Linear consecutive-k-out-of-n: F system reliability with common-mode forced outages

This paper presents a model of consecutive-k-out-of-n: F system subject to common-mode forced outages, whose interarrival times are independent and exponentially distributed. The objective is to analytically derive the mean operating time between failures for a non-repairable component system. The average system failure time and the system availability are also considered. Then, the model is extended to a system with repairable components and unrestricted repair, in which service times are exponentially distributed.     

Profit analysis of a k-out-of-n trichotomous system

This paper studies the profit function of a mathematical model of a k-out-of-n trichotomous system. Failure times are assumed to follow negative exponential distributions with different parameters whereas the repair time distributions are taken to be general. The analysis is carried out by using the supplementary variable technique.     

Reliability analysis of a repairable k-out-of-n system with some components being suspended when the system is down

A k-out-of-n system with independent exponential components is investigated. It is assumed that some working components are suspended as soon as the system is down, repair starts immediately when a component fails and repair times are independent and exponentially distributed. Formulas for various reliability indices of the system including mean time between failures, mean working time in a failure–repair cycle, and mean down time in a failure–repair cycle are derived.     

Application and analysis for a consecutive-k-out-of-n:G structure

A consecutive-k-out-of-n:G system consists of an ordered sequence of n components such that the system works whenever at least k consecutive components in the system function. This study elaborates on the concept of consecutive-k-out-of-n:G systems and demonstrates an application of the basic theory of consecutive-k-out-of-n:G systems reliability to a railroad operation. The results can help in scheduling the incoming trains to utilize the station resources better.     

Optimization problems in k-out-of-n systems with minimal repair

The k-out-of-n system is a system with n independent components such that the system operates if and only if at least k of these n components operate succesfully. Such a system is an important complex system and is used for mass transit, safety systems and computer systems. This paper considers what is the most economical k-out-of-n system with minimal repair. We solve two problems to minimise the long term expected cost rate per unit time; (i) the optimal number of elements, and (ii) the optimal replacement time before system failure. Finally, a numerical example is given.     

A consecutive k-out-of-n:G system with dependent elements—a matrix formulation and solution

The reliability of a consecutive k-out-of-n:G system is studied. The system may involve dependent trials, so that the probability of success of a certain test is increased after carrying out corrective actions. The problem is solved via a matrix formulation, using state space methods. Results are shown to correlate for the case of independent trials with those in the literature.     

Configuration determination for k-out-of-n partially redundant systems

The k-out-of-n configuration is a widely adopted structure for partially redundant safety systems. This configuration ensures a high level of reliability and safety with limited financial and space resources. It also facilitates on-line Test and Maintenance (T&M) without having to shut the system down. One question a decision maker needs to answer when adopting k-out-of-n systems is: what is the best configuration for the application, i.e. how many channels in total need to be utilized and among these channels, how many channels need to function simultaneously in order for the system to function. There are various factors to consider in order to make this decision. This paper looks at this problem from a reliability engineers point of view. A quantitative analysis is performed for both unavailability and probability of spurious operation due to independent failure. In particular, the relative gain and/or loss of these quantities that occurs due to changing from one configuration to another are compared through rigorous mathematical analysis. The results provide important information that can be used when choosing system configurations to meet regulatory requirements and financial constraints. The two different configurations for shutdown systems in Nuclear Power Plants, the 2-out-of-3 system and the 2-out-of-4 system, are utilized as an example to illustrate the theoretical results.     

System-level load–strength interference based reliability modeling of k-out-of-n system

The issue of information loss in the process of system reliability modeling through conventional load–strength interference analysis is discussed first, and the reason why it is impossible to construct dependent system reliability model simply by means of component reliability index is demonstrated. Then, an approach to modeling the reliability of dependent system with common cause failure (CCF) is presented. The approach is based on system-level load–strength interference analysis and a concept of ‘conditional failure probability of component’ as well. With the opinion that load randomness is the direct cause of failure dependence, a discrete type system reliability model is developed via the conditional component failure probability concept. At last, the model's capabilities to estimate system reliability with CCF effect and to predict high multiplicity failure probability based on low multiplicity failure event data are proved.     

Minimal cut sets of s–t networks with k-out-of-n nodes

In this paper, we extend traditional directed s–t network by letting nodes have k-out-of-n property: To generate output flows, a node must receive at least k flows from its n input links, where k is an integer assigned for the node and its value can be any number from 1 to n. To evaluate the system reliability, minimal cut sets for the extended network are defined for nodes. Under this definition, an extended network and its sink node have the same minimal cut sets. A new algorithm is designed to generate minimal cut sets for all nodes, starting with the source node and ending with the sink node. With different initializations, the algorithm can be applied for extended s–t networks with or without node failures.     

Exact reliability formula and bounds for general k-out-of-n systems

The paper deals with reliability of general k-out-of-n systems whose component failures need not be independent and identically distributed. The result is an exact closed form reliability formula which is based on Feller's result. The formula is efficient and easy to use for manual and computer computations. The approximations for the system reliability are given and are useful when dealing with large systems. Two examples illustrate the use of the results.     

A simple algorithm for evaluating the k-out-of-n network reliability

Evaluating the network reliability is an important topic in the planning, designing, and control of systems. The minimal cut set (MC, an edge set) is one of the major and fundamental tools for evaluating network reliability. A k-out-of-n MC is a special MC in a k-out-of-n network in which some nodes must receive at least k flows from their n input edges, where k is an integer number between 1 and n. In this study, an alternative method is given first to define a MC using a node set (called MCN) in k-out-of-n networks. A very simple algorithm based on some intuitive theorems that characterize the structure of the MCN and the relationship between MC and MCN is developed to solve the k-out-of-n network reliability by finding the k-out-of-n MCs between two special nodes. The proposed algorithm is not only easier to understand and implement, but is also better than the existing algorithm. The correctness of the proposed algorithm will be analyzed and proven. Two examples are illustrated to show how all k-out-of-n MCs are generated and verified in a k-out-of-n network using the proposed algorithm. The reliability of one example is then computing using one example.     

New insights into the assessment of k-out-of-n and related systems

k-out-of-n and related systems have received much attention in the recent past years. Hundreds of articles were devoted to various methods to assess them. In this article, we show that there exist very efficient algorithms to compute the reliability of k-out-of-n, l-to-h-out-of-n and consecutive k-out-of-n systems. k-within-r-out-of-n systems are intrinsically much harder. We study the performance of binary decision diagrams (BDDs) on these systems. Then, we propose a new approximation scheme. This algorithm is much more efficient in practice than already proposed methods.     

Optimal design of multi-state weighted k-out-of-n systems based on component design

This paper presents a study on design optimization of multi-state weighted k-out-of-n systems. The studied system reliability model is more general than the traditional k-out-of-n system model. The system and its components are capable of assuming a whole range of performance levels, varying from perfect functioning to complete failure. A utility value corresponding to each state is used to indicate the corresponding performance level. A widely studied reliability optimization problem is the “component selection problem”, which involves selection of components with known reliability and cost characteristics. Less adequately addressed has been the problem of determining system cost and utility based on the relationships between component reliability, cost and utility. This paper addresses this topic. All the optimization problems dealt with in this paper can be categorized as either minimizing the expected total system cost subject to system reliability requirements, or maximizing system reliability subject to total system cost limitation. The resulting optimization problems are too complicated to be solved by traditional optimization approaches; therefore, genetic algorithm (GA) is used to solve them. Our results show that GA is a powerful tool for solving these kinds of problems.     

A path-based algorithm for evaluating the k-out-of-n flow network reliability

Evaluating the network reliability is an important topic in the planning, designing and control of systems. The minimal path (MP, an edge set) set is one of the major, fundamental tools for evaluating network reliability. A k-out-of-n MP is a union of some MPs in a k-out-of-n flow network in which some nodes must receive flows from their k input distinctive edges (each input edge has one flow) to generate one flow, where k is an integer number between 2 and n. In this study, a very simple a-lgorithm based on some intuitive theorems that characterize the k-out-of-n MP structure and the relationship between k-out-of-n MPs and k-out-of-n MP candidates is developed to solve the k-out-of-n flow network reliability by finding the k-out-of-n MPs between two special nodes. The proposed algorithm is easier to understand and implement. The correctness of the proposed algorithm will be analyzed and proven. One example is illustrated to show how all k-out-of-n MPs are generated and verified in a k-out-of-n flow network using the proposed algorithm. The reliability of one example is then computing using the same example.     

A new algorithm for generating minimal cut sets in k-out-of-n networks

Evaluating the network reliability is an important topic in the planning, designing, and control of systems. A k-out-of-n network is a special network in that some nodes must receive at least k (>1) flows from all of their input edges (n). In real-life cases, many networks such as computer and telecommunications include k-out-of-n nodes for redundancy. The minimal-cut-node-set (MCN) is the major and fundamental tools for evaluating the k-out-of-n network reliability. In this study, a very simple algorithm based on some intuitive theorems that characterize the structure of the MCN is developed to solve the k-out-of-n network reliability. Compared to the existing algorithms, the proposed algorithm generates all k-out-of-n MCs without duplication based on fewer MCNs and fewer (k-out-of-n MC) candidates. The proposed algorithm is not only easier to understand and implement, but is also better than the existing algorithms. The correctness of the proposed algorithm will be analyzed and proven. One example is illustrated to show how all k-out-of-n MCs are generated, verified, and implemented to evaluate the k-out-of-n network reliability using the proposed algorithm.     

Reliability analysis of K-out-of-N: G systems with dependent failures and imperfect coverage

This paper presents a Markov model for reliability analysis of K-out-of-N: G systems subject to dependent failures with imperfect coverage. Closed form solutions of the probabilities are used to obtain the reliability and the mean time to failure (MTTF). A numerical example is provided to illustrate the results.     

Exact equation and an algorithm for reliability evaluation of K-out-of-N:G system

An expression for reliability of K-out-of-N:G system is proposed. An algorithm for computing reliability of K-out-of-N system is given. It is an easy to implement, fast and memory efficient algorithm and helps to improve the computational efficiency considerably.     

New bounds on the reliability of the consecutive k-out-of-r-from-n:F system

The main goal of this article is to show that the lower and upper bounds on the failing probability resp. reliability of consecutive k-out-of-r-from-n:F reliability systems developed by Sfakianakis, Kounias and Hillaris [M. Sfakianakis, S. Kounias, A. Hillaris, IEEE Trans. Reliability 1992;R-41:442–447] can be improved by applying fourth order Boole–Bonferroni bounds. Further we propose the application of the Hunter–Worsley bound in the framework of reliability system analysis, too. Numerical results of the formerly published test examples and harder problems are given. The computer code was written in and is available on request from the authors.     

Transient analysis of reliability with and without repair for K-out-of-N :G systems with M failure modes

This paper represents Markov models for transient analysis of reliability with and without repair for K-out-of-N :G systems subject to M failure modes. The reliability and the mean time between failures of repairable systems can be calculated as a result of numerical solution of simultaneous set of linear differential equations. Closed form solutions of the transient probabilities are used to obtain the reliability and the mean time to failure for nonrepairable systems.