Shock model in Markovian environment

A Markov modulated shock models is studied in this paper. In this model, both the interarrival time and the magnitude of the shock are determined by a Markov process. The system fails whenever a shock magnitude exceeds a pre‐specified level η. Nonexponential bounds of the reliability are given when the interarrival time has heavy‐tailed distribution. The exponential decay of the reliability function and the asymptotic failure rate are also considered for the light‐tailed case. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005     

Optimal replacement under additive damage and other failure models

A machine or production system is subject to random failure. Upon failure the system is replaced by a new one, and the process repeats. A cost is associated with each replacement, and an additional cost is incurred at each failure in service. Thus, there is an incentive for a controller to attempt to replace before failure occurs. The problem is to find an optimal control strategy that balances the cost of replacement with the cost of failure and results in a minimum total long-run average cost per unit time. We attack this problem under the cumulative damage model for system failure. In this failure model, shocks occur to the system in accordance with a Poisson process. Each shock causes a random amount of damage or wear and these damages accumulate additively. At any given shock, the system fails with a known probability that depends on the total damage accumulated to date. We assume that the cumulative damage is observable by the controller and that his decisions may be based on its current value. Supposing that the shock failure probability is an increasing function of the cumulative damage, we show that an optimal policy is to replace either upon failure or when this damage first exceeds a critical control level, and we give an equation which implicitly defines the optimal control level in terms of the cost and other system parameters. Also treated are some more general models that allow for income lost during repair time and other extensions.     

Preventive maintenance and replacement under additive damage

A system deteriorates due to shocks received at random times, each shock causing a random amount of damage which accumulates over time and may result in a system failure. Replacement of a failed system is mandatory, while an operable one may also be replaced. In addition, the shock process causing system deterioration may be controlled by continuous preventive maintenance expenditures. The joint problem of optimal maintenance and replacement is analyzed and it is shown that, under reasonable conditions, optimal maintenance rate is decreasing in the cumulative damage level and that beyond a certain critical level the system should be replaced. Meaningful bounds are established on the optimal policies and an illustrative example is provided.     

On failure modeling

A promising approach to failure modeling, in particular to developing failure-time distributions, is discussed. Under this approach, system state or wear and tear is modeled by an appropriately chosen random process—for example, a diffusion process—and the occurrences of fatal shocks are modeled by a Poisson process whose rate function is state dependent. The system is said to fail when either wear and tear accumulates beyond an acceptable or safe level or a fatal shock occurs. This approach has significant merit. First, it provides revealing new insights into most of the famous and frequently used lifetime distributions in reliability theory. Moreover, it suggests intuitively appealing ways for enhancing those standard models. Indeed, this approach provides a means of representing the underlying dynamics inherent in failure processes. Reasonable postulates for the dynamics of failure should lend credence to the prediction and estimation of reliability, maintainability, and availability. In other words, accuracy of representation could lead to better, more reliable prediction of failure.     

Shock models with phase type survival and shock resistance

New closure theorems for shock models in reliability theory are presented. If the number of shocks to failure and the times between the arrivals of shocks have probability distributions of phase type, then so has the time to failure. PH-distributions are highly versatile and may be used to model many qualitative features of practical interest. They are also well-suited for algorithmic implementation. The computational aspects of our results are discussed in some detail.     

Optimal maintenance policy for stochastically failing equipment: A diffusion approximation

A system receives shocks at random points of time. Each shock causes a random amount of damage which accumulates over time. The system fails when the accumulated damage exceeds a fixed threshold. Upon failure the system is replaced by a new one. The damage process is controlled by means of a maintenance policy. There are M possible maintenance actions. Given that a maintenance action m is employed, then the cumulative damage decreases at rate r^m. Replacement costs and maintenance costs are considered. The objective is to determine an optimal maintenance policy under the following optimality criteria: (1) long-run average cost; (2) total expected discounted cost over an infinite horizon. For a diffusion approximation, we show that the optimal maintenance expenditure rate is monotonically increasing in the cumulative damage level.     

考虑载荷作用次数的多失效模式扭力轴灵敏度分析

机械构件的不同的失效模式之间具有一定的相关性,而且随机载荷作用次数对机械构件的可靠性有一定程度的影响。因此对机械构件进行可靠性灵敏度分析时,需要充分考虑其不同失效模式和载荷作用次数的影响。通过运用顺序统计量理论考虑载荷多次作用以及多种失效模式条件下机械构件可靠性及可靠性灵敏度的变化规律,运用随机摄动理论和四阶矩技术,建立一种考虑载荷作用次数的多失效模式机械构件可靠性灵敏度分析数值方法的应力强度干涉模型。在随机变量前四阶矩已知的情况下,结合灵敏度分析的梯度算法,推导出关于随机变量均值和方差的灵敏度计算公式。以某履带车辆底盘扭力轴为例进行计算,得到其可靠度随载荷作用次数、随机变量均值和方差而改变的可靠性灵敏度变化曲线,为扭力轴的可靠性优化提供一定的理论依据。研究成果可以推广到相关机械可靠性灵敏度设计和结构优化领域,具有非常重要的实用意义。     

Model parameter estimation and residual life prediction for a partially observable failing system

We consider a partially observable degrading system subject to condition monitoring and random failure. The system's condition is categorized into one of three states: a healthy state, a warning state, and a failure state. Only the failure state is observable. While the system is operational, vector data that is stochastically related to the system state is obtained through condition monitoring at regular sampling epochs. The state process evolution follows a hidden semi‐Markov model (HSMM) and Erlang distribution is used for modeling the system's sojourn time in each of its operational states. The Expectation‐maximization (EM) algorithm is applied to estimate the state and observation parameters of the HSMM. Explicit formulas for several important quantities for the system residual life estimation such as the conditional reliability function and the mean residual life are derived in terms of the posterior probability that the system is in the warning state. Numerical examples are presented to demonstrate the applicability of the estimation procedure and failure prediction method. A comparison results with hidden Markov modeling are provided to illustrate the effectiveness of the proposed model. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 190–205, 2015     

Optimal replacement under additive damage in randomly varying environments

A system is subject to a sequence of randomly occurring shocks. Each shock causes a random amount of damage which accumulates additively. Any of the shocks might cause the system to fail. The shock process is in some sense related to an environmental process in order to describe randomly varying external factors of an economical and/or technical nature as well as internal factors of a statistical nature. A discrete time formulation of the problem is given. Sufficient conditions are found for optimality of a generalized control-limit rule with respect to the total cost criterion: Whenever the accumulated damage s is not less than a specified critical number t(i), depending on the environmental state i, replace the system by a new one; otherwise do not replace it. Moreover, bounds are given for these critical numbers.     

Replacement models under additive damage

A production system which generates income is subject to random failure. Upon failure, the system is replaced by a new identical one and the replacement cycles are repeated indefinitely. In our breakdown model, shocks occur to the system in a Poisson stream. Each shock causes a random amount of damage, and these damages accumulate additively. The failure time depends on the accumulated damage in the system. The income from the system and the cost associated with a planned replacement depend on the accumulated damage in the system. An additional cost is incurred at each failure in service. We allow a controller to replace the system at any stopping time T before failure time. We will consider the problem of specifying a replacement rule that is optimal under the following criteria: maximum total long-run average net income per unit time, and maximum total long-run expected discounted net income. Our primary goal is to introduce conditions under which an optimal policy is a control limit policy and to investigate how the optimal policy can be obtained. Examples will be presented to illustrate computational procedures.     

Optimal redesign decisions through failure rate estimates

To reduce the time-to-market of newly developed systems, manufacturers increasingly adopt strategies where systems are brought to market while system field reliability is still uncertain. These systems are typically sold under performance-based contracts, which incentivizes potential customers to invest in them despite reliability uncertainty. Such contracts make the manufacturer (partly) responsible for the availability of the system. Subsequently, when field reliability is lower than anticipated, the manufacturer may choose to redesign the system to avoid high contract penalties. Redesign is a costly effort which may substantially increase field reliability. Deciding when to redesign is challenging, especially because the initial failure rate estimate by the system's engineers is refined over time as failure data accrues. We propose a model that endogenizes the failure rate updating to analyze this tactical redesign decision. We study additive and multiplicative redesigns and show that the optimal policy has a control limit structure. We benchmark our optimal policy against a static counterpart numerically, and conclude that basing redesign decisions on the updated estimate of the failure rate can substantially reduce costs.     

屏蔽数据在新型截尾样本下系统的可靠性分析

当系统含有屏蔽数据时,在具有随机移走逐步增加型截尾模型下,讨论了部件寿命服从双参数指数分布的串联系统可靠性估计问题。设随机移走系统数服从二项分布,利用极大似然方法,Bayes理论及方法,推导出双参数指数部件参数、系统可靠性函数、失效率函数及移走概率的极大似然估计和Bayes估计。并利用Monte Carlo方法对两种估计结果进行了比较,表明Bayes估计较极大似然估计效果更优。     

Linear programming and the reliability of multicomponent systems

Several problems in the assignment of parallel redundant components to systems composed of elements subject to failure are considered. In each case the problem is to make an assignment which maximizes the system reliability subject to system constraints. Three distinct problems; are treated. The first is the classical problem of maximizing system reliability under total cost or weight constraints when components are subject to a single type of failure. The second problem deals with components which are subject to two types of failure and minimizes the probability of one mode of system failure subject to a constraint on the probability of the other mode of system failure. The third problem deals with components which may either fail to operate or may operate prematurely. System reliability is maximized subject to a constraint ori system safety. In each case the problem is formulated as an integer linear program. This has an advantage over alternative dynamic programming formulations in that standard algorithms may be employed to obtain numerical results.     

Transient analysis of Markov models of fault‐tolerant systems with deferred repair using split regenerative randomization

The (standard) randomization method is an attractive alternative for the transient analysis of continuous time Markov models. The main advantages of the method are numerical stability, well‐controlled computation error, and ability to specify the computation error in advance. However, the fact that the method can be computationally very expensive limits its applicability. In this paper, we develop a new method called split regenerative randomization, which, having the same good properties as standard randomization, can be significantly more efficient. The method covers reliability‐like models with a particular but quite general structure and requires the selection of a subset of states and a regenerative state satisfying some conditions. For a class of continuous time Markov models, model class C₂, including typical failure/repair reliability‐like models with exponential failure and repair time distributions and deferred repair, natural selections are available for both the subset of states and the regenerative state and, for those natural selections, theoretical results are available assessing the efficiency of the method in terms of “visible” model characteristics. Those results can be used to anticipate when the method can be expected to be competitive. We illustrate the application of the method using a large class C₂ model and show that for models in that class the method can indeed be significantly more efficient than previously available randomization‐based methods. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006     

Reliability of a two-dimensional demand-based networked system with multistate components

Existing research on multistate system reliability has mainly focused on one-dimensional systems such as parallel systems, linear sliding window systems, and linearly consecutively connected systems. However, two-dimensional networked systems widely exist in real-world applications such as lighting systems, monitoring systems, and computer network systems. This research considers a two-dimensional networked system consisting of multistate components. The system fails if the cumulative performance of any row or any column cannot meet a predetermined demand. A novel reliability evaluation algorithm is proposed for the considered two-dimensional networked system by extending the universal generating function technique. Furthermore, the proposed model and reliability evaluation algorithm are extended to a two-dimensional networked system with phased missions. The proposed models and algorithms are illustrated by a matrix heating system in a thermoforming machine.     

稳态随机过程激励下基于首次超越破坏的结构优化设计

把结构系统动力可靠性分析与最优化设计结合起来 ,以结构系统的最小质量为目标函数 ,给出了考虑在平稳随机过程激励下多自由度线性系统总的可靠性的结构优化设计方法。运用谱分析理论 ,推导了结构系统在平稳随机过程激励下响应的统计特征 ,同时结合首次超越破坏的Possion模型计算结构系统的可靠性 ,最终采用广义乘子法得到结构系统设计变量的最优值。计算结果表明该方法是可行的     

Sequential determination of inspection epochs for reliability systems with general lifetime distributions

The problem of determining the optimal inspection epoch is studied for reliability systems in which N components operate in parallel. Lifetime distribution is arbitrary, but known. The optimization is carried with respect to two cost factors: the cost of inspecting a component and the cost of failure. The inspection epochs are determined so that the expected cost of the whole system per time unit per cycle will be minimized. The optimization process depends in the general case on the whole failure history of the system. This dependence is characterized. The cases of Weibull lifetime distributions are elaborated and illustrated numerically. The characteristics of the optimal inspection intervals are studied theoretically.     

可修系统的一个模型

本文研究一种具有4种故障形式的可修系统。利用向量Markov过程方法,求出了系统的可靠性指标。     

A general model of heterogeneous system lifetimes and conditions for system burn‐in

Burn‐in is a technique to enhance reliability by eliminating weak items from a population of items having heterogeneous lifetimes. System burn‐in can improve system reliability, but the conditions for system burn‐in to be performed after component burn‐in remain a little understood mathematical challenge. To derive such conditions, we first introduce a general model of heterogeneous system lifetimes, in which the component burn‐in information and assembly problems are related to the prediction of system burn‐in. Many existing system burn‐in models become special cases and two important results are identified. First, heterogeneous system lifetimes can be understood naturally as a consequence of heterogeneous component lifetimes and heterogeneous assembly quality. Second, system burn‐in is effective if assembly quality variation in the components and connections which are arranged in series is greater than a threshold, where the threshold depends on the system structure and component failure rates. © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 364–380, 2003.     

Probabilistic models of decreasing failure rate processes

The “infant mortality” effect observed in the statistical treatment of reliability consists of a decreasing with age of the conditional probability of equipment failure (failure rate). One widely applicable explanatory hypothesis is that of population heterogeneity. This is developed here as a basis for several specific models of decreasing failure rate processes. Since, in the case of repairable devices, decreasing failure rate is often observed after the occurrence of failure and repair, consideration is extended to include repair in an explicit way. This union of failure and repair models is a fruitful one in view of the interaction between the two processes and gives a complete picture of the life of the device in terms of a stochastic process, usually with non-independent interfailure times. Four models, of particular significance due to their plausibility, mathematical tractability, and frugality of parameterization, are presented.