A reliability system under different types of shock governed by a Markovian arrival process and maintenance policy K

A reliability system subject to shocks producing damage and failure is considered. The source of shocks producing failures is governed by a Markovian arrival process. All the shocks produce deterioration and some of them failures, which can be repairable or non-repairable. Repair times are governed by a phase-type distribution. The number of deteriorating shocks that the system can stand is fixed. After a fatal failure the system is replaced by another identical one. For this model the availability, the reliability, and the rate of occurrence of the different types of failures are calculated. It is shown that this model extends other previously published in the literature.     

Determining optimal replacement time for metal cutting tools

Traditional tool life models do not take into account the variation inherent in metal cutting processes. As a consequence, the real tool life rarely matches the predicted values. To compensate for this uncertainty, tools are usually replaced prematurely, which leads to unnecessarily high tool costs. In some cases, however, wear-out occurs earlier than predicted, which imposes a risk of workpiece damage or rework and can lead to other extra charges. To balance these costs, this paper proposes an age replacement model. It is assumed that penalty costs are incurred each time a tool fails before the planned replacement. The probability of such an event is determined from the tool reliability function, which models the wear-out by a mixture of Weibull distributions, while failures due to external stresses are accounted for by a homogeneous Poisson process. The optimal replacement time is then determined from a total time on test (TTT) plot. The adequacy of the proposed approach for practical application is tested and confirmed in a case study on turning of Inconel 718 with cubic boron nitride (CBN) tools.     

Identifiability issues in dynamic stress–strength modeling

In many real-life scenarios, system reliability depends on dynamic stress–strength interference, where strength degrades and stress accumulates concurrently over time. In some other cases, shocks appear at random time points, causing damage which is only effective at the instant of shock arrival. In this paper, we consider the identifiability problem of a system under deterministic strength degradation and stochastic damage due to shocks arriving according to a homogeneous Poisson process. We provide conditions under which the models are identifiable with respect to lifetime data only. We also consider current status data and suggest to collect additional information and discuss the issues of model identifiability under different data configurations.     

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.     

The optimal replacement for additive damage models in discrete setting

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A replacement model for an additive damage model with restoration

A system existing in a random environment receives shocks at random points of time. Each shock causes a random amount of damage which accumulates over time. A breakdown can occur only upon the occurrence of a shock according to a known failure probability function. Upon failure the system is replaced by a new identical one with a given cost. When the system is replaced before failure, a smaller cost is incurred. Thus, there is an incentive to attempt to replace the system before failure. The damage process is controlled by means of a maintenance policy which causes the accumulated damage to decrease at a known restoration rate. We introduce sufficient conditions under which an optimal replacement policy which minimizes the total expected discounted cost is a control limit policy. The relationship between the undiscounted case and the discounted case is examined. Finally, an example is given illustrating computational procedures.     

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.     

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.     

Optimal replacement for discrete additive damage models: Algorithm

A system receives shocks at successive random points of discrete time, and each shock causes a positive integer-valued random amount of damage which accumulates on the system one after another. The system is subject to failure and it fails once the total cumulative damage level first exceeds a fixed threshold. Upon failure the system must be replaced by a new and identical one and a cost is incurred. If the system is replaced before failure, a smaller cost is incurred. In previous work, under some assumptions, we specified a replacement rule which minimizes the long-run (expected) average cost per unit time and possesses control limit property. In this paper, a general algorithm for such models is developed. This research has been jointly supported by ITDC, contract No.105-82150 and the National Natural Science Foundation of China.     

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.     

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.     

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.     

A mathematical analysis of tension shocks in long belt conveyors

This paper presents a mathematical study of the development and propagation of tension shock waves in the topside of long belt conveyor systems. Employing conservation laws for mass and momentum we construct a system of nonlinear hyperbolic equations governing the behavior of tension and momentum per unit length. An important feature of this construction is the nonlinear relationship between tension and overdensity due to the geometric sag of the belt between the idlers. Numerical profiles of the tension are computed using the moving finite element method which is especially suited for the numerical study of shocks. With our model we believe we have identified two of the sources of the large-scale tension shocks which sometimes severely damage such long belt conveyor systems.     

Survival of systems with protection subject to two types of external attacks

We consider a large system that, due to its importance and (or) large economic value, should be protected from possible harmful attacks. The attacker executes two types of attacks modeled by the corresponding shock processes: those that target the repairable defense system and those that target the non-repairable ‘main’ system itself. The quality of the performance of the defense system, i.e., the ability to neutralize the attacks on the main system depends on its state defined by the accumulated damage. In order to obtain the survival probabilities for the main system, we develop a new class of state-dependent extreme shock models, where the accumulated damage caused by the shock process of one type affects the probability of failure from shocks of the other type.     

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.     

A Random Shock Model with Mixed Effect,Including Competing Soft and Sudden Failures,and Dependence

A system is considered, which is subject to external and possibly fatal shocks, with dependence between the fatality of a shock and the system age. Apart from these shocks, the system suffers from competing soft and sudden failures, where soft failures refer to the reaching of a given threshold for the degradation level, and sudden failures to accidental failures, characterized by a failure rate. A non-fatal shock increases both degradation level and failure rate of a random amount, with possible dependence between the two increments. The system reliability is calculated by four different methods. Conditions under which the system lifetime is New Better than Used are proposed. The influence of various parameters of the shocks environment on the system lifetime is studied.     

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.     

Transonic shocks for steady Euler flows with cylindrical symmetry

In this paper, we prove the existence of transonic shocks adjacent to a uniform one for the full Euler system for steady compressible fluids with cylindrical symmetry in a cylinder, and consequently show the stability of such uniform transonic shocks. Mathematically we solve a free boundary problem for a quasi-linear elliptic–hyperbolic composite system. This reveals that the boundary conditions and equations interact in a subtle way. The key point is to “separate” in a suitable way the elliptic and hyperbolic parts of the system. The approach developed here can be applied to deal with certain multidimensional problems concerning stability of transonic shocks for the full Euler system.     

A note on replacement policy for a system subject to non-homogeneous pure birth shocks

A system is subject to shocks that arrive according to a non-homogeneous pure birth process. As shocks occur, the system has two types of failures. Type-I failure (minor failure) is removed by a general repair, whereas type-II failure (catastrophic failure) is removed by an unplanned replacement. The occurrence of the failure type is based on some random mechanism which depends on the number of shocks occurred since the last replacement. Under an age replacement policy, a planned (or scheduled) replacement happens whenever an operating system reaches age T. The aim of this note is to derive the expected cost functions and characterize the structure of the optimal replacement policy for such a general setting. We show that many previous models are special cases of our general model. A numerical example is presented to show the application of the algorithm and several useful insights.     

Maintenance of a deteriorating single server system with Markovian arrivals and random shocks

We consider a single server queue in which the arrivals occur according to a Markovian arrival process. The system is subject to external shocks causing the server to deteriorate and possibly fail. The maintenance of the server is provided either as a preventive one or for a complete failure so as to bring back to normal. Under the assumptions of Poisson shocks, exponential services and exponential maintenance with rates depending on the state of the server, and a general (discrete) probability distribution for the intensity of the shocks, the model is analyzed in steady-state. Some interesting theoretical results along with a few illustrative numerical examples are reported. An optimization problem involving various costs is studied numerically.