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In this paper, state space of a time-homogeneous Markov process is split into several ordered subspaces. Then, there are three kinds of transitions between states—transitions from a higher-order subspace to a lower-order subspace, transitions within the same subspace, and transitions from a lower-order subspace to a higher-order subspace. Considering time interval omission problem and first passage time considered, we define some related counting processes for the Markov process and discuss their associated probabilities, expectations and generating functions by using Laplace transform. The main results are presented in matrix forms. The relationships among counting processes and their special cases are also discussed briefly. Some simple numerical examples are presented to illustrate the established results. These results will be useful in different problems arising in reliability, economics and social science fields. [ABSTRACT FROM AUTHOR]

A warranty product often has an interval structure, resulting in different warranty periods for each component due to their distinct degradation paths. From the manufacturer's perspective, it is reasonable to establish appropriate warranty periods for each part to achieve cost savings. This paper introduces a dependent series configuration consisting of a main component and an auxiliary component. The failure of the main component leads to system failure, while the failure of the auxiliary component only causes a random damage to the main component. These two components follow different Wiener degradation paths, and then their first passage times (FPTs) follow disparate inverse Gaussian distributions. Assuming a pre-deterministic warranty period, W , for the main component, and η W , 0 < η < 1 , for the auxiliary component, the warranty period can be divided into two stages: [ 0 , η W ] and [ η W , W ]. In this context, the paper proposes warranty claim policies for replacement, objective-oriented imperfect repair (OOIR), and refund. The corresponding warranty costs sustained by manufacturer and consumer within these two stages are considered. Finally, a numerical example is presented to illustrate the warranty costs sustained by manufacturer and consumer under different scenarios, as well as the total warranty costs. • The degradation paths, warranty periods and warranty costs of the series system is analyzed. • The warranty costs sustained by manufacturer and consumer under different scenarios is analyzed. • The interdependence between various components is considered. [ABSTRACT FROM AUTHOR]

• Two-kind combined reliability systems are introduced in terms of two reliability systems: (n , f , k) and < n , f , k >. • Some important concepts such as irreducible structure and structure equivalence are introduced. • We present an important criterion: four-class combined reliability systems are needed to be considered in essential. • A coordinate system is introduced for representing the working and failure regions for the combined reliability systems. • A procedure for calculations of the combined reliability systems is given. In the paper, two-element (f , k) and < f , k > combined systems are developed, and two-kind combined reliability systems and some related new concepts such as irreducible structure and structure equivalence are introduced, which can be used not only in real practical situations, but also enrich the reliability contents. We proved that four-class combined reliability systems need to be considered actually, which facilitates the handling of these combined reliability systems. Meanwhile, a coordinate system with two axes, the number of total failed components and the number of maximal consecutive failed components, is introduced in the paper for representing the working and failed areas for the combined reliability systems, which brings out a useful tool for understanding the working and failure mechanisms, and system state transition ways, for the combined reliability systems. To summarize the reliability calculation, a procedure for calculations of the combined reliability systems is presented, which is an efficient guidance for these reliability computations. Finally, two numerical examples are given to illustrate the computation procedure in which the detailed steps, special on using Finite Markov Chain Imbedding Approach (FMCIA), are shown. We believe that the present paper can shed light on researches on combined reliability systems. [ABSTRACT FROM AUTHOR]

• A new optimal maintenance model is proposed for a three-unit parallel production system. • The goal of maintenance policy is to maximize the system's yields. • We obtained the expectation yields of the system and found the optimal maintenance policy by comparing the expectation yields. • A procedure for calculations of the optimal maintenance policy is given. Many production facilities consist of multiple parallel units to facilitate the completion of production plans, yet these devices are inevitably prone to malfunctions with increased operating time. Therefore, developing the optimal maintenance policy to maximize the production capacity of production facilities is a crucial issue. This paper focuses on determining the optimal maintenance policy for a three-unit production system, aiming to maximize its yield. The optimal solution is provided based on the given maintenance policy, and a general optimal maintenance policy for the system is derived. The properties related to the optimal maintenance policy are analyzed. The proposed model offers two notable contributions when compared to existing research. Firstly, a maintenance policy based on the number of visits and yields is developed. Secondly, the parallel assumption of the system and the assumption that the subsystems satisfy any distribution are in line with reality, enriching the research on maintenance strategies for production systems. To illustrate the results of the model, a numerical example is presented. [ABSTRACT FROM AUTHOR]

• In a shock environment, the system's self-healing mechanism is considered. • The damage evolution effect of shock is proposed. • In a dynamic self-healing environment, two competing failure modes are presented. • The change in degradation rates is considered. We consider a system subject to two distinct failure modes, i.e., a soft failure subject to gradual degradation, and a hard failure caused by cumulative shocks, whichever comes first. This type of systems has wide applications and has been extensively studied in the literature. However, little attention has been paid to the dynamic self-healing mechanism in such systems. With the emergence of intelligent systems and complex architectures, materials with intrinsic self-healing mechanism have the ability to recover damage under external random shocks. In order to illustrate practical significance of the proposed model, reinforced concrete beams (RC-beam) with self-healing capsules in dependent competing failure processes due to cumulative crack evolution effect and rebar performance degradation are taken as the application background. In this paper, the change of degradation rates and damage evolution effect are considered based on the design of critical failure threshold and self-healing mechanism. This study proposes a novel reliability model for a system experiencing dependent competing degradation and cumulative shock failure in a dynamic self-healing environment. A procedure is developed to evaluate the reliability based on this model. We demonstrate the fitness of proposed model through numerical simulation and sensitivity analysis. [ABSTRACT FROM AUTHOR]

• A system subject to both shocks and environment-affected degradation is investigated. • Maintenance actions related to randomly evolving environments are considered. • Explicit formulas for the cumulative distribution function of first passage time and system availability are derived. • A Monte Carlo simulation algorithm for computing the system first passage time is provided. • A detailed case study is conducted to demonstrate derived formulas. Many systems or devices may experience degradation and shocks simultaneously whose behaviors may have discrepancies in different system operating environments. In this paper, a reliability model is developed for systems subject to both sudden shocks and natural wear processes in randomly evolving environments. The natural wear behavior of the system under different environment is governed by a distinct Gamma process. The system fails when the overall degradation which contains the natural wear and the cumulative damage caused by previous arrival shocks hits a preset threshold. To calculate the cumulative distribution function of the first passage time, the explicit computation formula based on analytical methods and the simulation algorithm based on Monte Carlo simulation methods are provided, which could verify each other. Further, a corrective replacement policy is considered in the case where the environment switching process takes place only when the system is functioning, and then the formula for the system availability is analytically derived. Finally, a study case of the lithium-ion battery is given to illustrate the proposed model and obtained results. [ABSTRACT FROM AUTHOR]

Hawkes processes have been widely studied in both theory and applications because of their self-exciting effects, but there are still some questions and room left for exploring the self-exciting point processes. In this paper, an ephemeral self-exciting Hawkes process is developed, which contains 1-memory and 0-memory self-exciting point processes. After giving the definition of the ephemeral self-exciting Hawkes process, some explanations on this new point process, properties, declustering of the occurrences of events, positive and negative self-exciting effects are presented. Moments of this point process are derived as well in the paper by using the Laplace transforms and the elementary method. Some special cases are considered and the computation methods of moments are also presented by the difference method and inverse Laplace transforms, respectively. Combining dynamics of the self-exciting point process and the delayed renewal process is an important feature of the proposed ephemeral self-exciting Hawkes process, which may shed light on the related studies in the future. [ABSTRACT FROM AUTHOR]

Exploring on reliability modeling and analysis on a marine equipment in a dynamic environment is a meaningful and challenging issue, because the system commonly carries out the task at sea away from land and suffers a distinct influence of environment. Thus, a reliability model of a multi‐state repairable system operating in dynamic environment is developed by introducing the background of the marine power system in this paper. The novelty of the research lies in the modeling and computing methods are relatively innovative by employing the aggregated stochastic processes, Hadamard production and matrix‐analytic method. First, the working modes of the marine power system under several different kinds of conditions are introduced. Then, the evolution of both the system states and environment are described as discrete‐time Markov chains with multiple and different transition probability matrices. The failure probability and repair probability of components are also distinct in different environments. Furthermore, some performance indexes, especially the index relevant to the environment, are derived, respectively. Finally, the conclusion is obtained by a numerical example of the marine power system, which also illustrates the validity and applicability of the proposed model. [ABSTRACT FROM AUTHOR]

Compared to binary-state system models, multistate (MS) system models are more practical and useful when describing complex systems in applications, such as communication systems, oil pipeline systems, and grid or cloud computing infrastructures. This paper models a generalized sparsely connected MS consecutive- ${k}$ -out-of- ${n}$ : ${G}$ system. The proposed model is regarded as a natural extension of the consecutive- ${k}$ -out-of- ${n}$ system. Moreover, the proposed system model could cover three special MS submodels, i.e., a decreasing MS: ${G}$ system model, an increasing MS: ${G}$ system model, and a nonmonotonic MS: ${G}$ system model. Three corresponding real-world examples are given to illustrate the applications of the model. Meanwhile, by utilizing the well-known finite Markov chain imbedding method, the explicit system reliability and state distribution for the three kinds of submodels are presented, respectively. Finally, the developed indexes and analysis method are demonstrated by numerical examples. [ABSTRACT FROM AUTHOR]

We study a model of a two-stage degradation process in a dynamic environment. The two stages, the normal stage and the defective stage, are separated by the first hitting time of the alarm threshold by the degradation level. Wiener processes with piecewise linear drift are used in each stage to describe the degradation level in a dynamic environment. System failure is triggered in two ways: the system degradation level reaches the defect-based failure threshold; the duration in the defective stage is larger than the duration-based failure threshold. Explicit expressions for the system reliability for different duration-based failure thresholds are obtained. These include when the duration-based failure threshold is zero, when it is a positive constant and when it tends to infinity. A simulation procedure is described for the case in which the duration-based failure threshold is a random variable. Finally, some numerical examples are presented to illustrate the proposed reliability assessment method. The modelling method and the results can be used for reliability evaluation, residual life prediction and maintenance decision optimization of a system with two-stage degradation in a dynamic environment. [ABSTRACT FROM AUTHOR]

Linear Hawkes processes are widely used in many fields and means are the basic and critical information of them. However, there is little research on linear Hawkes processes' means. In this paper, we present a numerical method based on the Laplace transform and inverse Laplace transform for means of linear Hawkes processes. The advantage of this method is that whatever the kernel function is, we can always obtain the numerical solutions of means for a linear Hawkes process. In addition, this numerical method provides the basic information of linear Hawkes processes by means. As an application, the numerical method is applied in a WeChat network model. [ABSTRACT FROM AUTHOR]

In this paper, we consider four different kinds of multistate consecutive- ${\boldsymbol{k}}$ systems, namely a multistate linear ${\boldsymbol{m}}$ -consecutive- ${\boldsymbol{k}}$ -out-of- $n$ : $G$ system, a multistate linear consecutive- ${\boldsymbol{k}}$ -out-of- $n$ : $G$ system with sparse ${\boldsymbol{d}}$ , a multistate linear ${\boldsymbol{m}}$ -consecutive- ${\boldsymbol{k}}$ -out-of- $n$ : $G$ system with sparse ${\boldsymbol{d}}$ , and a multistate linear $ < n,{\boldsymbol{f}},{\boldsymbol{k}} > :G$ system. The reliability formulas for these systems are derived by using the finite Markov chain imbedding approach. Unlike some previous research, the vectors ${\boldsymbol{m}},{\boldsymbol{k}},{\boldsymbol{d}}$ , and ${\boldsymbol{f}}$ do not need to be monotonic in this paper. Some illustrative examples are also presented to show how the reliabilities of the four multistate consecutive-k systems are calculated. [ABSTRACT FROM AUTHOR]

• A two-stage degradation system with two performance characterizes is considered. • A correlated bivariate Wiener process is adopted in modeling. • The explicit expressions of system reliabilities and availabilities are derived. • Two algorithms for the simulations of system reliabilities are given. It is still a challenge to study the degradation mechanisms of complex systems with multiple performance characteristics. This paper develops a two-stage stochastic degradation model. The degeneration processes of two correlated performance characteristics are described by a correlated bivariate Wiener process in the first stage, in which neither of the degradation levels of performance characteristics reaches their thresholds. When one of the degradation levels of performance characteristics reaches its threshold, the system operates defectively in the second stage, and the system degradation process is described by the other performance characteristic which is modeled by a univariate Wiener process. The system fails completely when both the degradation levels of performance characteristics reach their thresholds. Explicit expressions of the system reliability and availabilities are given based on different inspection and maintenance policies. Simulation is also presented, and it shows that the analytical results and the simulation results are in good agreement. Finally, a detailed case study of the rail track geometry degradation is given to demonstrate the results obtained in the paper. [ABSTRACT FROM AUTHOR]

It is a well-known definition that the time until entering an absorbing state in a finite state Markov process follows a phase-type distribution. In this article we extend this distribution through adding two new events: one is that the number of transitions among states reaches a specified threshold; the other is that the sojourn time in a specified subset of states exceeds a given threshold. The system fails when it enters the absorbing state or two new events happen, whichever occurs first. We develop three models in terms of three circumstances: (i) the two thresholds are constants; (ii) the number of transitions is random while the sojourn time in the specified states is constant; and (iii) the sojourn time in the specified states is random while the number of transitions is constant. To the performance of such systems, we employ the theory of aggregated stochastic processes and obtain closed-form expressions for all reliability indexes, such as point-wise availabilities, various interval availabilities, and distributions of lifetimes. We select special distributions of two thresholds for models 2 and 3, which are the exponential and geometric distributions. The corresponding formulas are presented. Finally, some numerical examples are given to demonstrate the proposed formulas. [ABSTRACT FROM AUTHOR]

• A repairable multi-state two-phase mission system with finite number of phase switches is proposed. • The probability density function for the time to first failure of the two-phase mission system is presented. • Some reliability indexes, such as the point-wise and interval availabilities, are given. In this article, a repairable multi-state two-phase mission system with finite number of phase switches is considered. System states can be divided into a subset of working states and a subset of failed states, and the system can be repaired from the failed subset to the working subset. However, when the number of switches between two phases exceeds a specified integer, the system will be permanently damaged and cannot be repaired any more. The closed-form formulas of the probability density function of the time to first failure of the system, the point-wise availability, the interval availability and other reliability indexes are obtained by using the theory of aggregated stochastic processes. Finally, a detailed example of lithium-ion batteries is given to illustrate the proposed model and obtained results. [ABSTRACT FROM AUTHOR]

A partial self-exciting point process is proposed based on some phenomena in real world, which is a new kind of Hawkes process with partial self-exciting effect. The features, properties and moments of the partial self-exciting point process are investigated in terms of Dynkin formula and the method proposed by Cui et al. (2017), and the relationship between the new and conventional Hawkes processes is given. The maximum-likelihood estimations (MLEs) for the parameters in the partial self-exciting point process for two data situations are given via some sets of equations that can be solved numerically by suing the commend fslove in Matlab. Some numerical examples are presented based on simulation data to illustrate the MLE method, and an algorithm to generate the random event times for the partial self-exciting point process is presented as well. A primary goal of the work is to extend the applicability of Hawkes process and to present the MLE of parameters for partial self-exciting point process. [ABSTRACT FROM AUTHOR]

• Characterizing the impact of shocks on system failure behavior via random virtual age increments in normal and defective stages. • Developing a duration-based mission abort policy based on the two-stage failure process. • Analytically deriving mission success probability and system survivability under the proposed stochastic failure process and mission abort policy. • Investigating the optimal mission abort policy balancing the tradeoff between mission success probability. Safety critical systems such as aircrafts, submarines and space stations are required to perform various missions. To enhance the survivability of such systems, a mission can be aborted when a certain malfunction or incident condition is satisfied and a rescue procedure should be activated. This paper develops a novel mission abort policy for systems experiencing both internal failure and external shocks. Failure process of the system can be divided into two stages from new to the initialization of a defect, and from that to failure. Motivated by the virtual age concept, the impact of external shocks is characterized by random virtual age increment in the two stages. We consider a policy where a mission is aborted if the duration in defective state is larger than a given threshold. Under the stochastic failure model and mission abort policy, mission success probability and system survivability are evaluated and the optimal abort threshold balancing the tradeoff between the system survivability and the mission success probability is investigated. A numerical example on a cooling system in chemical reactors is given to illustrate the applicability of the abort policy. [ABSTRACT FROM AUTHOR]

Research on reliability for balanced systems has become an important issue in the reliability field; it is being intensively studied both in theory and applications. The concept of balance has many implications, which results in various balanced systems needing to be studied in reliability modelling, analysis and optimization and so on. In the present article, two balanced reliability systems are introduced and their reliability models are developed, then formulas for system reliability, probability density functions, and moments of lifetimes under two models are obtained using the results in aggregated stochastic processes or phase-type distributions for two-dimensional and high-dimensional situations. In addition, a matrix operator, the Kronecker product operator, is used to create concise forms for the related formulas. An algorithm for computing these reliability indexes is given to facilitate use. Some examples including symbolic and numerical ones for the simplest case, the two-dimensional situation, are shown to illustrate the presented results. The presented work may shed light on future research on reliability and safety for balanced systems under different concepts of balance. [ABSTRACT FROM AUTHOR]

Many practical systems are subject to neglected down time, i.e., if the down time is less than a given threshold, then the system can be considered as working during the down time. Otherwise, the system is considered as in the failure state. This paper investigates the availability and maintenance policy for inspected systems with neglected down time. The instantaneous and steady-state availabilities of the system are derived analytically. Furthermore, a maintenance model is developed to find the optimal inspection interval, , which minimizes the long-run average cost rate. Based on the developed maintenance model, the analytical properties of the optimal inspection policy are explored. Numerical examples on ventilator system calculating the availability and optimal inspection policy are given to illustrate the applicability of the proposed model. [ABSTRACT FROM AUTHOR]

Highlights • Degradation-shock dependences are considered by two SEPs. • The general path model and the Wiener process model are applied to the dependent competing failure processes. • Analytical results of the reliability indexes are achieved for models 1 and 2. • Simulations are adopted to obtain the system reliability and lifetime pdf for models 3 and 4. Abstract In this research, reliability models are developed for systems or devices subject to dependent competing soft and hard failure processes with degradation-shock dependences. The new models in this paper extend previous researches by considering novel shock effect patterns (SEPs) resulting from multiple species of external shocks. A soft failure occurs when overall degradation level exceeds the soft failure threshold. Meanwhile, a hard failure occurs when the transmitted shock size exceeds the hard failure threshold. The dependences between the soft and hard failure processes are considered in terms of two different SEPs, degradation level and degradation rate. For SEP I, the shocks can increase the degradation level. Thus, the soft failure is indicated by the overall degradation level comprising of the internal continuous degradation and the additional damage caused by shocks. For SEP II, the shocks can increase the degradation rate of the soft failure process. Specifically, in this research four reliability models focusing on different stochastic processes and SEPs are developed. Closed-form reliability formulas are derived for the general path models, while simulation methods are adopted to get the reliability for the Wiener process models. Finally, numerical examples are given to illustrate the models and the approaches. [ABSTRACT FROM AUTHOR]

This paper develops reliability and maintenance models for a single‐unit system subject to hard failures under random environment of external shocks. Motivated by the observations of shot‐noise process in practice, the impact of shock damage on system failure behavior is characterized by random hazard rate increments. To remove such negative impact, imperfect preventive repair is performed periodically, and preventive replacement is performed after several repairs. Considering the joint effects of both random shocks and imperfect repair on the system hazard rate, we derive recursive equations for the system reliability function. Furthermore, we investigate the optimal maintenance policy that minimizes the expected cost per unit time of the system. The applicability of the reliability and maintenance model is validated by a case study on a wind turbine system. [ABSTRACT FROM AUTHOR]

Availability analysis is an important issue in many practical fields. This paper investigates the availability for general repairable systems with repair time threshold. Based on practical applications, a repair time threshold is introduced. If the period of a repair is less than a predefined time threshold, then the system may be considered as working during this period, i.e., the effect of the repair could be neglected. Otherwise, if the period of a repair is longer than the given threshold, then the system is considered as working from the beginning of the system failure until the repair time exceeding the threshold, i.e., the time point of the system down could be delayed. We consider both constant and random repair time threshold. This paper valuates the user-perceived availability, when the user does not experience any service interruption because the duration of repair is too short. The results can be applied in reliability engineering, queueing theory and many other fields. A numerical example for ventilator system is presented to demonstrate the application of the developed approach. [ABSTRACT FROM AUTHOR]

Highlights • Two extended Phase-type models based on Markov repairable systems are proposed. • The distribution functions for two extended Phase-type distributions are presented. • Some reliability indexes under two models, such as the point-wise availabilities, various interval availabilities, are given. Abstract Two extended Phase-type models with competing risks for failures of Markov repairable systems with absorbing states are presented in this article. We divide the states into three subsets: perfect, imperfect and failure (absorption) for a Markov repairable system. Two models are developed in terms of the proposed failure criteria of the repairable system. For model 1, the system failure criteria of the repairable system are, whichever occurs first, (1) when the system goes into the failure state (absorption state), or (2) when the system transfers from imperfect states to other states, and before that time the transitions from perfect to imperfect states reached a specified number. For model 2, there still are two criteria whichever occurs first. The first one is the same as failure criterion (1) in model 1, but failure criterion (2) is replaced by (3) which is when the sojourn time in imperfect states exceeds a given threshold. Under two models, two distributions are proposed, which are extensions of the well-known Phase-type distribution. Some reliability indexes under two models, such as the distributions of lifetimes, the point-wise availabilities, various interval availabilities, are given. Finally, some numerical examples are presented to illustrate the results obtained in this article. [ABSTRACT FROM AUTHOR]

Highlights • Three types of thresholds including alarm line and two different failure thresholds are used to model the system reliability. • Two degradation process models, Wiener-Gamma and Gamma-Gamma, are developed. • Explicit formulas for the system reliability in terms of duration thresholds for the degradation models are presented. Abstract Stochastic degradation process models are developed in terms of cumulative degradation signals of systems under three types of thresholds including alarm line and two different failure thresholds. One of the failure thresholds corresponds to the degradation amount, and the other corresponds to the duration. We assume that the degradation process can be separated into two distinctive stages by a change-point which is the first passage time of the degradation process with respect to the alarm line. For simplicity, the first stage is regarded as an age-dependent (or non-stationary) Wiener process or an age-dependent (or non-stationary) Gamma process, and the second stage is regarded as an age-dependent (or non-stationary) Gamma process; thus, two special degradation models, Wiener-Gamma and Gamma-Gamma, are constructed. Specially, two-stage stationary Wiener-Gamma and Gamma-Gamma models can be viewed as special cases. The system reliability is defined as the probability that the degradation signals do not exceed a failure threshold and the duration of exceeding the alarm line is less than the duration threshold. Some reliability results including two cases in terms of duration thresholds (constants and random variables) are derived. The moments of lifetime are also given based on the results. In addition, simulation is carried out to verify the given results. And some numerical examples are presented to illustrate the results obtained in the paper. [ABSTRACT FROM AUTHOR]

Highlights • A traffic accident model based on self-exciting point processes is proposed. • The stationary condition, reliability and safety analysis under a traffic model are investigated. • The related probability and characteristics are presented by the simulation methods. • The MLEs for the parameters in a traffic accident model for two data situations are given. Abstract Traffic accidents may pose a high risk with respect to human travel safety. In order to improve the safety of travelling, an insight into the traffic accidents is needed. This requires a practical and specifically approach when it comes to modelling for traffic accidents. This paper presents a traffic accident model based on self-exciting processes, which describes the specific phenomena and problems in traffic accidents. The research focuses on the stationary condition, reliability and safety analysis of the traffic network in a certain city or the traffic situation in a certain area. First, a traffic accident model based on self-exciting processes is developed, and secondly, the stationary condition of the model is studied, which is used to analyze the reliability and safety of the traffic network in a certain city or the traffic situation in a certain area. In addition, the probability and related characteristics for traffic process are evaluated by simulation methods. The traffic accident model is developed with the use of self-exciting processes. Maximum Likelihood Estimate (MLE) is used for the parameter estimations of the traffic accident model. Finally, a numerical example is presented, in which the traffic accident model developed is applied to the simulation data. [ABSTRACT FROM AUTHOR]

Highlights • A novel dependent two-stage failure process model is introduced by considering the shared external shocks. • The impact of shocks on the two-stage failure process is characterized via the random hazard rate increment. • Multiple failure criterions for systems subject to two-stage failure process are considered. • Explicit analytical expression for system reliability is derived based on the stochastic failure model. • An illustrative example of oil pipeline systems is presented to validate the application of the reliability model. Abstract This paper evaluates system reliability performance based on a dependent two-stage failure process with competing failures. The failure process of the system can be divided into two stages, i.e., the defect initialization stage, and the defect development stage. Dependence between these two stages is reflected in the fact that they share the same shock process modeled by a nonhomogeneous Poisson process. The impact of shock damage on system failure behavior is characterized by random hazard rate increments of the two stages. Based on practical failure behavior of industrial systems, we consider two typical and competing failure modes, defect-based failure and duration-based failure. Defect-based failure occurs when a defect reaches the damage threshold and duration-based failure is triggered when the duration in defective state is larger than a time threshold. We derive some results on system reliability and show that, with different parameter settings, our model reduces to several classic competing risk models. Finally, a detailed illustrative example of an oil pipeline system is given to demonstrate the applicability of the proposed model. [ABSTRACT FROM AUTHOR]

In the present paper, a discrete-time semi-Markovian system is studied for both theoretical and practical demands. The state space of a semi-Markov chain is divided into three subsets, namely working state subset, changeable state subset and failure state subset, resulting from applications for this kind of models in degradation analysis on system performances. Performances of changeable states depend on recent evolutions of the system, including the immediate history and future. The Z-transform is used to investigate stochastic properties of the semi-Markovian system and distributions of some sojourn time lengths, based on which, for example, the working period length and failure period length, are discussed. Reliability measures such as reliability, point availability and interval availability are derived for the system. Besides, two new reliability measures, point covering availability and interval covering availability, are discussed for discrete-time semi-Markovian systems for the first time. Preventive maintenance and corrective maintenance from outside of the system are also investigated with respect to related costs. Finally, a numerical example is given to illustrate the results obtained in the paper. Those models and results presented in the paper can be used in reliability, maintainability, supply chain management and other areas. [ABSTRACT FROM AUTHOR]

In the field of reliability, there are many indexes, such as reliability, point availability, interval availability, and others, which have been widely used to describe the properties of repairable systems. However, some situations of interest cannot be covered by the existing indexes. In this paper, we introduce the definitions of multipoint covering availability and multi-interval covering availability, and formulas are derived in matrix forms for aggregated Markov repairable systems especially when the covering working intervals can overlap. The formulas of binary-point and binary-interval covering availabilities are given by an enumeration method. For multipoint and multi-interval problems, a recursive method is employed. Numerical examples are shown to illustrate the results derived in this paper and finally some conclusions and discussions are presented. [ABSTRACT FROM AUTHOR]

Wiener degradation processes with fixed failure threshold have recently received extensive applications due to their desirable mathematical properties. However, a system may experience various environments during the overall degradation life cycle. This may lead to abrupt changes of both the failure thresholds and drift parameters due to the fluctuation of endurance of the deterioration system. Motivated by the practical needs of such multi-phase degradation systems, in this paper, a multi-phase degradation model based on the Wiener process is developed to describe the phenomenon of abrupt changes of failure thresholds and drift parameters. Both the two-phase and multi-phase ( n  ≥ 3) cases are considered under changing failure thresholds and drift parameters in this paper. Analytical expression of system reliability is given for the Wiener process model with two successive different failure thresholds. Meanwhile, a recursive formula is developed to get the analytical expression of system reliability for the Wiener process model with multiple successive different failure thresholds ( n  ≥ 3). In addition, simulation procedures are carried out to verify the given results. And some numerical examples are presented to illustrate the proposed model. [ABSTRACT FROM AUTHOR]

A concept of Sequential Probability Series System (SPSS) is developed in this paper, which widely exists in many practical sectors such as power plants, inventory management and security management. In SPSS the failure states of each unit are divided into two classes according to their consequences: dangerous failure and safe failure, where the former results in system failure while the latter has no impact on the system. Suppose that when a failure unit appears in SPSS, the system fails with probability p while the other units in SPSS can continue working with probability 1 − p . This paper treats the problem of achieving optimal allocation of units in SPSSs that maximizes expected total working time of all units. Three optimal allocation models are formulated. We derive the analytical expressions for the optimal allocation solutions under certain assumptions. A genetic algorithm and a Monte Carlo method are provided to solve the allocation problems whose analytical solutions are difficult to obtain. An application can be found in Remote Power Feeding System (RPFS). Numerical examples for a RPFS are presented to demonstrate the application of the developed approach in each model. [ABSTRACT FROM AUTHOR]

Modern computing systems typically use a large number of independent, non-identical computing nodes to perform a set of coordinated computations in parallel. The computing system and its constituent computing nodes often exhibit more than two performance levels or states corresponding to different computing powers. This paper models and evaluates performability of large-scale multi-state computing systems, which is the probability that a computing system performs at a particular performance level. The heterogeneity in the constituent components of different nodes (due to factors such as different model generations, model suppliers, and operating environments) makes performability analysis difficult and challenging. In this paper a specification method for system performance level (SPL) is first introduced. A multi-valued decision diagram (MDD) based approach is then proposed for performability analysis of multi-state computing systems consisting of nodes with different state occupation probabilities, which encompasses novel and efficient MDD model generation procedures. Example and benchmark studies are performed to show that the proposed approach can offer efficient performability analysis of large-scale computing systems. [ABSTRACT FROM PUBLISHER]

In maintenance field, there are many indexes such as instantaneous availability, steady state availability, interval availability and others, which have been widely used to describe the various properties and performance of repairable systems on maintenance. However, all the present availability indexes still cannot cover some situations which people are interested in. In the paper, two new interval availability indexes are introduced for Markov repairable systems, which is named as an availability with a given window length and containing a specified point or interval. The new interval availability is the probability that the system is working during a given window length and this window must contains a specified point or interval. Their calculation formulas are presented in matrix forms by using the Laplace transform. Some properties on the two indexes are discussed briefly, and numerical examples are shown to illustrate the features of the two new indexes, finally conclusions are given. These two new indices can become the conventional interval availability when the given window length is zero or equal to the specified interval length. The results in the paper may be used to measure the performance of repairable system more deeply and detail. [ABSTRACT FROM AUTHOR]

The main focus of this paper is on the development of reliability measures for a repairable multi-state system which operates under dynamic regimes under the discrete-time hypothesis. The switching process of regimes is governed by a Markov chain, and the functioning process of the system follows another Markov chain with different transition probability matrices under different regimes. In terms of two chains as above, a new Markov chain is constructed to depict the evolution process of the dynamic system. For the regime consideration, some novel reliability indices are essential and firstly introduced in this paper. By means of hierarchical partitions for the new state space, Ion-Channel modeling theory and discrete-time Markov chain, the traditional and novel reliability and availability functions for the system under random regimes are easily obtained with the closed form solutions, such as two types of system reliabilities, two types of system point availabilities, two types of system multiple-point availabilities and the associated system multi-interval availabilities and so on. In addition, some probability distributions of sojourn times we are interested in are discussed and computed here. Finally, a numerical example is given to illustrate the results obtained in the paper. [ABSTRACT FROM AUTHOR]

Discrete repairable degradation systems can be modeled by a homogeneous discrete-time second-order semi-Markov chain with finite state space for theoretical and practical sake. In the present paper, the state space can be divided into three subclasses, namely excellent states, good states (middle states) and failure states. The transitions among states may be driven by degradations and inside shocks of the system, repair actions, self-healings, energy supplement and other recovery ways. The Z-transform is used to give distributions for some interesting problems such as distributions of an I-period (oscillation & working time) and a gap between I-periods, and instantaneous and steady availabilities of the system. Finally, some numerical examples are given to illustrate the results obtained in the paper. The work may be used in reliability and maintenance analysis of discrete time repairable systems. [ABSTRACT FROM AUTHOR]

In this article, a warm standby n-unit system is studied. The system is operational as long as there is one unit normal. The unit online, which has a lifetime distribution governed by a phase-type distribution, is also attacked by a shock from some external causes. Assume that shocks arrive according to a Poisson process. Whenever an interarrival time of shock is less than a threshold, the unit online fails. The lifetimes of the units in warm standby is exponentially distributed. A repair man who can take multiple vacations repairs the failed units based on the "first-in-first-out" rule. The repair times and the vacation times of repairman are governed by different phase-type distributions. For this system, theMarkov process governing the system is constructed. The system is studied in a transient and stationary regime; the availability, the reliability, the rates of occurrence of the different types of failures, and the working probability of the repairman are calculated. A numerical application is performed to illustrate the calculations. [ABSTRACT FROM AUTHOR]

The article focuses mainly on a conditional imputation algorithm of quantile-filling to analyze a new kind of censored data, mixed interval censored and complete data related to interval-censored sample. With the algorithm, the imputed failure times, which are the conditional quantiles, are obtained within the censoring intervals in which some exact failure times are. The algorithm is viable and feasible for the parameter estimation with general distributions, for instance, a case of Weibull distribution that has a moment estimation of closed form by log-transformation. Furthermore, interval-censored sample is a special case of the new censored sample, and the conditional imputation algorithm can also be used to deal with the failure data of interval censored. By comparing the interval-censored data and the new censored data, using the imputation algorithm, in the view of the bias of estimation, we find that the performance of new censored data is better than that of interval censored. [ABSTRACT FROM AUTHOR]

Different regimes, for example, different seasons, bull market/bear market, multiple missions etc., affect the performance of systems or products greatly in the real world. Thus, a dynamic reliability model, with cyclic multiple mission periods, is developed for non-repairable discrete state systems in this paper. By taking Laplace transforms and aggregated stochastic theory for describing the conditional probabilities of transitions among different mission periods and system states, we obtain closed form solutions of some reliability indexes and distributions of sojourn times with perfect and imperfect function. Three models of valve systems are considered in details based on different reliability structures, and a numerical example of marine reactor safety water injection system is also presented by applying our proposed methods for two special cases: (i) when the mission durations are fixed and; (ii) when the mission durations have exponential distributions. [ABSTRACT FROM AUTHOR]

We propose the consecutive-$k_r $-out-of- $n_r $:F linear zigzag structure and circular polygon structure, which extend the consecutive-k-out-of-n:F system model. By employing the finite Markov chain imbedding approach, we have proven the equations for calculating the reliability of consecutive-k-out-of-n:F system with different patterns given the state of the first and(or) the last component(s). Based on those reliability equations, we provide the rules for obtaining the reliability of the linear zigzag structure and circular polygon structure. Numerical examples are presented as an illustration of our new system model and demonstrate the use of the equations and rules proposed in this paper. [ABSTRACT FROM AUTHOR]

Multi-state system models are more practical and valuable when describing complex engineering structures than binary-state system models. Modelling of linear consecutive- (k , l) -out-of- n : F systems with shared components under non-homogeneous Markov dependence is discussed in this work. The proposed model is an extension of both "consecutive- k -out-of- n : F system" and "consecutive- k -out-of- n : F system with shared components between adjacent subsystems". All the components in the system are assumed to have three states and are non-homogeneous Markov-dependent. Using the method of sum of disjoint products (SDP), the system reliability formula is derived by summing the reliability values for all disjoint cases. The well-known finite Markov chain imbedding approach (FMCIA) is used for deriving the reliability formula for each disjoint case. For the special cases of systems with homogeneous Markov-dependent or independent components, the corresponding results are deduced. Finally, some numerical examples are presented to illustrate all the results developed here. • Consecutive-(k,l)-out-of-n: F systems with shared components are modelled. • Using the method of SDP and FMCIA, the system reliability formula is derived. • The method is also fit for homogeneous Markov-dependent or independent cases. • Numerical examples demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Cascading failure models have been studied more recently as for electric power systems, epidemics, and interdependent networks. The current paper presents three cascading failure models in the point-view of system reliability based on the normalized CASCADE model. The corresponding system reliability indices are developed including system reliability, expectation lifetimes of systems, distributions of total number of failures, distributions of cascading failure propagation generations, and other related probabilities. Two numerical examples are discussed, and the conclusions are summarized. [ABSTRACT FROM AUTHOR]

In this paper, two degradation models are developed in terms of degradation signals of systems with some Wiener diffusion processes under pre-specified periodical calibrations. Some detailed results on two models are given by solving some recursive equations. The system reliability is defined as the probability that the degradation signals do not exceed a threshold value by time t. Two methods for getting the system reliability are presented: one is a stochastic process-based method in terms of recent results on the distribution of the first passage time, and the other is based on partial differential equations with absorbing boundary conditions. The moments of lifetime (first passage time) also are given based on both methods. Some extended questions and results are discussed for more general situations, and finally discussions and conclusions are presented. [ABSTRACT FROM AUTHOR]

To accurately describe the performance of repairable systems operating under alternative environments, for example, mild/harsh, working/idling, maximum/minimum level demand etc., a Semi-Markov process with a finite state space and two different Semi-Markov kernels is introduced. The state set of the system which is regarded as acceptable might depend on the environments. Two important reliability indices, the availability and time to the first system failure, are obtained via Markov renewal theory, transform and matrix methods. The results and numerical examples are also provided for two special cases: (1) when sojourn times under alternative environments are constants and (2) when sojourn times under environments have exponential distributions. [ABSTRACT FROM AUTHOR]

The environment in which a system operates can have a crucial impact on its performance; for example, a machine operating in mild or harsh environments or the flow of a river changing between seasons. In this article, we consider a dynamic reliability system operating under a cycle ofKregimes, which is modeled as a continuous-time Markov process withKdifferent transition rate matrices being used to describe the various regimes. Results for the availability of such a system and probability distributions of the first uptime are given. Three special cases, which occur due to situations where the durations of the regime are constant and where the number of up states in different regimes are identical or increasing, are considered in detail. Finally, some numerical examples are shown to validate the proposed approach. [ABSTRACT FROM PUBLISHER]

In this article, a new attributes double sampling plan for three-class products (ADSPTP) is presented, and the corresponding operating characteristic function is constructed based on the given procedure of performing ADSPTP. Average sample numbers (ASN) for complete inspection and curtailed inspection of the second sample are derived and the extreme point is discussed on the three-dimensional ASN surface. In addition, the differences arising from different parameters are studied. Sampling plans are designed by a nonlinear optimization model. Finally, numerical examples and discussions are given to illustrate the obtained results, and tables of the designed plans under various conditions are provided. [ABSTRACT FROM AUTHOR]

This paper studies a cold standby repairable system with two identical components and one repairman having multiple vacations applying matrix-analytic methods. The lifetime of the component follows a phase-type distribution. The repair times and the vacation times of the repairman are governed by different phase-type distributions, respectively. For this system, the Markov process governing the system is constructed. The system is studied in a transient and stationary regime, the availability, the reliability, the rates of occurrence of the different types of failures, and the working probability of the repairman are calculated, respectively. A numerical application is performed to illustrate the calculations. [ABSTRACT FROM AUTHOR]

Brassinosteroids (Brs) are a newly recognized group of active steroidal hormones that occur at low concentrations in all plant parts and one of the active and stable forms is 24-epibrassinolide (EBR). We investigated the effect of EBR on tomato ( Lycopersicon esculentum Mill.) and its mechanism when seedlings were exposed to low temperature and poor light stress conditions. Leaves of stress-tolerant 'Zhongza9' and stress-sensitive 'Zhongshu4' cultivars were pre-treated with spray solutions containing either 0.1 μM EBR or no EBR (control). The plants were then transferred to chambers where they were exposed to low temperatures of 12 °C/6 °C (day/night) under a low light (LL) level of 80 μmol · m · s. Exogenous application of EBR significantly increased the antioxidant activity of superoxide dismutase, catalase and peroxidase, and decreased the rate of O · formation and HO and malondialdehyde contents. Additionally, the ATP synthase β subunit content was increased by exogenous hormone application. Based on these results, we conclude that exogenous EBR can elicit synergism between the antioxidant enzyme systems and the ATP synthase β subunit so that scavenging of reactive oxygen species becomes more efficient. These activities enable plants to cope better under combined low temperature and poor light stresses. [ABSTRACT FROM AUTHOR]

A consecutive-k-out-of- n : F ( G ) system with sparse d consists of n components ordered in a line or a circle, while the system fails (works) iff, there exist at least k consecutive failed (working) components with sparse d for 0 \le d \le n - k . In this paper, a circular consecutive- k -out-of- n system with sparse d is considered. Some equations for system reliability and Birnbaum importance are derived by means of the finite Markov chain imbedding approach. Then the Birnbaum importance of components is compared in the situations where the system is under an IID model, and where one of the components is known to be failed, respectively. Finally, some numerical examples are followed to illustrate the results obtained in the paper. [ABSTRACT FROM PUBLISHER]

This paper studies a cold standby repairable system with working vacations and vacation interruptions. The repairman׳s multiple vacation, the working vacation policy and the vacation interruption policy are considered simultaneously. The lifetime of component follows a Phase-type (PH) distribution. The repair time in the regular repair period and the working vacation period are other two Phase-type distributions with different representations, and the successive vacation times are governed by a Markovian arrival process (MAP). For this system, a vector-valued Markov process is constructed. We obtain several important performance measures for the system in transient and stationary regimes applying the matrix-analytic method. Finally, a numerical example is given to illustrate the results obtained in the paper. [ABSTRACT FROM AUTHOR]

In this article, two new approaches are introduced to design attributes single plans, and the corresponding models are constructed separately. For Approach I, an algorithm is proposed to design sampling plans by setting a goal function to fulfill the two-point conditions on the operating characteristic curve. For Approach II, the plan parameters are solved by a nonlinear optimization model which minimizes the integration of the probability of acceptance in the interval from the producer's risk quality to the consumer's risk quality. Then numerical examples and discussions based on numerical computation results are given to illustrate the approaches, and tables of the designed plans under various conditions are provided. Moreover, a fact is given to be proved that there is a relation between the conventional design and the new approaches. [ABSTRACT FROM AUTHOR]