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We introduce a bivariate tempered space-fractional Poisson process (BTSFPP) by time-changing the bivariate Poisson process with an independent tempered $\alpha$ -stable subordinator. We study its distributional properties and its connection to differential equations. The Lévy measure for the BTSFPP is also derived. A bivariate competing risks and shock model based on the BTSFPP for predicting the failure times of items that undergo two random shocks is also explored. The system is supposed to break when the sum of two types of shock reaches a certain random threshold. Various results related to reliability, such as reliability function, hazard rates, failure density, and the probability that failure occurs due to a certain type of shock, are studied. We show that for a general Lévy subordinator, the failure time of the system is exponentially distributed with mean depending on the Laplace exponent of the Lévy subordinator when the threshold has a geometric distribution. Some special cases and several typical examples are also demonstrated. [ABSTRACT FROM AUTHOR]

In this study, we define a series system with n$n$ non‐independent and non‐identical components using a shock model with n+1$n+1$ sources of fatal shocks. Here, it is assumed that the shocks happen randomly and independently, following a Weibull distribution with various scale and shape parameters. A dependability model with 2(n+1)$2(n+1)$ unknown parameters is produced by this process. Making statistical conclusions about the model parameters is the main objective of this research. We apply the maximum likelihood and Bayes approaches to determine the model parameters' point and interval estimates. We shall demonstrate that no analytical solutions to the likelihood equations must be solved to obtain the parameters' maximum likelihood estimates. As a result, we will use the R program to approximate the parameter point and interval estimates. Additionally, we will use the bootstrap‐t and bootstrap‐p methods to approximate the confidence intervals. About the Bayesian approach, we presume that each model parameter is independent and follows a gamma prior distribution with a range of attached hyperparameter values. The model parameters' posterior distribution does not take a practical form. We are unable to derive the Bayes estimates in closed forms as a result. To solve this issue, we use the Gibbs sampler from the Metropolis‐Hasting algorithm based on the Markov chain Monte Carlo method to condense the posterior distribution. To demonstrate the relevance of this research, a real data set application is detailed. [ABSTRACT FROM AUTHOR]

We investigate a bivariate replacement policy for a system under shocks effect. A system including two units experiences to one of two types of shocks. Whenever a type I shock arrives, unit 1 has a minor failure which can be removed through a minimal repair; while a type II shock leads to a catastrophic failure of the system. The occurrence probabilities of shock types depend on the number of shocks since the last replacement. Whenever unit 1 suffers a minor failure, it causes some additive damage to unit 2. Once cumulative damage of unit 2 reaches a threshold level L, unit 2 will fail, which will also cause unit 1 to fail at the same time, causing a catastrophic failure of the system. Furthermore, unit 2 whose cumulative damage is x may fail minorly with probability π(x) at the instant of minor failure of unit 1, and requires minimal repair when it fails. It is assumed that the system implements preventive replacement as the system's age reaches T, or the mth type I shock occurs, or corrective replacement as a type II shock occurs or the cumulative damage to unit 2 reaches a threshold level L, whichever comes first. For this model, we derive the formula for the mean cost rate and determine analytically and numerically the corresponding optimal policy. Finally, it is also shown that the policy can be regarded as a generalization of existing policies. This generalization makes our policy more flexible and applicable in real situations than existing policies. [ABSTRACT FROM AUTHOR]

In this article, a new δ$$ \delta $$‐shock model that takes into account the magnitude of shocks is introduced and studied from reliability perspective. According to the new model, the system breaks down if either a shock after non‐critical shock occurs in a time length less than δ1$$ {\delta}_1 $$ or a shock after a critical shock occurs in a time length less than δ2,$$ {\delta}_2, $$ where δ1<δ2$$ {\delta}_1<{\delta}_2 $$. The distribution of the system's lifetime is studied for both discrete and continuous intershock time distributions. It is shown that a new model is useful to describe a certain cold standby repairable system. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new class of continuous bivariate distributions based on a shock model. The proposed class is general in the sense that numerous families of distributions can be generated from it. Distributional properties of the proposed class of distributions are studied. The bivariate ageing property and the dependence structure are also analysed. It is shown that the developed bivariate distributions exhibit positive dependency and a well-defined subclass of the proposed class of bivariate distributions satisfy the bivariate lack of memory property. Specific families of distributions are derived and applied to a real data set. [ABSTRACT FROM AUTHOR]

The economic design of control charts depends on the process shock model distribution due to difficulties from both theoretical and practical aspects. This paper pursues to develop the economic design of X ̄ control chart for monitoring continuous flow processes under Bur XII shock model. The Burr XII distribution is flexible and its hazard rate can take many forms like fixed, increasing, decreasing, single mode or even U-shaped. Continuous flow processes find applications in batch processes like filtration, chemical processing, and more. A sensitivity analysis is executed and numeric examples are given to illustrate the effects of changing the parameters of the shock model distribution on the optimum values of the economic design models as discussed in this paper. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the shock model with two kinds of shocks from competing risks outlook. A component or system having two failure modes is considered. The two crude probabilities of failures are modelled and assessed. For assessment of the crude probabilities, Type I (time) censoring life test is considered. Monte-Carlo simulation experiments are used to validate the results derived. [ABSTRACT FROM AUTHOR]

The leading-edge discontinuity problem is addressed for numerical solutions of the 2-D axisymmetric flow equations coupled to an analytical blading model formulation. While this discontinuity affects any 2-D solution because of the incidence angle, it needs a specific treatment under supersonic inflow conditions so that both the started and the unstarted rotor regimes (i.e., with either still supersonic or subsonic flow into the blade passage) can be properly described by the analytical blading model. The new flux splitter includes Levine's theory for evaluation of the unique incidence at started regime, and generalizes it to the unstarted for evaluation of detached shock loss. A discontinuous Riemann solution through the leading edge determines whether the blade geometry and backpressure admit started operation; if they do not, the splitter automatically switches to the unstarted regime. The started mode, however, can be disabled when its computed mass flow exceeds the unstarted choke limit and does not make the rotor start. Capture of operation type as well as prediction of choke mass flow, shock loss, and impact on rotor performance are stressed through comparison to experimental and 3-D numerical results for the NASA Rotor 67 and Stage 37 test cases. [ABSTRACT FROM AUTHOR]

A generalized mixed shock model, which mixes two run shock models, is developed and analyzed. According to the model, the system subject to both internal degradation and external shocks fails upon the occurrence of k 1 consecutive shocks whose magnitude is between predefined critical values of d 1 and d 2 such that d 1 < d 2 , or k 2 consecutive shocks whose magnitude is above d 2 . The system's reliability, mean time to failure, and mean residual lifetime are all calculated under the assumption that the lifetime of the system due to internal wear and external shock arrival times follows a phase-type distribution. The best policy for replacement is also discussed. There are also graphical representations and numerical examples for the proposed model, in which both lifetime distribution of internal degradation and the interarrival periods between external shocks follow the Erlang distribution. [ABSTRACT FROM AUTHOR]

Shock process that can increase the failure rate of the systems working under a stochastic environment is known as the shot-noise process. In the existing literature focusing on the shot-noise process, it is usually assumed that the failure rate increments caused by shocks are the same constants or identically distributed variables. However, such an assumption has limitations if the external environment is constantly changing or the shocks can weaken the system gradually. This article proposes an extended shock model in which the failure rate increments of systems caused by shocks need not be identical or identically distributed. The reliability functions of the systems as well as some reliability indices are derived explicitly. Some special cases are discussed. Besides, a preventive maintenance model is also investigated in which the system is replaced when the number of shocks reaches a fixed threshold. Finally, an example on a refrigeration system is provided to illustrate the proposed model. [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]

For the k -out-of- n system consisting of components that have different weights, the system is in a good state if the total weight of working components is at least k. Such a system is known to be weighted k -out-of- n :G system. Although the weighted k -out-of- n system that has continuously distributed components' lifetimes has been extensively studied, the discrete weighted k -out-of- n :G system has not been considered yet. The present paper fills this gap by modeling and analyzing the weighted k -out-of- n :G system that consists of discretely distributed components' lifetimes. In particular, the behavior of the total capacity/weight of the system with respect to the component failures is evaluated. An optimization problem that is concerned with the determination of optimal number of spare components is also formulated by utilizing the mean lost capacity of the system. • Discrete weighted k-out-of-n system is studied. • The lost capacity by the system is investigated. • The dynamic performance over the failures is analyzed. [ABSTRACT FROM AUTHOR]

This paper is devoted to study censored δ -shock models for both cases when the intershock times have discrete and continuous distributions. In particular, the distribution and moments of the system's lifetime are studied via probability generating functions and Laplace transforms. For discrete intershock time distributions, several recursions for evaluating the probability mass function, the survival function and the moments of the system's lifetime are given. As it is shown for the discrete case, the distribution of the system's lifetime is directly linked with matrix-geometric distributions for particular classes of intershock time distributions, such as phase-type distributions. Thus, matrix-based expressions are readily obtained for the exact distribution of the system's lifetime under discrete setup. Also, discrete uniform intershock time distributions are examined. For the case of continuous intershock time distributions, it is shown that the shifted lifetime has a compound geometric distribution, and based on this, the distribution of the system's lifetime is approximated via discrete mixture distributions having a mass at δ and matrix-exponential distributions for the continuous part. Both for the discrete and the continuous case, Lundberg-type bounds and asymptotics for the survival function of system's lifetime are given. To illustrate the results, some numerical examples, both for the discrete and the continuous case, are also given. [ABSTRACT FROM AUTHOR]

Safety-critical systems are commonly required to complete specific missions in shock environments, and their failures may lead to severe economic losses and significant safety hazards. To enhance system reliability, protective devices are commonly equipped to resist external shocks. The existing literature focuses mainly on the maintenance policy of safety-critical systems, ignoring the system reliability analysis considering the effect of protective devices and the self-exciting mechanism of shocks. This paper considers multi-state systems equipped with a protective device in shock environments where valid shocks and invalid shocks occur stochastically. The system state degenerates due to valid shocks or the self-exciting behavior of invalid shocks. The self-exciting mechanism is triggered when the number of cumulative or consecutive invalid shocks suffered by the system exceeds a certain threshold, leading the system to a worse state. The protective device can be triggered to protect the system from the damage of external shocks when the state is worse than a predetermined threshold. The protective effect is characterized by reducing the probability of valid shocks. A finite Markov chain embedding approach is used to evaluate the system reliability index. In addition, an optimization model is constructed to determine the optimal triggering threshold of the protective device. The numerical results indicate that protective devices can significantly improve the reliability of the system and incorporating the self-exciting mechanism of shocks into reliability modeling contributes to accurate reliability evaluation. [ABSTRACT FROM AUTHOR]

In this paper, delta and extreme shock models and a mixed shock model which combines delta-shock and extreme shock models are studied. In particular, the interarrival times between successive shocks are assumed to belong to a class of matrix-exponential distributions which is larger than the class of phase-type distributions. The Laplace -Stieltjes transforms of the systems' lifetimes are obtained in a matrix form. Survival functions of the systems are approximated based on the Laplace-Stieltjes transforms. The results are applied for the reliability evaluation of a certain repairable system consisting of two components. [ABSTRACT FROM AUTHOR]

In this paper, the extreme and δ shock models are studied, in which systems are subject to dependent hard failure and soft failure processes. Under the extreme shock model, a hard failure occurs if the magnitude of a single shock surpasses a critical level, while under the δ shock model, a hard failure occurs if the interval time between two successive shocks is smaller than a threshold. Under both shock models, soft failure of a system happens if the total volume of degradation surpasses a given soft failure threshold. The hard and soft failure processes are interdependent owing to the fact that external shock will bring abrupt increment in the degradation path of the system, and on the other hand, the amount of total degradation will affect the hard failure threshold of the system. The failure of the system occurs if a hard failure or a soft failure occurs, whichever happens first. The reliability expressions of the systems under two shock models are derived explicitly. Some reliability indices of the system are calculated by utilizing the probability analysis method. A case study of the sliding spool is provided to illustrate the proposed model. [ABSTRACT FROM AUTHOR]

In this paper, we develop an exact test for testing exponentiality against renewal increasing mean residual life class. The critical values are presented for different sample sizes. Asymptotic properties of the test statistic are also studied. Some numerical results are presented for evaluating the performance of the proposed test. The test procedure is illustrated by using two real data sets. The proposed method is then modified to incorporate right censored observations and the efficiency loss due to censoring is evaluated using Pitman's asymptotic efficacy value. [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]

We show that each infinite exchangeable sequence τ1, τ2, . . . of random variables of the generalised Marshall–Olkin kind can be uniquely linked to an additive subordinator via its deFinetti representation. This is useful for simulation, model estimation, and model building.

Statistical inference for extremes has been a subject of intensive research during the last years. In this paper, we generalize the Marshall-Olkin bivariate Pareto distribution. In this case, a new bivariate distribution is introduced by compounding the Pareto Type $\rm{II}$ and geometric distributions. This new bivariate distribution has natural interpretations and can be applied in fatal shock models or in competing risks models. We call the new proposed model Marshall-Olkin bivariate Pareto-geometric (MOBPG) distribution, and then investigate various properties of the new distribution. This model has five unknown parameters and the maximum likelihood estimators cannot be afforded in explicit structure. We suggest to use the EM algorithm to calculate the maximum likelihood estimators of the unknown parameters, and this structure is quite flexible. Also, Monte Carlo simulations are performed to investigate the effectiveness of the proposed algorithm. Finally, we analyze a real data set to investigate our purposes.

Although bivariate imprecise copulas have recently attracted substantial attention, the multivariate case seems still to be open. So, it is natural to test it first on shock model induced copulas, a family which might be the most useful in various applications. We investigate a model in which some of the shocks are assumed imprecise and develop the corresponding set of copulas. In the Marshall's case we get a coherent set of distributions and a coherent set of copulas, where the bounds are naturally corresponding to each other. The situation in the other two groups of multivariate imprecise shock model induced copulas, i.e., the maxmin and the reflected maxmin (RMM) copulas, is substantially more involved, but we are still able to exhibit their properties. These are the main results of the paper that serves as the first step into a theory that should develop in this direction. Also, the theory of imprecise RMM copulas seems to be new even in the bivariate case. It turns out that all imprecise copulas under consideration are coherent. [ABSTRACT FROM AUTHOR]

We study the extreme points (in the Krein-Milman sense) of the class of semilinear copulas and provide their characterization. Related results into the more general setting of conjunctive aggregation functions (i.e., semi–copulas and quasi–copulas) are also presented. [ABSTRACT FROM AUTHOR]

We consider a system consisting of two units (A and B), which is subject to two types of shocks (I and II) that occur according to a non-homogeneous Poisson process. The probabilities of these two shock types are age-dependent. Each type-I shock causes a minor failure of unit A, which can be corrected by a minimal repair. Meanwhile, this type of shock results in a certain amount of damage to unit B. These damages to unit B are accumulated to trigger a preventive replacement (PM) or a corrective replacement (CM) action. In addition, a minor failure for unit B with the cumulative damage of z will occur with probability at a type-I shock instant. Type-II shock is a major one that causes system replacement. We consider a two-dimensional PM policy, which prescribes that the system is preventively replaced at age T, or at the time when the total damage to unit B exceeds a prespecified level Z (but less than the failure level K where K > Z) or is replaced correctively at first type-II shock or when the total damage to unit B exceeds a failure level K, whichever occurs first. Thus, both PM and CM actions may be performed in our model. To minimise the expected cost per unit time, the optimal policy (,) is derived analytically and determined numerically. We also show that our model is a generalisation of many previous maintenance models in the literature. [ABSTRACT FROM AUTHOR]

The traditional shock model generally describes the magnitude of the cumulative damage caused by a random shock sequence and compares the magnitude with a predetermined threshold to obtain the failure time of a component. There are two limitations in this kind of models in practice: First, the statistical characteristics of the damage due to a single shock may be difficult to obtain, which means the magnitude of the damage may not be described by an appropriate distribution; Second, the cumulative shock magnitude may be difficult to measure, or it may be difficult for a failure mode to be described by a threshold, meaning that the magnitude of the damage and the threshold may not be compared with each other. Considering both failure and censored samples, a reliability modeling method is proposed in this work to address the above problems. The shock model is first established by using both continuous and discrete phase-type (PH) distributions. Then the parameter estimation method of the shock model is derived based on EM method and the identifiability of the parameters in PH distributions is also given. Finally, the adaptability of the model is analyzed using three different types of simulation cases.

In the study of the reliability of systems in reliability engineering, it has been defined several measures in the reliability and survival analysis literature. The reliability function, the mean residual lifetime and the hazard rate are helpful tools to analyze the maintenance policies and burn-in. In this paper, we consider a network consisting of n components having the property that the network has two states up and down (connected and disconnected). Suppose that the network is subject to shocks that each may cause the component failures. We further suppose that the number of failures at each shock follows a truncated binomial distribution and the process of shocks is nonhomogeneous Poisson process. This paper investigates the reliability function, the mean residual lifetime and the hazard rate of the network under shock model. An example and illustrative graph is also provided.

Safety-critical systems are commonly required to complete specific missions in shock environments, and their failures may lead to severe economic losses and significant safety hazards. To enhance system reliability, protective devices are commonly equipped to resist external shocks. The existing literature focuses mainly on the maintenance policy of safety-critical systems, ignoring the system reliability analysis considering the effect of protective devices and the self-exciting mechanism of shocks. This paper considers multi-state systems equipped with a protective device in shock environments where valid shocks and invalid shocks occur stochastically. The system state degenerates due to valid shocks or the self-exciting behavior of invalid shocks. The self-exciting mechanism is triggered when the number of cumulative or consecutive invalid shocks suffered by the system exceeds a certain threshold, leading the system to a worse state. The protective device can be triggered to protect the system from the damage of external shocks when the state is worse than a predetermined threshold. The protective effect is characterized by reducing the probability of valid shocks. A finite Markov chain embedding approach is used to evaluate the system reliability index. In addition, an optimization model is constructed to determine the optimal triggering threshold of the protective device. The numerical results indicate that protective devices can significantly improve the reliability of the system and incorporating the self-exciting mechanism of shocks into reliability modeling contributes to accurate reliability evaluation.

The well-known Marshall–Olkin model is known for its extension of exponential distribution preserving lack of memory property. Based on shock models, a new generalization of the bivariate Marshall–Olkin exponential distribution is given. The proposed model allows wider range tail dependence which is appealing in modeling risky events. Moreover, a stochastic comparison according to this shock model and also some properties, such as association measures, tail dependence and Kendall distribution, are presented. The new shock model is analytically quite tractable, and it can be used quite effectively, to analyze discrete–continuous data. This has been shown on real data. Finally, we propose the multivariate extension of the Marshall–Olkin model that has some intersection with the well-known multivariate Archimax copulas. [ABSTRACT FROM AUTHOR]

For a system that is subject to shocks, it is assumed that the distribution of the magnitudes of shocks changes after the first shock of size at least d1, and the system fails upon the occurrence of the first shock above a critical level d2 (> d1). In this paper, the distribution of the lifetime of such a system is studied when the times between successive shocks follow matrix-exponential distribution. In particular, it is shown that the system's lifetime has matrix-exponential distribution when the intershock times follow Erlang distribution. The model is extended to the case when the system fails upon the occurrence of l consecutive critical shocks. [ABSTRACT FROM AUTHOR]

• A maintenance model based on the shot-noise process is studied. • The failure rate increments caused by shocks needn't be identically distributed. • Periodical PM policy is adopted for the system subject to shocks. • Expression of the long-run average cost rate is derived explicitly. • Existence and uniqueness of the optimal PM policy are proved theoretically. External shock process, known as the shot-noise process, will lead to the failure rate increasing of systems operating under random environment. In this paper, assuming that each shock may cause a random increment of the system failure rate, the optimal periodic preventive maintenance (PM) policy for the system is investigated. The system is preventively maintained at periodic time points n T , n = 1 , 2 , ... , N − 1 , and is preventively replaced at time point N T. The PM is not perfect, however, it can remove the accumulated failure rate increments caused by external shocks occurred during the last PM interval. When the system fails during the PM intervals, a minimal repair is performed to restore the function of the system. The expression of the average cost rate of the system is derived explicitly. The existence and uniqueness of the optimal bivariate PM policy (N , T) that minimizes the average cost rate are proved theoretically. Numerical examples are presented to illustrate the maintenance model and show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, particular shock models are studied for the case when the times between successive shocks and the magnitudes of shocks have discrete phase‐type distributions. The well‐known shock models such as delta shock model, extreme shock model, and the mixed shock model which is obtained by combining delta and extreme shock models are considered. The probability generating function and recursive equation for the distribution of the system's lifetime are obtained for the cases when the interarrival times between shocks and the magnitudes of shocks are independent and when they are dependent. System reliability is computed for particular interarrival distributions such as geometric, negative Binomial and generalized geometric distributions. [ABSTRACT FROM AUTHOR]

We propose an extended maintenance overtime policy for the cumulative damage model where an operating unit suffers some damage due to shocks. It is assumed that the total damage is additive, and the unit fails when the total damage has exceeded a prespecified damage level. It is supposed that we start to observe occurrence of shocks after time T, and the unit is replaced at Nth (N=1,2,…) shock over time T or at failure, whichever occurs first. That is, we propose a new policy by extending maintenance overtime policy. One example is a rental of system such as industrial equipment with some reservations. For such systems, they should be maintained or replaced at a prespecified number of uses over a scheduled time. For such a model, we obtain the mean time to replacement and the expected costs rate. Further, we discuss about optimal number N^* and time T^* which minimizes the expected cost rate when shocks occur in a Poisson process. Finally, numerical examples are given, and suitable discussions are made.

In this paper, two bivariate models based on the proposed methods of Marshall and Olkin are introduced. In the first model, the new bivariate distribution is presented based on the proposed method of Marshall and Olkin (1967) which has natural interpretations, and it can be applied in fatal shock models or in competing risks models. In the second model, the proposed method of Marshall and Olkin (1997) is generalized to bivariate case and a new bivariate distribution is introduced. We call these new distributions as the bivariate Gompertz (BGP) distribution and bivariate Gompertz-geometric (BGPG) distribution, respectively. Moreover, the BGP model can be obtained as a special case of the BGPG model. Then, we present various properties of the new bivariate models. In this regard, the joint and conditional density functions, the joint cumulative distribution function can be obtained in compact forms. Also, the aging properties and the bivariate hazard gradient are discussed. This model has five unknown parameters and the maximum likelihood estimators cannot be obtained in explicit form. We propose to use the EM algorithm to compute the maximum likelihood estimators of the unknown parameters, and it is computationally quite tractable. Also, Monte Carlo simulations are performed to investigate the effectiveness of the proposed algorithm. Finally, we analyze three real data sets for illustrative purposes. [ABSTRACT FROM AUTHOR]

In the classical extreme shock model, the system fails due to a single catastrophic shock. In this paper, by assuming different arrival patterns of the shocks, two new types of extreme shock models are introduced. In these models, m possible sources may exert shocks on the system. Both models reduce to the classical extreme shock model when m = 1. Assuming phase-type distribution for times between successive shocks, we obtain survival functions and mean time to failure values of the system under new models. Two different optimization problems are also considered to determine the optimal number of sources. [ABSTRACT FROM AUTHOR]

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Series and parallel systems consisting of two dependent components are studied under bivariate shock models. The random variables N1 and N2 that represent respectively the number of shocks until failure of component 1 and component 2 are assumed to be dependent and phase-type. The times between successive shocks are assumed to follow a continuous phase-type distribution, and survival functions and mean time to failure values of series and parallel systems are obtained in matrix forms. An upper bound for the joint survival function of the components is also provided under the particular case when the times between shocks follow exponential distribution. [ABSTRACT FROM AUTHOR]

Abstract A new mixed shock model is introduced and studied. According to the model, for two fixed critical values d 1 and d 2 such that d 1 < d 2 , the system under concern fails upon the occurrence of k consecutive shocks of size at least d 1 or a single large shock of size at least d 2. The new model combines run and extreme shock models. Reliability properties of the system are studied under two cases: when the interarrival time X i between the (i − 1) th and i th shock, and the magnitude of the i th shock Y i are independent for all i , and when the interarrival time between the (i − 1) th and i th shock, and the magnitude of the i th shock are dependent for all i. [ABSTRACT FROM AUTHOR]

This paper is devoted to redundancy allocation in series (parallel) systems subject to random shocks. The non-homogeneous Poisson process is used to describe the shock process, and redundant series (parallel) system reliability is given. The majorization order allows for stochastic comparison between random lifetimes of systems under two redundancy allocation policies. Then, the redundancy allocation policy for maximizing (minimizing) the series (parallel) system reliability is presented. Finally, the effect of the number of subsystems on system reliability is analyzed. [ABSTRACT FROM AUTHOR]

The economic design of control charts depends on the process shock model distribution and due to difficulties from both theoretical and practical aspects, one of the proper alternatives may be the Burr XII (Burr) distribution which is quite flexible and its hazard rate can be fixed, increasing, decreasing, single mode or even U-shaped. Yet, in Burr distribution context, only one assignable cause model were considered where more realistic approach may be to consider multiple assignable causes. Thus, in this paper, an economic design of the control charts under the Burr shock model and multiple assignable causes were considered and compared with three types of prior distribution for the mean shift parameter. Also, a numerical example and a sensitivity analysis for more illustration and studying the effect of changing parameters of the shock model distribution on the optimum values of proposed design models were conducted, respectively. The results indicate that when the prior distribution is of negative exponential distribution, the multiple assignable cause models under Burr XII (Burr) distribution have minimum loss-cost. [ABSTRACT FROM AUTHOR]

Many engineering systems suffer gradual deterioration due to both external environmental damage and internal stress caused by working loads. System degradation is directly related to its working load, providing opportunities to control degradation by adjusting the workload. However, most existing research neglects the effect of environmental factors on system failure behavior and maintenance decisions. This paper addresses this research gap by investigating the optimal joint inspection interval, condition-based maintenance, and loading policies for systems operating in a random shock environment. We formulated the problem as a Markov decision process aimed at minimizing the long-run discounted cost, utilizing the value iteration algorithm to find optimal integrated policies while analyzing the corresponding structural properties of the policy. We extended our model by characterizing the shock arrival process with a non-homogeneous Poisson process, conducting comprehensive policy comparison and parameter sensitivity analyses through a numerical example. Our results illustrate that dynamic working load adjustment significantly impacts system degradation and the long-run expected cost. Moreover, the optimal joint policy is highly dependent on the relationship between the working load and system state deterioration. Finally, we derived some managerial implications for the joint development of load regulation and maintenance implementation to support decision-making. • Joint optimization of inspection, dynamic loading, and maintenance are studied. • A universal shock model is used to characterize the impact of external environment. • The optimal maintenance policy is proved to be a control limit policy. • The optimal loading policy balancing degradation and revenue is investigated. [ABSTRACT FROM AUTHOR]

• Systems subject to environmental shocks with limited self-healing resources are studied. • Two self-healing policies are proposed to allocate the limited resources over time. Accordingly, two new reliability models are developed. • Reliabilities of the system under different self-healing policies are derived in a recursive way. • Four new indexes are proposed and derived, based on which the performance of the self-healing policies could be evaluated, compared and further optimized. For artificial systems, the rising cost and inaccessible scenes for maintenance have increased the need of self-healing, which intends to give systems the ability to heal the damage and faults without external intervention like living organisms. Although self-healing techniques have been widely studied and applied in many engineering systems, reliability research for such systems is limited. Besides, the self-healing ability of artificial systems is generally derived from self-healing resources embedded in the system, while the limitations of embedded resources have less been considered in related research. To fill the research gap, in this paper we consider systems operating under shock environments with limited self-healing resources. Two self-healing policies are designed for allocating the limited resources and enhancing the system reliability. Under different policies, new reliability models are developed and a recursive method is proposed to evaluate the system reliability. Further, some new indexes are introduced and derived for evaluating the performance of the self-healing policies. Based on the new indexes, numerical examples are presented in the end to compare the two policies. [ABSTRACT FROM AUTHOR]

A shift-dependent information measure is favorable to handle in some specific applied contexts such as mathematical neurobiology and survival analysis. For this reason, the weighted differential entropy has been introduced in the literature. In accordance with this measure, we propose the weighted generalized cumulative residual entropy as well. Despite existing apparent similarities between these measures, however, there are quite substantial and subtle differences between them because of their different metrics. In this paper, particularly, we show that the proposed measure is equivalent to the generalized cumulative residual entropy of the cumulative weighted random variable. Thus, we first provide expressions for the variance and the new measure in terms of the weighted mean residual life function and then elaborate on some characteristics of such measures, including equivalent expressions, stochastic comparisons, bounds, and connection with the excess wealth transform. Finally, we also illustrate some applications of interest in system reliability with reference to shock models and random minima.