject to a workload which can be expressed in terms of the arrival rate of the work and the workload is equally shared by surviving components in the system. We assume that an increased workload induces a higher failure rate of each remaining component. The time consumed for the completion of each work is assumed to be a constant or a random quantity following an Exponential distribution. Under this model, as a measure for system performance, we derive the long-run average number of works performed per unit time and consider optimal workload which maximizes the system performance.

An exact and an approximated reliabilities of a 2-dimensional consecutive k-out-of-n:F system are discussed. Although analysis to obtain exact reliability requires many calculation resources for a system with a large number of components, the proposed method obtains the reliability lower bound by using a combinatorial equation that does not depend on the system size. The method has an assumption on the maximum number of failed components in an operable system. The reliability is exact when the total number of failed components is less than the assumed maximum number. The accuracy of the method is confirmed by numerical examples.

In an optical fiber ring topology network such as FDDI (Fiber Distributed Data Interface) rings and SONET (Synchronous Optical Network) rings, the number of consecutively bypassed failed stations is limited by the optical power loss constraint. In recent years, this situation was represented as a consecutive k-out-of-n:F system and the two-terminal reliability was presented in the literature, but K-terminal reliability has not been presented. In this paper, we obtain K-terminal reliability expressions for dual-counter rotating networks (DR's) that use both self-heal and station-bypass switches in which all components (stations, links and bypass switches) can fail. The results are useful in evaluating the reliabilities of FDDI ring networks parametrically and making reliability comparisons. This method can be used to obtain a closed-form reliability expression in a more general ring-network such as 'ring of trees. '

A generalized class of consecutive-k-out-of-n:G systems, referred to as Con/k*/n:G systems, is studied. A Con/k*/n:G system has n ordered components and is good if and only if ki good consecutive components that originate at component i are all good, where ki is a function of i. Theorem 1 gives an O(n) time equation to compute the reliability of a linear system and Theorem 2 gives an O(n2) time equation for a circular system. A distributed computing system with a linear (ring) topology is an example of such system. This application is very important, since for other classes of topologies, such as general graphs, planar graphs, series-parallel graphs, tree graphs, and star graphs, this problem has been proven to be NP-hard.

Mean Time Between Failures (MTBF) is an important measure of practical repairable systems, but it has not been obtained for a repairable linear consecutive-k-out-of-n: F system. We first present a general formula for the (steady-state) availability of a repairable linear consecutive-k-out-of-n: F system with nonidentical components by employing the cut set approach or a topological availability method. Second, we present a general formula for frequency of system failures of a repairable linear consecutive-k-out-of-n: F system with nonidentical components. Then the MTBF for the repairable linear consecutive-k-out-of-n: F system is shown by using the frequency of system failure and availability. Lastly, we derive some figures which show the relationship between the MTBF and repair rate µorρ(=λ/µ) in the repairable linear consecutive-k-out-of-n: F system. The figures can be easily used and are useful for reliability design.

For systems in which the probability that an incorrect output is observed differs with input values, we adopt the redundant usage of n copies of identical systems which we call the n-redundant system. This paper presents a method to find the optimal redundancy of systems for minimizing the probability of dangerous errors. First, it is proved that a k-out-of-n redundancy or a mixture of two kinds of k-out-of-n redundancies minimizes the probability of D-errors under the condition that the probability of output errors including both dangerous errors and safe errors is below a specified value. Next, an algorithm is given to find the optimal series-parallel redundancy of systems by using the properties of the distance between two structure functions.