Extreme Shock Model Research Articles

Shock models governed by an inverse gamma mixed Poisson process

AbstractWe study three classes of shock models governed by an inverse gamma mixed Poisson process (IGMP), namely a mixed Poisson process with an inverse gamma mixing distribution. In particular, we analyze (1) the extreme shock model, (2) the δ-shock model, and the (3) cumulative shock model. For the latter, we assume a constant and an exponentially distributed random threshold and consider different choices for the distribution of the amount of damage caused by a single shock. For all the treated cases, we obtain the survival function, together with the expected value and the variance of the failure time. Some properties of the inverse gamma mixed Poisson process are also disclosed.

Reliability modeling based on multiple wiener degradation-shock competing failure process and dynamic failure threshold

Given multiple degradation failure and shock failure processes within the complex system during operation, a reliability model that combines the degradation-shock competing failure process and dynamic failure threshold is modeled. The Wiener process with random effects is considered the degradation process model, which includes random effects to account for the heterogeneity among system units. Then, the extreme shock model with a dynamic failure threshold is used to depict the random shock. The copula function is carried out to illustrate the correlation between multiple degradation processes, the reliability model is constructed further. To verify the application of this model, a numerical case and a micro-electro-mechanical system comprising two micro-mechanical resonators are employed. The parameter sensitivity of the proposed model is analyzed. The study results show that the reliability model, which combines the Wiener process with random effects and the dynamic failure threshold, more accurately reflects the actual operational state of the complex system.

Reliability analysis of the multi-state system with nonlinear degradation model under Markov environment

Research conducted on multi-state systems has garnered significant attention as a result of their engineering applications in the aerospace industry. The paper explores the intricacies of multi-state systems operating in complex environments and experiencing degeneration processes. A mixed shock model is introduced, which consists of multiple run shock models and extreme shock models. A state transition matrix is constructed using a Markov chain embedding method to establish a reliability model of hard failure process. In contrast to the traditional reliability modeling approach of hard failure process, this method incorporates time-dependencies. A numerical integration method is employed to derive the reliability function of the soft failure process based on the Gamma process. To validate the effectiveness of the developed methodology, an application of a micro-engine is utilized. Finally, compared with the Monte Carlo simulation, the developed model can be considered as a useful tool for the system manager to assess the reliability of the multi-state system with the nonlinear degradation process.

A general class of shock models with dependent inter-arrival times

We introduce and study a general class of shock models with dependent inter-arrival times of shocks that occur according to the homogeneous Poisson generalized gamma process. A lifetime of a system affected by a shock process from this class is represented by the convolution of inter-arrival times of shocks. This class contains many popular shock models, namely the extreme shock model, the generalized extreme shock model, the run shock model, the generalized run shock model, specific mixed shock models, etc. For systems operating under shocks, we derive and discuss the main reliability characteristics (namely the survival function, the failure rate function, the mean residual lifetime function and the mean lifetime) and study relevant stochastic comparisons. Finally, we provide some numerical examples and illustrate our findings by the application that considers an optimal mission duration policy.

On mixed censored [formula omitted]-shock models

In this paper we introduce the mixed censored δ-shock model which combines the censored δ-shock model and the classical extreme shock model. Under the mixed censored δ-shock model, the system fails whenever no shock occurs within a δ-length time period from the last shock, or the magnitude of the shock is larger than another critical threshold γ>0. For the discrete-time case of occurrence of shocks, by assuming the dependence between intershock times and the corresponding magnitudes of shocks we derive the probability generating function (pgf) of the lifetime of the system, and a matrix-based expression is obtained for the exact distribution of the system’s lifetime when the distribution of intershock times and the magnitudes of shocks have a discrete bivariate phase-type distribution. Similar results are obtained by assuming the independence between intershock times and the corresponding magnitudes of shocks, and by proving that the distribution of the shifted lifetime at δ, is the convolution of a discrete compound geometric distribution and a discrete compound Bernoulli distribution, we get several results concerning the distribution of system’s lifetime, like as, simple and efficient recursions for evaluating the survival function and the probability mass function (pmf), the mean and the variance of system’s lifetime, as well as discrete Lundberg-type upper bounds for the reliability function. For the continuous-time case of occurrence of shocks we obtain an exact formula for the reliability function and the Laplace–Stieltjes transform of system’s lifetime by assuming the dependence between intershock times and the corresponding magnitudes of shocks, and under the independence setup, we obtain the reliability function when the intershock times have the uniform distribution, and we give an asymptotic result under the Poisson process for the arrival of shocks. Similar to the discrete-time case, it is shown that the distribution of the shifted lifetime at δ, is the convolution of a compound geometric distribution and a compound Bernoulli distribution and using this we obtain two-sided Lundberg-type bounds for the survival function. Finally, some numerical examples to illustrate our results, are also given.

Reliability of a mixed δ-shock model with a random change point in shock magnitude distribution and an optimal replacement policy

A mixed δ-shock model when there is a change in the distribution of the magnitudes of shocks is defined and studied. Such a model which is a combination of the δ-shock model and the extreme shock model with a random change point (studied by Eryilmaz and Kan, 2019), is useful in practice since a sudden change in environmental conditions may cause a larger shock. In particular, the reliability and the mean time to failure of the system are evaluated by assuming that the random change point has a discrete phase-type distribution. Analytical results for evaluating the reliability function of the system for several joint distributions of the interarrival times and the magnitudes of shocks, are also given. The optimal replacement policy that is based on a control limit is also proposed when the number of shocks until the change point follows geometric distribution. The results are illustrated by numerical examples.

Reliability analysis of series and parallel systems with heterogeneous components under random shock environment

This article considers the reliability optimization for series and parallel systems under random shock environment, where the components come from a heterogeneous population consisting of some distinct subpopulations. We propose the extreme and cumulative shock models, where the random shock environment for each subsystem is modeled by the nonhomogeneous Poisson process. We analyze the reliability of systems with different components allocation policies, and then obtain the optimal policy that optimizes the system reliability. At the same time, we discuss the influence of the heterogeneity on the system reliability. An illustrative example is also presented.

Reliability analysis for [formula omitted]-out-of-[formula omitted](G) systems subject to dependent competing failure processes

In this paper, reliability analysis for k-out-of-n(G) systems suffering from interdependent competing failure processes is conducted. The system consisting n different components fails if and only if the number of operating components is less than k. Each component degrades linearly with time. A soft failure happens if the total degradation performance exceeds the soft failure threshold of the corresponding component. All components in the system are influenced by random shocks which arrive according to a homogeneous Poisson process. The hard failure of a component occurs when a single shock with large shock load arrives (extreme shock model), or the accumulated shock load of a sequential shocks surpasses the hard failure threshold of the component (cumulative shock model). A component fails if a hard failure or a soft failure occurs. Since the external shock may cause an abrupt degradation increment to a component, the hard failure and soft failure processes of each component are relevant. Moreover, because a shock has impact on all working components in the system, the failure processes of these components are also relevant. Using multivariate analysis methods and techniques, the closed forms of the reliability functions of the k-out-of-n(G) system under the extreme shock model and the cumulative shock model are obtained. Monte Carlo simulation is used as an auxiliary method to compute the system reliability. As a numerical example, the reliability of the micro-electro-mechanical systems is calculated to illustrate the developed theory.

A new mixed δ-shock model and associated reliability properties

In reliability literature, shock models and multi-state systems have both been discussed extensively. Shock models have attracted a great deal of attention due to their important role in engineering systems. In the δ-shock model, a system fails if the interval time between two successive shocks is less than a pre-fixed threshold δ. In this study, a new version of mixed δ-shock models is defined by combining δ-shock and extreme shock models, thus causing the failure of a multi-state system in two ways: first, when k interarrival times between two consecutive shocks with magnitude larger than the critical threshold γ are in and second, when the interarrival time between two consecutive shocks is less than δ 1. The survival function is obtained for three cases: (i) system’s lifetime, (ii) when the system performs completely, and (iii) when the system performs partially. A Monte Carlo simulation study is carried out to validate the analytical results established here.

On stochastic ordering among extreme shock models

AbstractIn the usual shock models, the shocks arrive from a single source. Bozbulut and Eryilmaz [(2020). Generalized extreme shock models and their applications. Communications in Statistics – Simulation and Computation49(1): 110–120] introduced two types of extreme shock models when the shocks arrive from one of $m\geq 1$ possible sources. In Model 1, the shocks arrive from different sources over time. In Model 2, initially, the shocks randomly come from one of $m$ sources, and shocks continue to arrive from the same source. In this paper, we prove that the lifetime of Model 1 is less than Model 2 in the usual stochastic ordering. We further show that if the inter-arrival times of shocks have increasing failure rate distributions, then the usual stochastic ordering can be generalized to the hazard rate ordering. We study the stochastic behavior of the lifetime of Model 2 with respect to the severity of shocks using the notion of majorization. We apply the new stochastic ordering results to show that the age replacement policy under Model 1 is more costly than Model 2.

Multi-State Reliability Analysis Based on General Wiener Degradation Process and Random Shock

A multi-state reliability analysis model suffering from a dependent competing failure process is developed in this study, where the soft failure is described by general nonlinearity random effect. Wiener degradation process with measurement error and the hard failure is caused by random shock. Considering that the shock process not only may cause abrupt damage but also can accelerate degradation, there are some correlations between soft failure and hard failure. Based on the proposed new model, the multi-state functions are obtained under the cumulative shock model and the extreme shock model, and the system state probabilities are given under the different degradation state points. The fatigue cracks growth data example and MEMS oscillator example are given to demonstrate the proposed new model. At last, some sensitivity analyses are given to illustrate the influence of parameters on the state probability and the system reliability.

Reliability analysis for systems with interactive competing degradation processes and mixed shock effects

In this article, an extended mixed shock model is investigated, in which the system suffers from mutually dependent competing soft failure and hard failure processes. The system studied degrades gradually with time and the number of shocks coming from the environment. A soft failure occurs when the overall degradation amount surpasses the soft failure threshold of the system. The extended mixed shock model is a combination of a system-state-dependent extreme shock model and a system-state-dependent δ shock model. A hard failure occurs when the magnitude of a single shock exceeds the threshold D(t), or an inter-arrival time between shocks is less than the recovery time threshold The mutual dependency between the soft failure and hard failure processes is reflected in the fact that: (1) the external shock will accelerate the degradation process of the system by causing an abrupt increment of its degradation amount; (2) the system degradation amount determines the thresholds of the extreme shock model and the δ shock model. The system fails when a hard failure or a soft failure occurs. By using the stochastic process theory, the reliability expression and some reliability indices of the system are obtained. Some special cases of the model are also discussed. Finally, a case study of the pier columns of a sea bridge is presented to illustrate the developed model.

Two shock models for single‐component systems subject to mutually dependent failure processes

AbstractIn this paper, the extreme and 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 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.

Reliability modeling for dependent competing failure processes with phase-type distribution considering changing degradation rate

In this paper, a system reliability model subject to Dependent Competing Failure Processes (DCFP) with phase-type (PH) distribution considering changing degradation rate is proposed. When the sum of continuous degradation and sudden degradation exceeds the soft failure threshold, soft failure occurs. The interarrival time between two successive shocks and total number of shocks before hard failure occurring follow the continuous PH distribution and discrete PH distribution, respectively. The hard failure reliability is calculated using the PH distribution survival function. Due to the shock on soft failure process, the degradation rate of soft failure will increase. When the number of shocks reaches a specific value, degradation rate changes. The hard failure is calculated by the extreme shock model, cumulative shock model, and run shock model, respectively. The closed-form reliability function is derived combining with the hard and soft failure reliability model. Finally, a Micro-Electro-Mechanical System (MEMS) demonstrates the effectiveness of the proposed model

On history-dependent mixed shock models

In this paper, we consider a history-dependent mixed shock model which is a combination of the history-dependent extreme shock model and the history-dependent $\delta$-shock model. We assume that shocks occur according to the generalized Pólya process that contains the homogeneous Poisson process, the non-homogeneous Poisson process and the Pólya process as the particular cases. For the defined survival model, we derive the corresponding survival function, the mean lifetime and the failure rate. Further, we study the asymptotic and monotonicity properties of the failure rate. Finally, some applications of the proposed model have also been included with relevant numerical examples.

Reliability analysis for the fractional-order circuit system subject to the uncertain random fractional-order model with Caputo type

IntroductionAccording to the competing failure theorem, the fractional-order Resistor Capacitance (RC) circuit system suffers not only from internal degradation but also from external shocks. However, due to the general differences of each failure type in the data availability and cognitive uncertainty, a better model is needed to describe the degradation process within the system. Also, a new reliability analysis method is needed for the circuit system under internal degradation and external shocks. ObjectivesTo demonstrate this problem, this paper proposes a novel class of Caputo-type uncertain random fractional-order model that focuses on the reliability analysis of a fractional-order RC circuit system. MethodsFirst, an uncertain Liu process is used to describe the internal degradation of soft faults and a stochastic process is used to describe the external random shocks of hard faults. Secondly, taking into account the correlation and competition among the fault types, an extreme shock model and a cumulative shock model are constructed, and chance theory is introduced to further explore the fault mechanisms, from which the corresponding reliability indices are derived. Finally, the predictor–corrector method is applied and numerical examples are given. ResultsThis paper presents a reliability analysis of a fractional-order RC circuit system with internal failure obeying an uncertain process and external failure obeying a stochastic process, and gives the calculation of the reliability indexes for different cases and the corresponding numerical simulations. ConclusionA new competing failure model for a fractional-order RC circuit system is presented and analyzed for reliability, which is proved to be of practical importance by numerical simulations.

Two novel critical shock models based on Markov renewal processes

AbstractIn 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.

Assessment of Shock Models for a Particular Class of Intershock Time Distributions

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.

Reliability analysis for multi-component systems with interdependent competing failure processes

In this paper, two reliability models for multi-component systems suffering from dependent competing failure processes is studied. All components in the systems are subject to two types of failures, i.e., soft failure owing to internal degradation and hard failure due to external shocks. A soft failure happens when the total degradation amount of a component exceeds its soft failure threshold, and a hard failure of a component occurs when a single shock load or the accumulated shock load exceeds its hard failure threshold. A component fails when a soft failure or a hard failure happens, whichever occurs first. The external shock arriving according to a non-homogeneous Poisson process has impact on all components of the systems simultaneously. The soft failure and hard failure processes are dependent because the external shock can not only break down components stochastically, but also accelerate the degradation process of the components. By using the multivariate analysis technique, the analytic forms of the reliability functions for systems under cumulative shock model and extreme shock model are derived. The random point method in Monte Carlo simulation and vector program in Matlab are applied to calculate multiple integral in the expressions of the reliability functions. Finally, a case study on the Micro-Electro Mechanical Systems with multiple different components is provided to illustrate the proposed model.

Reliability assessment for discrete time shock models via phase‐type distributions

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.