New failure and minimal repair processes for repairable systems in a random environment

Most often, minimal repair is defined as a replacement of a failed item by an operable item that has the same distribution of the remaining lifetime as the failed one just prior a failure. This is the so‐called statistical minimal repair extensively explored in the literature. Another well‐known type of minimal repair takes into account the state of a system prior to a failure (the information‐based minimal repair). In this paper, we suggest the new type of minimal repair to be called conditional statistical minimal repair. Our approach goes further and deals with the corresponding minimal repair processes for systems operating in a random environment. Moreover, we also consider heterogeneous populations of items, which makes the model more realistic. Both of these aspects that affect the failure mechanism of items are studied. Environment is modeled by the nonhomogeneous Poisson shock process. Two models for the failure mechanism defined by the extreme shock model and the cumulative shock model, respectively, are considered. Some examples illustrating our findings are presented.     

Two shock and wear systems under repair standing a finite number of shocks

A shock and wear system standing a finite number of shocks and subject to two types of repairs is considered. The failure of the system can be due to wear or to a fatal shock. Associated to these failures there are two repair types: normal and severe. Repairs are as good as new. The shocks arrive following a Markovian arrival process, and the lifetime of the system follows a continuous phase-type distribution. The repair times follow different continuous phase-type distributions, depending on the type of failure. Under these assumptions, two systems are studied, depending on the finite number of shocks that the system can stand before a fatal failure that can be random or fixed. In the first case, the number of shocks is governed by a discrete phase-type distribution. After a finite (random or fixed) number of non-fatal shocks the system is repaired (severe repair). The repair due to wear is a normal repair. For these systems, general Markov models are constructed and the following elements are studied: the stationary probability vector; the transient rate of occurrence of failures; the renewal process associated to the repairs, including the distribution of the period between replacements and the number of non-fatal shocks in this period. Special cases of the model with random number of shocks are presented. An application illustrating the numerical calculations is given. The systems are studied in such a way that several particular cases can be deduced from the general ones straightaway. We apply the matrix-analytic methods for studying these models showing their versatility.     

Lifetime properties of a cumulative shock model with a cluster structure

This paper studies a generalized cumulative shock model with a cluster shock structure. The system considered is subject to two types of shocks, called primary shocks and secondary shocks, where each primary shock causes a series of secondary shocks. The lifetime behavior of such a system becomes more complicated than that of a classical model with only one class of shocks. Under a non-homogeneous Poisson process of primary shocks, we analyze the lifetime behavior of the system with light-tailed and heavy-tailed distributed secondary shocks. We show some important characteristics of lifetime of this type of system. Our model, as an extension of the classical shock models, has wide applications in maintenance engineering, operations management, and insurance risk assessment.     

Reliability modeling for mutually dependent competing failure processes due to degradation and random shocks

Many systems are subject to two mutually dependent competing risks namely degradation and random shocks, and they can fail due to two competing modes of failure, soft and hard failure. Soft failure occurs when the total degradation performance, including continuous degradation and sudden degradation increments caused by random shocks, exceeds a certain critical threshold level. Hard failure occurs when the magnitude of any shock (extreme shock model) or the accumulated damage of shocks (cumulative shock model) is beyond some strength threshold level, which is affected by the temporal degradation performance. From viewpoints of Stress-Strength models and Cumulative damage/shock model, a realistic reliability model is developed in this article for mutually dependent competing failure processes due to degradation and random shocks. Finally, a numerical example of Micro-Electro-Mechanical System (MEMS) is conducted to illustrate the implementation and effectiveness of the proposed model.     

On a single discrete scale for preventive maintenance with two shock processes affecting a complex system

A new approach to optimal maintenance of systems (networks) is suggested. It is applied to systems subject to two external independent shock processes. A system ‘consists’ of two parts, and each shock process affects only its own part. A new notion of bivariate signature is suggested and used for obtaining survival characteristics of a system and further optimization of the preventive maintenance actions. The preventive maintenance optimization is considered in the univariate discrete scale that counts the overall numbers of shocks of both types. An example of a transportation network is considered. Copyright © 2016 John Wiley & Sons, Ltd.     

Availability of inspected systems subject to shocks – A matrix algorithmic approach

We examine the limiting average availability of a maintained system that deteriorates due to random shock process and as a response to its usage (wear out). System’s failures are not self-announcing, hence, failures must be detected via inspection. We consider randomly occurring shocks that arrive according to a Poisson process and cumulatively damage the system. Two models are considered: in Model 1 the shock and wear out processes are independent of the external environment and in Model 2, the shocks arrival rate, the shock magnitudes and the wear out rate are governed by a random environment which evolves as a Markov process. We obtain the system’s availability for both models.     

Structured replacement policies for a Markov-modulated shock model

We establish the optimality of structured replacement policies for a periodically inspected system that fails silently whenever the cumulative number of shocks, or the magnitude of a single shock it has received, exceeds a corresponding threshold. Shocks arrive according to a Markov-modulated Poisson process which represents the (controllable or uncontrollable) environment.     

泊松冲击下冷贮备可修系统的可靠性分析

本文研究了一类由有限个同质部件和一个修理工组成的冷贮备可修系统在随机冲击下的可靠性问题。假设冲击以泊松过程到达。当冲击到达时,它会独立地对系统中工作的部件产生影响,而不会对冷贮备部件产生影响。每次冲击的量都服从某一确定的分布,受冲击的部件以一定的概率发生故障,其故障概率是冲击量的函数,当工作的部件发生故障时,下一个冷贮备部件立即开始工作,当所有部件故障时,系统故障,故障部件按故障顺序进行修理,修理时间服从指数分布,故障部件能被修理如新。本文显式给出了系统首次故障前平均时间、稳态可用度、稳态故障频度等可靠性指标。     

基于累积损伤过程的不完全预防维护决策

基于累积损伤过程研究旧系统的不完全预防维护策略,冲击服从非时齐Poisson过程,并产生随机的损伤量,损伤量是加法累加的.系统在累积损伤量达到k或系统运行年龄达到T时进行计划内预防维护.在两次计划内预防维护之间,当累积损伤量达到预定水平K (k K)时,对系统进行计划外维护,其费用高于计划内预防维护,利用再生过程理论得到单位时间维护成本,讨论在时齐Poisson过程下的时间预防维护策略与水平预防维护策略,同时给出算例.     

Shock models under a Markovian arrival process

A device submitted to shocks arriving randomly and causing damage is considered. Every shock can be fatal or not. The shocks follow a Markovian arrival process. When the shock is fatal, the device is instantaneously replaced. The Markov process governing the shocks is constructed, and the stationary probability vector calculated. The probability of the number of replacements during a time is determined. A particular case in which the fatal shock occurs after a fixed number of shocks is introduced, and a numerical application is performed. The expressions are in algorithmic form due to the use of matrix-analytic methods. Computational aspects are introduced. This model extends others previously considered in the literature.     

System availability in a shock model under preventive repair and phase‐type distributions

A device that can fail by shocks or ageing under policy N of maintenance is presented. The interarrival times between shocks follow phase‐type distributions depending on the number of cumulated shocks. The successive shocks deteriorate the system, and some of them can be fatal. After a prefixed number k of nonfatal shocks, the device is preventively repaired. After a fatal shock the device is correctively repaired. Repairs are as good as new, and follow phase‐type distributions. The system is governed by a Markov process whose infinitesimal generator, stationary probability vector, and availability are calculated, obtaining well‐structured expressions due to the use of phase‐type distributions. The availability is optimized in terms of the number k of preventive repairs. Copyright © 2009 John Wiley & Sons, Ltd.     

Opportunistic Maintenance for Multi-component Shock Models

This paper is concerned with opportunistic maintenance on a multi-component cumulative damage shock model with stochastically dependent components. A component fails when its cumulative damage exceeds a given threshold, and any such a failure creates a maintenance opportunity, and triggers a simultaneous repair on all the components, including the non-failed ones, such that damages accumulated at various components are reduced to certain degrees. Utilizing the coupling method, stochastic maintenance comparisons on failure occurrences under different model parameters are obtained. Some positive dependence properties of this multi-component shock model are also presented.Lirong Cui: Supported by the NSF of China grant 70371048.Haijun Li: Supported in part by the NSF grant DMI 9812994.     

Extreme shock models: An alternative perspective

Extreme shock models have been introduced in Gut and Hüsler (1999) to study systems that at random times are subject to a shock of random magnitude. These systems break down when the shock overcomes a given resistance level.In this paper we propose an alternative approach to extreme shock models using reinforced urn processes. As a consequence of this we are able to look at the same problem under a Bayesian nonparametric perspective, providing the predictive distribution of systems’ defaults.     

Entropy Production and Admissibility of Shocks

In shock wave theory there are two considerations in selecting the physically relevant shock waves.There is the admissibility criterion for the well-posedness of hyperbolic conservation laws.Another consideraztion concerns the entropy production across the shochs.The latter is natural from the physical point of view,but is not sufficient in its straightforward formulation,if the system is not genuinely nonlinear.In this paper we propose the principles of increasing entropy production and that of the superposition of shocks.These principles arc shown to be equivalent to the admissibility criterion.     

Nonstationary shock models

This paper extends results obtained by Esary, Marshall and Proschan [10]. Life distribution properties of a device subject to shocks governed by a nonhomogeneous Poisson process are related to corresponding properties of the probability of failing after experiencing a given number of shocks. Physically motivated models are analyzed in which shocks cause damage to a set of components, the damages accumulate additively, and when the accumulated damage exceeds a critical threshold (possibly random) for any of the components, the device fails. Bounds are obtained on the moments of the life length of the device.     

中美经济冲击传播途径研究

本文应用近似因子模型以及因子增广的向量自回归模型,考察了1995～2009年间美国的经济冲击对我国经济的传播渠道.本文发现出口和消费者信息指数比进口以及金融市场对美国的经济冲击更为敏感;而就冲击的类型来看,美国经济的需求冲击比供给冲击对我国经济的影响更大.     

Slow mode shocks propagating in open and closed magnetic fields

A 2-D MHD model is used to investigate the propagation of slow mode shocks in the open and closed magnetic fields of the meridional plane near the sun. The solutions demonstrate that a forward slow shock could retain its slow shock characteristics into interplanetary space in the magnetically open region; however, it can evolve into an intermediate shock through the helmet-type current sheet to the open magnetic field. Project supported by the National Natural Science Foundation of China and partially the SIGMA Weather Project.     

An extended stochastic failure model for a system subject to random shocks

In this article, a stochastic failure model for a system subject to a random shock process is studied. It is assumed that a fatal shock results in an immediate system failure, whereas a non-fatal shock may increase the susceptibility of the system to failure. The lifetime distribution of the system and its failure rate function are derived, and the effect of environmental factors on the failure process of the system is also investigated. Lifetimes of systems operated under different environmental conditions are stochastically compared.