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Finite state Markov processes and their aggregated Markov processes have been extensively studied, especially in ion channel modeling and reliability modeling. In reliability field, the asymptotic behaviors of repairable systems modeled by both processes have been paid much attention to. For a Markov process, it is well-known that limiting measures such as availability and transition probability do not depend on the initial state of the process. However, for an aggregated Markov process, it is difficult to directly know whether this conclusion holds true or not from the limiting measure formulas expressed by the Laplace transforms. In this paper, four limiting measures expressed by Laplace transforms are proved to be independent of the initial state through Tauber's theorem. The proof is presented under the assumption that the rank of transition rate matrix is one less than the dimension of state space for the Markov process, which includes the case that all states communicate with each other. Some numerical examples and discussions based on these are presented to illustrate the results directly and to show future related research topics. Finally, the conclusion of the paper is given. [ABSTRACT FROM AUTHOR]

Dependency plays a key role in the design stage of multi-component manufacturing systems. To capture stochastic dependencies between subsystems, this paper considers a load-sharing k -out-of- n : G system which contains non-identical multi-state subsystems. The state evolution of each subsystem follows a continuous-time homogeneous Markov chain. Most existing research on k -out-of- n : G systems focuses on independent subsystems. However, a subsystem failure usually leads to a higher failure rate for each surviving subsystem within the system. This paper aims to study the interaction among subsystems and analyse the system reliability performance. The transition rate matrix for a given subsystem is assumed to depend on the total number of the remaining surviving subsystems, which can be represented as the sum of a baseline transition rate matrix and a stochastic dependency matrix. This paper proves that the transition rate matrix of the system is the generalised Kronecker sum of transition rate matrices of subsystems, and develops an explicit method to calculate the generalised Kronecker sum. The theory of aggregated stochastic processes is employed to obtain closed-form formulas for reliability indexes. A case study of multi-engine aircraft systems is provided where numerical examples are given to illustrate the developed model and obtained results. [ABSTRACT FROM AUTHOR]

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]

Reliability evaluation under shock models has attracted considerable attention in the literature. Existing research on shock models has focused primarily on damage effects of shocks treated as constants after the arrivals of shocks. Due to the presence of self‐healing, deterioration, and variation over time, the damage effects caused by the shocks may have some evolutions. In this work, we focus on arrivals of shocks that follow a continuous‐time renewal process, and a Markov process that describes the evolution of shock damage. In this regard, we develop here two kinds of shock models and then derive system reliabilities under these two kinds of shock models. The asymptotic behavior of the damage evolution process of each shock is then discussed, and the probabilities of each shock eventually disappearing and destroying the system are derived through the use of aggregated stochastic processes. Finally, the results and numerical examples are presented for three special cases, namely: (1) when the inter‐arrival times of the shocks follow exponential distribution, (2) when the inter‐arrival times of the shocks follow Erlang distribution, and (3) when the inter‐arrival times of the shocks follow uniform distribution. [ABSTRACT FROM AUTHOR]

Shock models have been widely studied in reliability area, because many real phenomena in reliability engineering can be depicted by the shock models. In general, the damage effects of shocks are treated as constants after the shocks arrive. In fact, the damage effects resulted from shocks may have some evolutions due to some self-healings, deteriorations and variations over time. In the present paper, three types of shock models with damage effect evolutions following Markov processes are developed in discrete time situations, then the system reliability formulas and the means and variances of their lifetimes are given. The relationship among the introduced shock models is discussed, and some special cases for three types of shock models are also studied. The asymptotic behaviors of damage evolution processes are presented for a special case. Some numerical examples are presented to illustrate the results obtained in the paper. [ABSTRACT FROM AUTHOR]

In this paper, we investigate systems subject to random shocks that are classified into critical and noncritical categories, and develop two novel critical shock models. Classical extreme shock models and run shock models are special cases of our developed models. The system fails when the total number of critical shocks reaches a predetermined threshold, or when the system stays in an environment that induces critical shocks for a preset threshold time, corresponding to failure mechanisms of the developed two critical shock models respectively. Markov renewal processes are employed to capture the magnitude and interarrival time dependency of environment‐induced shocks. Explicit formulas for systems under the two critical shock models are derived, including the reliability function, the mean time to failure and so on. Furthermore, the two critical shock models are extended to the random threshold case and the integrated case where formulas of the reliability indexes of the systems are provided. Finally, a case study of a lithium‐ion battery system is conducted to illustrate the proposed models and the obtained results. [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]

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]

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]

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]

Practically, many repairable systems are subject to neglected failures, i.e. if some failures can be repaired promptly they will not affect system availability. This paper investigates the availability and optimal maintenance policy for inspected Markov systems with down time threshold. Based on practical applications, a down time threshold is introduced. If a down time of the system is less than a given threshold, then the system may be considered as operating during the down time, i.e. the down time could be neglected. Otherwise, if a down time is longer than the given threshold, then the system is considered as operating from the beginning of the system failure until the down time exceeding the threshold, i.e. the down time could be delayed. Incorporating the down time threshold, the instantaneous and steady-state availabilities of the system are derived. Furthermore, a maintenance model is formulated to find the optimal inspection interval, T*, which minimizes the long-run average cost rate. 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]

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]

System signature is a crucial measure in the description and comparison of system structures in the field of reliability, and it does not depend on the component distributions under the assumption that all components are independent and identically distributed. A Markov process method is proposed to compute signatures for general coherent systems in this paper, which is more efficient for the computation of the system signature. As applications of the method, signature formulas of consecutive‐k‐out‐of‐n: F systems and related consecutive‐type systems are investigated, including both the linear and circular ones. Particularly, the related signature formulas for consecutive‐type systems with sparse d are derived for the first time. Illustrative examples are presented for methane reformer furnace systems. [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]

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]

A wireless sensor network is applied for detecting information, by nodes, then generates and transfers the packets to the clustering head for further transmission. In practice, due to the influence of environmental factors, traffic loads of nodes and barrier, a node may fail to detect the information which occurs within its sensing area. Thus, the redundant deployment is often conducted. There are some frequent questions need to be considered when deploying the WSN: (1) how many nodes are required for obtaining a confident coverage of the area; (2) how well does the WSN perform as time goes by, because it is an energy consumption technique; (3) when is the best opportunity to reallocate the nodes. To answer the above questions, we conduct this research. We use $$k$$ -out-of- $$n$$ model to calculate the $$k$$ -coverage probability, which determines the minimal number of nodes that are needed to be deployed in the monitored area. We formulate the node availability and WSN availability with the assumption that the information arrival is a Poisson process. By applying the age-based replacement model, we obtain the optimal reallocation interval with lowest long run expected cost rate. [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]

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]

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]

Phased-mission systems (PMS) can be widely found in a lot of practical application areas. Reliability evaluations and analysis for this kind of systems become important issues. The reliability of PMS is typically defined as the probability that the system successfully accomplishes the missions of all phases. However, thek-out-of-nsystem success criterion for PMS has not been investigated. In this paper, according to this criterion, we develop two new models, which are static and dynamic, respectively. The assumptions for these two models are described in detail as well. The system reliabilities for both models are presented for the first time by employing finite Markov chain imbedding approach (FMCIA). In terms of FMCIA, we define different state spaces for the two models, and transition probability matrices are obtained. Then some numerical examples are given to illustrate the application of FMCIA. Finally, some discussions are made and conclusions are summarized. [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]

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]

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]

Here we present the structure functions for the linear and circular m-consecutive-k, l-out-of- n:F (G) systems, which consists of n linearly or circularly ordered components such that the system fails (works) iff there are at least m times l-overlapping runs of consecutive k failed (working) components. The duality of these F and G systems is obtained, and the proofs are given. The generalized reliability formulae for the systems are obtained by using the finite Markov chain imbedding approach for statistically independent and Markov-dependent cases. Some miscellaneous results on the reliability of these systems and numerical examples are provided to illustrate the results obtained in the paper. [ABSTRACT FROM PUBLISHER]

In this paper, Markov models of repairable systems with repair time omission are considered whose finite state space is grouped into two sets, the set of working states, W, and the set of failed states, F. If the system enters failed states from a working state at any instance, and sojourns at the failed states F less than a given nonnegative critical value τ, then the repair interval can be omitted from downtime records. Otherwise, If the system enters failed states from a working state at any instance, and sojourns at the failed states F more than the given nonnegative critical value τ, then the repair interval cannot be omitted from downtime records. In terms of the assumption, a new model is developed. The focus of attention is the new model's availability, interval reliability and interval unreliability. Several results are derived for these reliability indexes for the new model. Some special cases and numerical examples are given to illustrate the results obtained by using Maple software in the paper. [ABSTRACT FROM AUTHOR]

Three models for the aggregated stochastic processes based on an underlying continuous-time Markov repairable system are developed in which two-part partition of states is used. Several availability measures such as interval availability, instantaneous availability and steady-state availability are presented. Some of these availabilities are derived by using Laplace transforms, which are more compact and concise. Other reliability-distributions for these three models are given as well. [ABSTRACT FROM AUTHOR]

Availability is an important measure of performance for repairable systems because it can describe the probability of a repairable system being in working states. Single point and interval availability measurements (availabilities) have been widely used, but they cannot meet the requirements of all applications. In this paper, multi-point, multi-interval, and mixed multi-point-interval availabilities are introduced, which are extensions of the single point and interval availabilities. The multi-interval availability formula was given by Csenki in 2007 by using the induction method in which the multi-interval availability was called the joint interval reliability, but a new simple technique is provided to present the formula of multi-interval availability in this paper. We also consider the steady-state situations for all new availabilities. For Markov repairable systems, the calculation formulae are presented for the new availabilities, and some properties of new availabilities are discussed too. Finally, some special cases and numerical examples are given to illustrate the results obtained in the paper. [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]

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]

Many multi-state systems settle on more than one operating levels and the elementary states having a common operating level can be treated as one. In this paper, one kind of such Markov repairable systems is introduced, which is named aggregated Markov repairable systems with multi-operating levels. In the system, the functioning states are lumped together according to their membership as a common operating level and each operating level is characterized by a performance rate. The systems degrade from a higher operating level to a lower one when no preventive maintenances (PMs) are performed. While the system is in some deteriorated operating levels that are easy to be recognized PMs are carried out. The PMs may be perfect or imperfect. If the system fails, the repairs may restore it to any of its operating levels. A multivariate semi-Markov process is build to describe performance properties of the system. Several reliability indices such as the availability, frequencies of repairs and failures are presented. Furthermore, the up time, the length of a cycle, the sojourn times in various operating levels and visiting numbers to them, the times that the system satisfies demands of customers and the output during a cycle are studied. Semi-Markov process theory, Laplace transform and matrix methods are employed in the study. A numerical example is given to illustrate the results in the paper. The impact of PM on the system is considered through the numerical illustration. [ABSTRACT FROM AUTHOR]

This article analyzes a single-unit repairable system consisting of an operating subsystem and a maintenance subsystem that is solely used to repair the operating subsystem in the event of it breaking down. Formulae for reliability indexes such as availability and distributions concerned with visits to certain subsets of states are presented in terms of state aggregations in which two partitions are used. The situation with exponentially distributed operational times is discussed not only as a special case of the approach used in this article but also using a Markov model. A numerical example is given to illustrate the results obtained in this article. [ABSTRACT FROM PUBLISHER]

This article studies reliability for a Markov repairable two-item cold standby system with neglected failures. In the system, if a failed time of the system is too short (less than a given critical value) to cause the system to fail, then the failed time may be omitted from the downtime record, i.e., the failure effect could be neglected. In ion-clwinnel modeling, this situation is called the time interval omission problem. The availability indices and the mean downtime are presented as two measures of reliability for this repairable system. Some numerical examples are shown to illustrate the results obtained in this article. [ABSTRACT FROM AUTHOR]

The dynamics of a system represented by a finite-state Markov process operating under two alternating regimes, for example, day/night, machine working/machine idling, etc., are modeled in this article. The transition rate matrices under the two regimes will usually be different. Also, the set of states of the system that are regarded as satisfactory may depend on the regime in operation: for example, a particular state of the system that may be regarded as satisfactory by day might not be tolerated at night (e.g., the headlights on a car not working). It is assumed that the regime durations are random variables and results are obtained for the availability of such a system and probability distributions for uptimes. Results and numerical examples are also given for two special cases: (i) when the regimes are of fixed duration; and (ii) when the regime durations have negative exponential distributions. [ABSTRACT FROM AUTHOR]

A linear connected-(r,s)-out-of-(m,n):F system consists of m\times n components arranged in m rows by n columns, and it fails iff there exists a r\times s subsystem in which all components are failed. The linear connected-(r,s)-out-of-(m,n):F system can be used for modeling engineering systems such as temperature feeler systems, supervision systems, etc. In this paper, a general method is proposed based on the finite Markov chain imbedding approach to study the exact reliability of a linear connected-(r,s)-out-of-(m,n):F system. Then a new more efficient method, which reduces the size of the state space by combining some states into one state, is presented to reduce the computing time. Furthermore, three numerical examples are given. The first two numerical examples show that the proposed algorithm is efficient not only when the component states are i.i.d., but also when the component states are statistically independent and non-identically distributed. And the last numerical example shows that our method can be used to compute not only the reliability, but also the component importance. [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]

A new system is defined based on a series Markov repairable system. In this new system, if a repair time of the system failure is too short (less than a given critical value) to cause the system to fail, then the repair interval may be omitted from the downtime record, i.e., the failure effect could be neglected. Otherwise, if a repair time is longer than the given critical value, then the system remains operating from the beginning of this repair until this repair time exceeding the critical value, i.e., the failure effect could be delayed. In Ion-Channel modeling, we call this situation the time interval omission problem. Incorporating this situation, the availability indices are presented as a measure of reliability. Some numerical examples are shown to illustrate the results obtained in the paper. [ABSTRACT FROM PUBLISHER]

In 1986, Fu proposed the finite Markov chain imbedding approach. Since then, it has played an important role in the analyses of system reliability, such as on reliability computations, importance analyses of components, failure rate function computations, the birth of new reliability system models, and analyses of new reliability tests. We present a survey on the developments and applications of this approach in the reliability field. [ABSTRACT FROM PUBLISHER]

This paper introduces a Markov model for a multi-state repairable system in which some states are changeable in the sense that whether those physical states are up or down depends on the immediately preceding state of the system evolution process. Several reliability indexes, such as availability, mean up time, and steady-state reliability are calculated using the matrix method. A sufficient condition under which the availabilities of the stochastically monotone repairable systems with history-dependent states can be compared is also obtained. Some examples are presented to illustrate the results in the paper. [ABSTRACT FROM AUTHOR]

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]

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]

• 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]

• 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]

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]