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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]

• 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]

• 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 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]

• 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]

• 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]

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]

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]

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]

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]

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]

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]

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]

Due to the complex structures and the multi-functionality of modern products, there are usually two or more performance characteristics which can reflect a product's degradation states. The degradation processes corresponding to these performance characteristics are dependent in general, which brings challenges to the degradation data analysis. In this paper, a gamma process based degradation model is developed for the bivariate dependent degradation data, where the dependency between the two degradation processes is captured by a common random effect naturally. The expectation maximization algorithm is employed to estimate the model parameters. Then, a real-time prediction method for a product's remaining useful life is proposed using the Bayesian method. Finally, both the simulation study and the case study are provided for illustration, whose results demonstrate that the proposed model as well as the corresponding inference methods does work well. • A gamma process based bivariate degradation model is proposed. • The dependency between the two PCs is captured by a common random effect naturally. • The MLEs of the model parameters are obtained via the EM algorithm. • Analytical formulas are presented for the RUL prediction. • The effectiveness is validated via simulation and case studies. [ABSTRACT FROM AUTHOR]

Reliability analysis for k -out-of- n systems has received much attention in the field of reliability due to its practical importance. In this work, models for linear and circular k -out-of- n : F systems with shared components, consisting of m subsystems, are discussed. The finite Markov chain imbedding approach cannot be used directly to derive reliability of the considered systems because of the common area existing between adjacent subsystems. For this reason, we first derive the probability that the number of failed components is a given value in the m common areas. Then, all disjoint cases, each of which is regarded as a series system consisting of m subsystems with different initial state probability distributions, are obtained, with which the n i th step transition rate matrix is calculated by using the Markov chain technique, for i = 1 , 2 , ... , m. Finally, the reliability of the system is obtained by summing the reliabilities of all cases along with the finite Markov chain imbedding approach. Some numerical examples are presented to illustrate the efficiency of the proposed reliability evaluation method. • Linear and circular k -out-of- n : F systems with shared components are modelled. • A method combining the finite Markov chain imbedding approach with some theoretical analysis is proposed. • Reliability calculation rules for both two considered systems are described based on our method. • Numerical examples are presented to illustrate the efficiency of the proposed reliability evaluation method. [ABSTRACT FROM AUTHOR]

Abstract: A system consisting of supply and demand is considered in this paper. Both the supply patterns and demand patterns are random. Thus, both the supply and the demand are modeled by Markov processes. The performance measure considered here is the probability that the demand is met by the supply during given time interval . A close form expression for the performance measure is obtained by using aggregated stochastic process theory and Kronecker matrix operations. In the meanwhile, the performance measures in a general interval and multiple intervals have also been given. Finally, a numerical example is given to illustrate the results obtained in this paper. [Copyright &y& Elsevier]

l First/second-order discrete-time aggregated semi-Markov processes are studied. l Multi-point/interval bounded-covering availabilities are introduced for discrete-time systems. l Formulas are derived of the new indices for these aggregated semi-Markov processes. Research on reliability indices has received considerable attention recently in the field of reliability. In this work, two new reliability indices — multi-point bounded-covering availability and multi-interval bounded-covering availability — are introduced for discrete-time systems. Their explicit formulas are derived for discrete-time first- and second-order aggregated semi-Markov systems, whose state spaces are divided into three subsets corresponding to perfect functioning states, imperfect functioning states and failure states, respectively. Finally, some numerical examples are presented to illustrate the developed results. The results established here will also be useful for many practical reliability systems. [ABSTRACT FROM AUTHOR]

Abstract: Some states in the aggregated semi-Markov repairable system with history-dependent up and down states are changeable in the sense that whether those physical states are up and down depends on the immediately preceding state of the system evolution process. Two reliability indices of the system, the frequency of failures and the time to the first system failure are given. The Laplace–Stieltjes transforms of several time distributions in a cycle, such as the up and down time, the total time the system is in the up, down and changeable states, the length of a single sojourn in the up, down and changeable states are derived. The means of them are also presented. Markov renewal theory, transform and matrix methods are employed for getting these performance measures. A numerical example is given to illustrate the results in the paper. [ABSTRACT FROM AUTHOR]

Abstract: This paper considers the one-sample sign test for population quantiles in general ranked set sampling, and proposes a weighted sign test because observations with different ranks are not identically distributed. It is shown analytically that optimal weight always improves the Pitman efficiency for all distributions. For each quantile, the sampling allocation that maximizes the sign test efficacy is identified and shown to not depend on the population distribution. Moreover, distribution-free confidence intervals for quantiles based on ordered values of optimal ranked set samples are discussed. [Copyright &y& Elsevier]

• A new balanced system with auto-balancing mechanisms is studied. • A new balanced system model is analyzed with the theory of aggregated stochastic process and semi-Markov process. • Formulas for the expectation lifetime and point-wise availabilities of the proposed systems are given. Reliability analysis of balanced systems has become one of the hot and significant topics in the reliability research field. As a new view for the research of balanced system, balanced mechanisms bring much room for research. Based on most practical systems, an auto-balancing mechanism for a multi-state balanced system composed of two subsystems is innovatively proposed for both theoretical and practical demands in this paper. The auto-balancing mechanism is activated to correct the temporary upcoming unbalance of system by adjusting the performance states of each subsystem and to ensure that the performance state difference between the two subsystems of the balanced system do not exceed a balance limitation. The aggregated stochastic processes and semi-Markov processes are used to obtain the reliability measures, and the related expressions of the expectation lifetime and point-wise availabilities of the system are derived. Finally, a numerical example is given to illustrate the results obtained in the paper. The model and the results presented in the paper can be used in reliability, transportation planning, supply chain management, and other areas. [ABSTRACT FROM AUTHOR]

Abstract: A proof on the optimal arrangement of a linear consecutive-k-out-of- system, with , was given by Jalali et al. [2005. The optimal consecutive-k-out-of- line for . J. Statist. Plann. Inference 128, 281–287]. However, a corollary for the case was still incomplete. In this note we point out what is missing from the proof of that corollary and present a new proof using a different approach. [Copyright &y& Elsevier]

Abstract: This paper discusses engineering systems with structural as well as failure dependence among their components. Using the Markov process approach, a coherent system of components with multivariate phase type life distribution is introduced to model the components working in a common random environment. The phase type model is flexible enough to comprise many realistic situations, and also manages to yield the explicit matrix representations of the system reliability measures. Some examples are also presented to illustrate the fact that ignoring the failure dependence among the components often results in over-estimating or under-estimating the system reliability. [Copyright &y& Elsevier]

The availability of systems undergoing periodic inspections is studied in this paper. A perfect repair or replacement of a failed system is carried out requiring either a constant or a random length of time. In Model A, the system is assumed to be as good as new on completion of inspection or repair. For Model B, no maintenance or corrective actions are taken at the time of inspection if the system is still working, and the condition of the system is assumed to be the same as that before the inspection. Unlike that studied in a related paper by Sarkar and Sarkar (J. Statist. Plann. Inference 91 (2000) 77.), our model assumes that the periodic inspections take place at fixed time points after repair or replacement in case of failure. Some general results on the instantaneous availability and the steady-state availability for the two models are presented under the assumption of random repair or replacement time. [Copyright &y& Elsevier]

Consecutive- k systems have well known for outstanding developments and extensive applications. In this paper, we propose consecutive- k -out-of- n : F systems with shared components between adjacent subsystems which extend the consecutive- k -out-of- n : F linear and circular models and the consecutive- k r -out-of- n r : F linear zigzag structure and circular polygon structure. They are referred to as an extension of linear zigzag structure and an extension of circular polygon structure respectively. The idea of computation for reliability is to cut the system into m subsystems and to get the reliability of every subsystem, then to obtain the system reliability in terms of reliabilities of all subsystems, the probability of occurrence of each event and the system structure. Based on this view, we obtain the reliability for the above two structures by combining the sum of disjoint products with finite Markov chain imbedding approach (FMCIA). A numerical example demonstrates the efficiency of the proposed method for our new models. • The consecutive- k -out-of- n : F systems with shared components between adjacent subsystems considering linear and circular structure are modelled. • A method combining sum of disjoint products with finite Markov chain imbedding approach is proposed. • Reliability calculation rules for both two models are given based on our method. • Numerical examples demonstrate the efficiency of the proposed method for our new models. [ABSTRACT FROM AUTHOR]

• Multi-state systems with multi-state components are investigated by their multi-state survival signatures. • State distributions of several multi-state consecutive type systems are derived. • A computation method for the multi-state survival signature is proposed. In this paper, the survival signature is introduced in a matrix form for a multi-state coherent system with multi-state components and then some related properties are discussed. Using the finite Markov chain imbedding approach and the relationship between survival signatures and state distributions, a computational method for survival signatures is proposed for multi-state consecutive type systems. Some illustrative examples are also presented to demonstrate how survival signatures are calculated by this method and as to how they are applied. [ABSTRACT FROM AUTHOR]

Reliability analysis of balanced system has become a key factor for the safe operation of some important equipment and the perfect completion of some important missions. As some real systems consisting of several subsystems with different levels of performance and under degradation processes when in operations are relatively common, they have received widely concerned in the recent years, but are rarely in the new reliability research field of balanced systems, the researches on the balanced systems by considering multi-state are rare, and even few researches are in combination with degradation processes. In this paper, a new balanced system for the multi-state system under degradation processes by considering new failure criteria is put forth. The system consists of two subsystems and fails when either subsystem enters into absorbing states, a new model is proposed and applied to two special cases by assuming that the thresholds of the sojourn time in upcoming unbalanced subsets are constant and random variables respectively, and this paper is the first to extend balanced systems into competing risks. The related expressions of the reliability indexes are derived by employing the theory of aggregated stochastic processes. Finally, numerical examples are presented and some future research topics are also discussed. [ABSTRACT FROM AUTHOR]

• A novel reliability model considering random shock damage and self-healing effect is established. • Dynamic mission abort policy is developed based on the time in mission and the number of minor failures. • A stochastic dynamic programming based optimal mission abort problem is formulated. • Structural properties of the optimal mission abort policy are investigated. Catastrophic failures of safety-critical systems could result in significant economic losses and damage. To improve the survival probability of safety-critical systems, self-healing mechanism is usually considered in the design stage and mission abort may be conducted if the failure risk becomes too high. We investigate the optimal mission abort policies for systems subject to a controllable shock process with self-healing mechanism. Minor failure and catastrophic failures are two competing failure modes of the system. Imperfect repair is carried out when a minor failure occurs and the external shock process is renewed after the imperfect repair. Mission abort decisions are considered based on the time in mission and the number of experienced minor failures. The optimal mission abort problem is formulated within the framework of stochastic dynamic programming to minimize the expected total cost of mission failure, system failure and imperfect repair. The structural properties of the optimal policy are investigated using the optimal stopping theory and a case study is presented to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

• Two Markov renewal shock models with multiple failure mechanisms are developed. • The reliability indexes for systems under two proposed models are analyzed. • Limiting behavior of mean time to failure of system under two models is discussed. In this article, two Markov renewal shock models with multiple failure mechanisms are developed based on the extreme shock model and cumulative shock model respectively. The shock size and interarrival time dependency is studied by assuming the sizes and interarrival times of shocks are governed by an absorbing Markov chain with finite state space. Under the competing risk extreme shock model, the system fails when a single shock of a large size occurs or when it enters the absorbing state whichever occurs first. Under the competing risk cumulative shock model, the system fails when it transfers to the absorbing state, or when the accumulation of sizes of the previous shocks exceeds a prespecified critical level, whichever occurs first. The methods to calculate the reliability functions and other reliability indexes under the proposed two Markov renewal shock models are provided. A special case that the shock arrival process is independent of the current state of the system is considered. The limiting behavior of the mean time to failure of the system under two models is discussed. Finally, a study case of lithium-ion battery systems is conducted to illustrate the proposed models and obtained results. [ABSTRACT FROM AUTHOR]

• Three types of competing risk models are developed for balanced systems. • New failure criteria of multi-state balanced systems are presented. • Formulas for some reliability indexes of proposed systems are given. The concept of balanced systems has been evolved with many meanings recently, and the research on this topic is a hot and significant issue in the reliability field. In this paper, the system consisting of two components does not fail immediately when it is unbalanced. An imminent unbalanced state is defined that the system experienced before reaching its final failure. According to the sojourn time of the system in the imminent unbalanced state and the number of transitions from the balanced to imminent unbalanced states, two new failure criteria are proposed. Combined with the situation where the system components may be permanently damaged, three types of competing risk models are developed for multi-state balanced systems with multiple failure criteria, which are time threshold models, count threshold models and the mixed threshold model. The closed-form formulas of reliability indexes of the multi-state balanced system under five proposed models are obtained by using the theory of aggregated stochastic processes, such as the probability density function of the system lifetime. The extensions to systems with multiple components are discussed. Finally, numerical examples are provided to illustrate the proposed model and the obtained results. [ABSTRACT FROM AUTHOR]

• A reliability balanced system with engines is proposed. • Two models of engines starting-up procedures are analyzed. • The explicit expressions of a sector and the system reliabilities are given. • Two special cases are considered under two models. The research on the reliability of balanced systems has become a hot topic recently. New structures, new balanced mechanisms, and new balanced models for the balanced systems have been developed to extend the study of balanced systems. In order to enrich the research of balanced systems, a new real practical engine balanced system based on the port system is proposed by considering the system structure and the balanced mechanism. This paper studies the reliability for the balanced engine system with the new structure consisting of m sectors in which each sector has n engines. Three models are developed by considering engine start-up rules as the balanced mechanism to re-balance the unbalanced system, then the formulas of reliability for the system, probability density functions and the means for system lifetimes under the two models are obtained by using Laplace transform. In addition, we discuss two special cases when the engine lifetime distribution follows exponential and gamma distributions respectively. Finally, numerical examples are given to illustrate the results obtained in the paper. [ABSTRACT FROM AUTHOR]

• A fixed change point model with a jump based on a Wiener process is developed. • The fixed change point model is extended to a random change point model. • Two phase changing patterns for the random change point model are proposed. • Analytical results of the system reliability indexes are given for Model 1 and the HPP case of PCP 2 in Model 2. A two-phase degradation system with a change point is commonly seen in many real-world degradation systems due to influence of internal mechanisms and external environments, such as lithium-ion batteries, light-emitting diodes (LEDs), plasma display panels (PDPs), and vacuum fluorescent displays (VFDs). Random jumps exist at the change points of the degradation processes. Motivated by this phenomenon, this paper attempts to model such kinds of systems based on the Wiener process. Specifically, In Model 1, a constant change point model is first considered. The analytical results of the system reliability, lifetime distribution, and the remaining useful lifetime (RUL) are obtained. Then, based on Model 1, a random change point model, i.e., Model 2, is proposed to explain the change point phenomenon which is influenced by the accumulative effect of a shock process. Two phase-change patterns (PCPs) are considered in Model 2. For PCP I, the change point is triggered when the number of shocks reaches a predetermined threshold. For PCP II, the change point is triggered when the accumulative shock damage exceeds the corresponding threshold. The system reliability indexes are studied by using analytical and simulation methods. Finally, several numerical examples are given to illustrate the approaches and results. [ABSTRACT FROM AUTHOR]

• A novel concept of repairable system's balanced condition is proposed. • The PH-distributions have been considered for Model 1. • Three effective maintenance policies are provided for case II of Model 2. • System reliability and availability indexes are derived for the two models. This paper considers a novel k -out-of- n : F repairable balanced system with m sectors. The characteristic of this balanced system is that the numbers of working components in the m sectors must keep the same at any time. Thus, some components should be shut down manually when the balance is broken. Specifically, two Markov repairable system models are proposed in this paper in terms of different restart mechanisms of shut down components. The excellent Markov process imbedding method is used to obtain the system reliabilities and availabilities. Phase-type (PH) distributions for the lifetime and repair time of the components in the first model are considered. For the second case of the second model, three different maintenance policies are proposed and their performances are compared through several numerical examples. To the best of our knowledge, the k -out-of- n : F repairable balanced system models proposed in the research of this paper are first reported in literature. Finally, numerical examples are given to illustrate the models and the approaches. [ABSTRACT FROM AUTHOR]

• Developing degradation-based and duration-based mission abort policies based on a two-stage degradation process. • Deriving mission reliability and system survivability under the proposed stochastic failure process and mission abort policies. • Investigating the optimal mission abort policies balancing the tradeoff between mission reliability and system survivability. • Providing a detailed comparison of the cost-saving performance of the two proposed abort policies numerically. Various safety-critical systems such as aircrafts, submarines and space stations are required to perform missions continually in harsh environment. To enhance the survivability of such systems, a mission can be aborted when the mission completion becomes problematic and a rescue procedure is immediately initiated. This paper investigates the optimal mission abort policy based on a two-stage Gamma process which can be separated into normal and defective stages. Degradation-based abort policy where a mission is aborted if the system degradation level is higher than a threshold and duration-based abort policy where a mission is aborted if the duration in defective stage is larger than a threshold are considered. The mission success probability and system survivability under these two policies are evaluated and the optimal mission abort thresholds balancing the tradeoff between the mission success probability and system survivability are investigated. A numerical example on an unmanned aerial vehicle (UAV) is presented to illustrate the applicability of the proposed abort policies. [ABSTRACT FROM AUTHOR]

• A competing risk model with restriction on transition times is developed. • Some reliability indexes of the proposed semi-Markov system are given. • A nonlinear integer programming with corresponding solution algorithm is presented. In this paper, a competing risk model with a restriction on transition times for semi-Markov multi-state repairable systems is proposed. The states of semi-Markov systems can be divided into three categories: a normal working subset, a defective working subset and a breakdown subset which contains an absorbing state where the system cannot escape once entering it. If the number of transitions between the normal and defective working subsets exceeds a given value, the system will be abandoned due to the high maintenance costs. The theory of aggregated stochastic process is employed to obtain the closed-form formulas of the probabilities of competing risks, distributions of survival times and point-wise availabilities. An integer nonlinear programming is presented which enables the system to satisfy the prescribed performance requirements and control its economic cost. A corresponding algorithm for solving the integer nonlinear programming is provided to find the optimal number of transition times. Two numerical examples including Markov process and semi-Markov process are conducted to illustrate the obtained results and proposed methods finally. [ABSTRACT FROM AUTHOR]

In engineering applications, system degradation is influenced by continuous degradation and random shocks simultaneously. In this paper, we study the reliability of a multi-component system with interacting components that are subject to both continuous degradation process and categorized shocks. By interacting components, we refer to the case where the degradation behavior of a particular component can influence that of another component. This is in contrast to previous works in the literature that typically assume independence in the degradation interaction in multi-component systems. Moreover, categorized shocks are assumed to selectively affect one or more components by either causing a sudden jump in the degradation level or accelerating the degradation rate, or both. Under these assumptions, we derive the reliability of a series system in a recursive way and propose a simulation method to approximate the failure time of k -out-of- n systems. An example of the MEMS (Micro-electro mechanical systems) oscillator, which is a typical system with interacting components subject to continuous degradation and categorized random shocks, is studied in the end to demonstrate the results obtained in this paper. [ABSTRACT FROM AUTHOR]

Motivated by many networked computing systems used in data centers, Grid or Cloud computing infrastructures, and spatially distributed wireless sensor networks configured in aerospace and military industries, this paper considers a multi-state series-parallel system where n heterogeneous components can possess multiple states with different performance rates. Performability is concerned with the probability of such a system performing at a cumulative system performance W characterized in terms of the minimum of all cumulative subsystem performance G j (1 ≤  j  ≤  m ), G j is the sum of the performance rates of components belonging to the j th subsystem. This paper proposes a multi-valued decision diagram (MDD)-based approach to model and evaluate the concerned performability. A single compact MDD model is constructed by sharing all isomorphic model structures involved in different cumulative system performance. The MDD model can be reused when different versions of component state probability distributions are evaluated, thus it can facilitate solutions to further problems that require executing numerous iterations of performability evaluations, for example, the redundancy and/or reliability optimization problems. [ABSTRACT FROM AUTHOR]

A multi-state linear consecutive- k -out-of- n : F system, MLC( k,n ) consists of n components ordered in a line, which fails if at least k consecutive components have failed. It abounds in real-world applications such as wireless sensor networks, microwave station networks, and oil pipeline systems. Performability of an MLC( k,n ) system is concerned with probability that the system performs at a performance state characterized in terms of the largest number of consecutive failed components. This paper proposes a multi-valued decision diagram (MDD)-based approach to model and evaluate performability of an MLC( k,n ) system with heterogeneous components following arbitrary lifetime distributions. The proposed approach encompasses a novel and efficient MDD model generation procedure. Both complexity analysis and illustrative examples are provided to show efficiency of the proposed MDD approach. As demonstrated through examples, the proposed MDD approach is also applicable to MLC( k,n ) systems with non-identical reparable components and component importance analysis. [ABSTRACT FROM AUTHOR]

Markov history-dependent repairable systems refer to the Markov repairable systems in which some states are changeable and dependent on recent evolutional history of the system. In practice, many Markov history-dependent repairable systems are subjected to neglected failures, i.e., some failures do not affect system performances if they can be repaired promptly. In this paper, we develop a model based on the theory of aggregated stochastic processes to describe the history-dependent behavior and the effect of neglected failures on the Markov history-dependent repairable systems. Based on the developed model, instantaneous and steady-state availabilities are derived to characterize the reliability of the system. Four reliability-related time distributions, i.e., distribution for the k th working period, distribution for the k th failure period, distribution for the real working time in an effective working period, distribution for the neglected failure time in an effective working period, are also derived to provide a more comprehensive description of the system's reliability. Thanks to the power of the theory of aggregated stochastic processes, closed-form expressions are obtained for all the reliability indexes and time distributions. Finally, the developed indexes and analysis methods are demonstrated by a numerical example. [ABSTRACT FROM AUTHOR]