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A common cause failure occurs when two or more elements fail due to a shared cause. The system with such failure often needs different repairmen and a multi-level maintenance strategy. The paper studies the reliability of k-out-of-n: G repairable systems considering common cause failure and multi-level maintenance strategy. The models with a multi-level maintenance strategy are established by considering the failure of all and partial components due to common cause failure. The reliability indices of the systems, such as availability and the mean time to first failure, are derived. An optimization maintenance model is established by minimizing the cost, and a three-level maintenance strategy is determined. Numerical examples are provided to illustrate the application. The results show that the multi-level maintenance strategy improves the system reliability and reduces the maintenance costs of systems with common cause failure.

Traditional binary-state systems have only two discrete system states: functional and non-functional. However, many practical facilities have more than two states and should be studied as a multi-state system (MSS). This chapter makes new contributions by modelling and analyzing the reliability and safety of MSSs with dependent component lifetimes. It discusses modelling and analyses of the reliability of dependent MSSs. The copula is used to model the dependent relationships of different components of an MSS. The copula connects a multivariate distribution to its marginal in such a way that it captures the entire dependence structure in the multivariate distribution. The copula selection problem can be split into three stages: get the lifetime data of components and the system; use a statistical method to estimate the parameters of the candidate copulas; and choose the appropriate copula from the candidates. The chapter presents a multi-state model, and the copula model is also applied to address the dependent relationship within the system.

The paper aims to find the relationship between conventional reliability and supply chain reliability, and introduce and adapt conventional reliability models to the field of supply chains, expanding the horizon of solving supply chain reliability problems. Based on a comprehensive literature review, the paper summarizes definitions of reliability in supply chain systems and presents reliability system structures and reliability indexes for supply chains. Relationship and differences between conventional reliability and supply chain reliability are shown. Illustrative examples such as the supply chain reliability problem in China are provided to show how to convert a supply chain reliability problem into a conventional reliability problem and then solve it using reliability techniques in conventional reliability

Copulas have proved to be very successful tools for the flexible modeling of dependence. Bivariate copulas have been deeply researched in recent years, while building higher-dimensional copulas is still recognized to be a difficult task. In this paper, we study the higher-dimensional dependent reliability systems using a type of decomposition called “vine,” by which a multivariate distribution can be decomposed into a cascade of bivariate copulas. Some equations of system reliability for parallel, series, and k-out-of-n systems are obtained and then decomposed based on C-vine and D-vine copulas. Finally, a shutdown system is considered to illustrate the results obtained in the paper.

A model for two-unit series multistate Markov repairable systems with repair time omission is developed based on a model for single-unit multistate Markov repairable systems. In the model, each component has three states: perfect operation, minor failure, and complete failure. Repair time for a complete failure, if it is sufficiently short (no longer than a critical value), does not affect the system operation. The critical value is first supposed to be a constant, and then to be a nonnegative random variable. Some reliability indexes, such as the instantaneous availability, the steady-state availability, and the mean time to first failure, are given. Finally, some numerical examples are presented to illustrate the results obtained in the paper.

In a parallel structure load-sharing system, the failure rate of the operating components will usually increase, due to the additional loading induced by the other components' failure. Hence failure dependency exists among components. To quantify the failure dependency, a dependence function is introduced. Under the assumptions that the repair time distributions of components are arbitrary and life times are exponential distributions whose failure rates vary with the number of operating components, a new load-sharing parallel system with failure dependency is proposed. To model the stochastic behavior of the system, the Semi-Markov process induced by it is given. The Semi-Markov kernel associated with the process is also presented. The availability and the time to the first system failure are obtained by employing Markov renewal theory. A numerical example is presented to illustrate the results obtained in the paper. The impact of the failure dependence on the system is also considered.

The computation of reliability characteristics of a system that consists of dependent components sometimes becomes difficult, especially when a specific type of dependence is not identified. In this paper, some systems with arbitrary dependent components are studied using copula. In the system, the components are dependent on each other and the dependent relations may be either linear or nonlinear correlation. The efficient formulas are presented to compute the reliability characteristics, such as reliability function, failure rate and meantime to failure of series, parallel and k-out-of-n systems. The reliability functions of dependant systems are compared with independent system. At last, the numerical examples are presented to illustrate the results obtained in this paper.

The problem of stochastically allocating redundant components to increase the system lifetime is an important topic of reliability. An optimal redundancy allocation is proposed, which maximizes the expected lifetime of a reliability system with subsystems consisting of components in parallel. The constraints are minimizing the total resources and the sizes of subsystems. In this system, each switching is independent with each other and works with probability p. Two optimization problems are studied by an incremental algorithm and dynamic programming technique respectively. The incremental algorithm proposed could obtain an approximate optimal solution, and the dynamic programming method could generate the optimal solution.

In multi-state systems with load-sharing, the load will be redistributed among working units while some units go to failure. Hence dependencies exit among units. To model the evolution of these systems, a multi-state Markov repairable system with redundant dependencies is introduced in this paper. More than two states are allowed for each unit of the system, including perfect working, deterioration and complete failure. The state transition rate of each working unit is a function of the number of functioning units and is quantified by a redundant dependence function. A two-dimensional vector, whose elements denote respectively the number of units that are in perfect and degraded operating states, is presented to describe accurately the performance of the system. The state space of the system is divided into distinct state sets according to the value of the vector. The visit probability to a specified state set (the acceptable, excellent, operating and warning state sets), the time to the first sys...

Supply chain management has received considerable attention in the literature and it is meaningful and important to be able to measure the reliability of supply chains. In the article, the suppliers in the supply chain systems are not independent of each other and the dependency relation may be either linear or nonlinear correlation. From the view of the distribution service process, a copula-based method is proposed for analyzing the reliability of supply chains. In this article, by introducing the model of k-out-of-n: G system into the studies of supply chains, an evaluation method is suggested and the reliability indexes are obtained. Finally, a numerical example is presented to illustrate the results obtained in this article.

In safety-critical systems, components with high reliability are required. Once a component's reliability fails to meet the minimum safety requirements, either the component should be replaced or, transferred to another system which is allowed to operate on a lower reliability level. Safety standards are not only related to the cost of maintenance, but also to the system's (wear-and-tear) and repair degrees. In this paper, two different repair degrees are discussed, and the best safety virtual age criterion is studied in order to minimize financial cost. These results are supported by numerical examples.

The computation of reliability characteristics of a system that consists of dependent components is sometimes difficult especially when the type of dependence is not known. This article introduces the copula method to calculate the reliability of dependent consecutive k-out-of-n: G systems. The components in these systems are dependent on each other and the dependency may be either linear or nonlinear. The copula is a popular tool for modeling the dependence structure of data. It contains the information about the dependency structure of a vector of random variables and can capture nonlinear dependence. Based on the copula theory, the article analyzes the consecutive k-out-of-n: G systems and gets the reliability indexes. Finally, some numerical examples are presented to illustrate the results obtained in this article.

From the viewpoint of a distributed service process, an ion-channel based method was proposed to analyze the reliability of supply chains. An analytic method was presented to determine the reliability of supply chains for logistics systems. New models were built to consider the following situations: when the nodes failed, different defect states had different measures and effects on the system. A Markov model was introduced to analyze the system safety and aggregate states to get the number of repairs and length of time before the system became dangerous by using ion-channel modeling theory. Finally a numerical example was presented to illustrate the results obtained in this paper.

In safety-critical systems, components must be highly reliable. Once a component's reliability is lower than the criterion, it will be replaced and transferred to be used in another system which does not need such high reliability. The safety virtual age criterion not only relates with the cost of repair actions, but also with the repair degrees and the systempsilas wearing degrees. In this paper, we considered different repair degrees and studied the best safety virtual age criterion to minimize the cost. Finally some numerical examples are presented to illustrate the results obtained in this paper.

From the point of the process of distribution service, a Markov chain based method is proposed for analyzing the reliability of supply chain. This paper presents an analytic method for determining the reliability of supply chains for logistics systems. By introducing the model of k -out-of- n :G system in reliability into the studies of supply chain, an evaluation method is suggested. And the reliability of distribution service is obtained when the distribution center operates steadily. These indexes have practical and significant meaning for evaluating the supply chain system reliability performance and can also be used to improve the reliability management.

Under the condition of stochastic default intensity, correlation of default intensity and default-free interest rate, the paper constructed the term structure model of credit risk. A Markov chain based method is proposed for analyzing the credit risk Based on reliability interference theory, we get the indexes such as the default probability, the mean time to the first default, steady probability vector of the credit rating and so on. By introducing the model series system in reliability into the studies of credit risk, the paper constructed a portfolio model with several bonds, and got the solution. The results from this study are of value in credit risk management. They can be used in evaluating and measuring the performance of credit risk and can provide significant help and guidance.

For analyzing the reliability of supply chains, there has been a renewed interest in Markov chain for its attractive properties during the recent years. However, it is far from generalization in terms of formulating a model for the first-order Markov chains. In this paper, from the view of the process of distribution service, a higher order Markov chain based method is proposed for analyzing the reliability of supply chains. This paper presents an analytic method for determining the reliability of supply chains for logistics systems. By introducing the model of k -out-of- n : F system in reliability into the studies of supply chains, an evaluation method is suggested, and the reliability indexes of distribution service are obtained.

In safety-critical systems, there are both dangerous and safe failure states. This paper presents different models of safety-critical systems including series and parallel systems whose lifetime distributions are general forms. Also the reliability aspects are derived such as the probabilities that the system is in working state, fail-safe state and fail-dangerous state, the mean times for the system working, fail-safe and fail-dangerous. Finally some numerical examples are presented to illustrate the results obtained in this paper. These indexes have practical and theoretical meanings for the systems. The derived formulas are new results and can be used regardless of the forms of component lifetime distributions and have significant meaning for evaluating the system safety performance and can also be used to improve the safety design of systems.

As a compromise between parametric regression and nonparametric regression, partially linear models are frequently used in statistical modelling. This article considers statistical inference for this semiparametric model when the linear covariate is measured with additive error and some additional linear restrictions on the parametric component are assumed to hold. We propose a restricted corrected profile least-squares estimator for the parametric component, and study the asymptotic normality of the estimator. To test hypothesis on the parametric component, we construct a Wald test statistic and obtain its limiting distribution. Some simulation studies are conducted to illustrate our approaches.