This paper presents two approaches for computing the reliability of complex networks subject to two kinds of failure, open failure and shorted failure. The reliabilities of some series-parallel networks are considered by many analysts. However a practical system is more complex. The methods given in this paper can be applied not only to a series-parallel network but also to a non-series-parallel network which is composed of non-identical and independent components subject to two kinds of failure. This paper also deals with a network subject to flow quantity constraint such as the one which is required to control j or more separate paths. For such a system it is difficult to obtain system reliability because the number of states to be considered in this system is extremely large compared to a conventional 2-state device system. In this paper we obtain the reliabilities for such systems by a combinatorial approach and by a simulation approach.

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.

This paper presents two man-machine reliability models. A system consists of one machine unit, one operator, and one event detecting monitor. The machine unit has three states, normal, abnormal, and failed. The event detecting monitor may fail in two ways. If a machine unit becomes abnormal, the event detecting monitor sends a signal, and the operator takes necessary actions. If the operator fails in the action in the cause of human error, the machine unit goes down. The condition of the operator is classified into two types, good and bad. The time to repair, and the human error rate both depend on the condition of the operator. The MTTF is obtained by using a Markov model and numerical computation. Furthermore, the optimal operating period which minimizes the overall cost is decided by using computer methods. Some numerical examples are shown.

This paper considers a two-echelon repair model where several series systems comprising multiple items are operated in each base. We propose a basic model and two modified models. For two models, approximation methods are developed to derive the system availability. The difference between the basic model and the first modified model is whether the normal items in failed series systems are available as spare or not. The second modified model relaxes the assumptions of the first modified model to reflect more realistic situation. We perform numerical analysis for the models to compare their system availabilities and verify the accuracy of the approximation methods.

In this paper, we study on an availability analysis for a multibase system with lateral resupply of spare items between bases. We construct a basic model that a spare item of a base is transported for operation to another base without spare upon occurrence of failure, and simultaneously, the base that supplies the spare item receives the failed item of the other base for repair. We propose an approximation method to obtain the availability of the system and show the accuracy of the solution through numerical experiments. Also, two modified models are constructed to show the efficiency of the basic model. The two models modify the assumption on the lateral resupply of spare items between bases in the basic model. We numerically illustrate that the basic model can increase the availability of the system compared with the two modified models through Monte Carlo simulation.

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.

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.

In this paper we discuss the system failure probability of a k-out-of-n system considering common-cause failures. The conventional implicit technique is first introduced. Then the failure probabilities are formulated when the independence between common-cause failure events is assumed. We also provide algorithms to enumerate all the cut sets and the minimal cut sets, and to calculate the system failure probability. These methods are extendable to the case of systems with non-identical components. We verify the effectiveness of our method by comparison with the exact solution obtained by numerical calculation.

A one-shot system is a system that can be used only once during its life, and whose failures are detected only through inspections. In this paper, we discuss an inspection policy problem of one-shot system composed of multi-unit in series. Failed units are minimally repaired when failures are detected and all units in the system are replaced when the nth failure is detected after the last replacement. We derive the expected cost rate approximately. Our goal is to determine the optimal inspection policy that minimizes the expected cost rate.

A lattice system in this paper is a system whose components are ordered like the elements of (m, n) matrix. A representative example of a lattice system is a connected-(r, s)-out-of-(m, n):F lattice system which is treated as a model of supervision system. It fails if and only if all components in an (r, s) sub lattice fail. We modify the lattice system so as to include a maintenance action and a restriction on the number of failed components. Then, this paper presents availability and MTBF of the repairable system, and reliability when the system stocks spare parts on hand to ensure the specified reliability level.

This paper discusses the resilience of networks based on graph theory and stochastic process. The electric power network where edges may fail simultaneously and the performance of the network is measured by the ratio of connected nodes is supposed for the target network. For the restoration, under the constraint that the resources are limited, the failed edges are repaired one by one, and the order of the repair for several failed edges is determined with the priority to the edge that the amount of increasing system performance is the largest after the completion of repair. Two types of resilience are discussed, one is resilience in the recovery stage according to the conventional definition of resilience and the other is steady state operational resilience considering the long-term operation in which the network state changes stochastically. The second represents a comprehensive capacity of resilience for a system and is analytically derived by Markov analysis. We assume that the large-scale disruption occurs due to the simultaneous failure of edges caused by the common cause failures in the analysis. Marshall-Olkin type shock model and α factor method are incorporated to model the common cause failures. Then two resilience measures, “operational resilience” and “operational resilience in recovery stage” are proposed. We also propose approximation methods to obtain these two operational resilience measures for complex networks.

A method of calculating the exact top event probability of a fault tree with dynamic gates and repeated basic events is proposed. The top event probability of such a dynamic fault tree is obtained by converting the tree into an equivalent Markov model. However, the Markov-based method is not realistic for a complex system model because the number of states that should be considered in the Markov analysis increases explosively as the number of basic events in the model increases. To overcome this shortcoming, we propose an alternative method in this paper. It is a hybrid of a Bayesian network (BN) and an algebraic technique. First, modularization is applied to a dynamic fault tree. The detected modules are classified into two types: one satisfies the parental Markov condition and the other does not. The module without the parental Markov condition is replaced with an equivalent single event. The occurrence probability of this event is obtained as the sum of disjoint sequence probabilities. After the contraction of modules without parent Markov condition, the BN algorithm is applied to the dynamic fault tree. The conditional probability tables for dynamic gates are presented. The BN is a standard one and has hierarchical and modular features. Numerical example shows that our method works well for complex systems.

In this paper an analysis of component and system reliability for lattice systems is proposed when component failures are not statistically independent. We deal the case that the failure rate of a component depends on the number of the adjacent failed components. And we discuss the maintainability of the system when a failed component is replaced by a spare component. At first we discuss the approximated reliability of each component. Then we estimate the mean number of failed components. Furthermore, the system reliability is approximated by using the component reliability.

A method of calculating the top event probability of a fault tree, where dynamic gates and repeated events are included and the occurrences of basic events follow nonexponential distributions, is proposed. The method is on the basis of the Bayesian network formulation for a DFT proposed by Yuge and Yanagi [1]. The formulation had a difficulty in calculating a sequence probability if components have nonexponential failure distributions. We propose an alternative method to obtain the sequence probability in this paper. First, a method in the case of the Erlang distribution is discussed. Then, Tijms's fitting procedure is applied to deal with a general distribution. The procedure gives a mixture of two Erlang distributions as an approximate distribution for a general distribution given the mean and standard deviation. A numerical example shows that our method works well for complex systems.

The maintenance of a system on a ship has limitations when the ship is engaged in a voyage because of limited maintenance resources. When a system fails, it is either repaired instantly on ship with probability p or remains unrepaired during the voyage with probability 1-p owing to the lack of maintenance resources. In the latter case, the system is repaired after the voyage. We propose two management policies for the overhaul interval of an IFR system: one manages the overhaul interval by number of voyages and the other manages it by the total voyage time. Our goal is to determine the optimal policy that ensures the required availability of the system and minimizes the expected cost rate.