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.

This paper presents an approximation method for deriving the availability of a parallel redundant system with preventive maintenance (PM) and common-cause failures. The system discussed is composed of two identical units. A single service facility is available for PM and repair. The repair times, the PM times and the failure times except for common-cause failures are all assumed to be arbitrarily distributed. The presented method formulates the problem of the availability analysis of a parallel redundant system as a Markov renewal process which represents the state transitions of one specified unit in the system. This method derives the availability easily and accurately. Further, the availability obtained by this method is exact in a special case.

A new method to obtain the availability of a cold standby series system with spare units is presented. Two models are considered. The first one is a series system with spare units. The other is m series systems with common spare units. The availabilities are solutions of nonlinear simultaneous equations and are obtained numerically.

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.

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.

This paper considers an availability analysis and an optimal inventory problem for a repairable 1-out-ofN:G system assuming an ordering policy of (s, s+1) type. The system consists of N identical subsystems which constitute 1-out-ofN:G, and each subsystem is a m units series system. Since the system is repairable, the exact evaluation of the system availability is extremely difficult. In this paper, the system availability is obtained by an approximation analysis. The results are reasonably accurate and are much easier than the exact evaluation. Then the optimal inventory problem is discussed. The numerical example shows that the solution is obtained relatively easily when the system consists of highly reliable units.

This paper considers a fleet system which consists of n equipments. The system is operated intermittently repeating a system operation period and a system standby period alternatively. In each system operation period, any c of n equipments are required to operate. Each equipment undergoes overhaul (O/H) after prespecified periods (or numbers) of operation. We obtain the availability of the system assuming the following disciplines for the O/H interval and the failure probability. The O/H interval is scheduled by calendar time (Case 1) or actual operating time (Case 2). The failure probability of an equipment is constant in each operation (Model 1) or depends on the number of operations after O/H (Model 2). This analysis aims at the determination of the rational O/H interval and the number of the equipments in the system. Also it can be applied to the problem of spares provisioning for a system with O/H.

Determination of spare quantity assuming a general failure distribution is discussed. A new method is presented which is an extension of a usual normal approximation method. The original method is essentially a Normal distribution approximation" to a Poisson process of failure occurrence. On the other hand, the new method assumes a general failure distribution. The mean and the variance of the number of failures within for a given period of time are necessary for determining the spare quantity. The mean is obtained in a simple form. The variance is give as a solution of an integral equation. The solution of this equation for a general case is obtained by applying a discrete approximation technique. Some numerical examples are provided to discuss the difference between when assuming a general failure distribution and when assuming the exponential failure distribution.

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.

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 considers a repair system which includes several repair stations, many user sites and a great many identical equipments operated in these sites. The means and the variances of the numbers of the equipments in these user sites and repair stations are obtained using an approximation method. The proposed repair model is applicable to a complex repair system such as a multi-echelon repair system.

This paper considers a two-unit warm standby redundant system. Preventive maintenance (PM) for an active unit is scheduled after a certain period. When an active unit fails or undergoes PM while the other is in standby, the operation is switched to the standby unit. Probability of success in switchover is constant. If the system fails in switchover, the system stops the operation, and resumes it after a certain time. MTTFF, MTBF and steady state availability are obtained. Numerical examples show the effects of PM interval on them.

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.