Approximating the Randomized Hitting Time Distribution of a Non-stationary Gamma Process

The non-stationary gamma process is a non-decreasing stochastic process with independent increments. By this monotonic behavior this stochastic process serves as a natural candidate for modelling time-dependent phenomena such as degradation. In condition-based maintenance the first time such a process exceeds a random threshold is used as a model for the lifetime of a device or for the random time between two successive imperfect maintenance actions. Therefore there is a need to investigate in detail the cumulative distribution function (cdf) of this so-called randomized hitting time. We first relate the cdf of the (randomized) hitting time of a non-stationary gamma process to the cdf of a related hitting time of a stationary gamma process. Even for a stationary gamma process this cdf has in general no elementary formula and its evaluation is time-consuming. Hence two approximations are proposed in this paper and both have a clear probabilistic interpretation. Numerical experiments show that these approximations are easy to evaluate and their accuracy depends on the scale parameter of the non-stationary gamma process. Finally, we also consider some special cases of randomized hitting times for which it is possible to give an elementary formula for its cdf.Approximation;Condition based maintenance;First hitting time;Non-stationary gamma process;Random threshold

Optimal maintenance of multi-component systems: a review

In this article we give an overview of the literature on multi-component maintenance optimization. We focus on work appearing since the 1991 survey "A survey of maintenance models for multi-unit systems" by Cho and Parlar. This paper builds forth on the review article by Dekker et al. (1996), which focusses on economic dependence, and the survey of maintenance policies by Wang (2002), in which some group maintenance and some opportunistic maintenance policies are considered. Our classification scheme is primarily based on the dependence between components (stochastic, structural or economic). Next, we also classify the papers on the basis of the planning aspect (short-term vs long-term), the grouping of maintenance activities (either grouping preventive or corrective maintenance, or opportunistic grouping) and the optimization approach used (heuristic, policy classes or exact algorithms). Finally, we pay attention to the applications of the models.literature review;economic dependence;failure interaction;maintenance policies;grouping maintenance;multi-component systems;opportunistic maintenance;maintencance optimization;structural dependence

Approximating the randomized hitting time distribution of a non-stationary gamma process

The non-stationary gamma process is a non-decreasing stochasticprocess with independent increments. By this monotonic behavior thisstochastic process serves as a natural candidate for modellingtime-dependent phenomena such as degradation. In condition-basedmaintenance the first time such a process exceeds a random thresholdis used as a model for the lifetime of a device or for the randomtime between two successive imperfect maintenance actions. Thereforethere is a need to investigate in detail the cumulative distributionfunction (cdf) of this so-called randomized hitting time. We firstrelate the cdf of the (randomized) hitting time of a non-stationarygamma process to the cdf of a related hitting time of a stationarygamma process. Even for a stationary gamma process this cdf has ingeneral no elementary formula and its evaluation is time-consuming.Hence two approximations are proposed in this paper and both have aclear probabilistic interpretation. Numerical experiments show thatthese approximations are easy to evaluate and their accuracy dependson the scale parameter of the non-stationary gamma process. Finally,we also consider some special cases of randomized hitting times forwhich it is possible to give an elementary formula for its cdf.approximation;condition based maintencance;first hitting time;non-stationary gamma process;random threshold

A general framework for statistical inference on discrete event systems.

We present a framework for statistical analysis of discrete event systems which combines tools such as simulation of marked point processes, likelihood methods, kernel density estimation and stochastic approximation to enable statistical analysis of the discrete event system, even if conventional approaches fail due to the mathematical intractability of the model.The approach is illustrated with an application to modelling and estimating corrosion of steel gates in the Dutch Haringvliet storm surge barrier.discrete event systems;kernel density estimation;optimization via simulation;parameter estimation;stochastic approximation;likelihood methods;market point process

Modelling and Optimizing Imperfect Maintenance of Coatings on Steel Structures

Steel structures such as bridges, tanks and pylons are exposed to outdoor weathering conditions. In order to prevent them from corrosion they are protected by an organic coating system. Unfortunately, the coating system itself is also subject to deterioration. Imperfect maintenance actions such as spot repair and repainting can be done to extend the lifetime of the coating. In this paper we consider the problem of finding the set of actions that minimizes the expected maintenance costs over a bounded horizon. To this end we model the size of the area affected by corrosion by a non-stationary gamma process. An imperfect maintenance action is to be done as soon as a fixed threshold is exceeded. The direct effect of such an action on the condition of the coating is assumed to be random. On the other hand, maintenance may also change the parameters of the gamma deterioration process. It is shown that the optimal maintenance decisions related to this problem are a solution of a continuous-time renewal-type dynamic programming equation. To solve this equation time is discretized and it is verified theoretically that this discretization induces only a small error. Finally, the model is illustrated with a numerical example.non-stationary gamma process;condition-based maintenance;degradation modelling;imperfect maintenance;life-cycle management;renewal-type dynamic programming equation

Maintenance Models for Systems subject to Measurable Deterioration

Complex engineering systems such as bridges, roads, flood defence structures, and power pylons play an important role in our society. Unfortunately such systems are subject to deterioration, meaning that in course of time their condition falls from higher to lower, and possibly even to unacceptable, levels. Maintenance actions such as inspection, local repair and replacement should be done to retain such systems in or restore them to acceptable operating conditions. After all, the economic consequences of malfunctioning infrastructure systems can be huge. In the life-cycle management of engineering systems, the decisions regarding the timing and the type of maintenance depend on the temporal uncertainty associated with the deterioration. Hence it is of importance to model this uncertainty. In the literature, deterioration models based on Brownian motion and gamma process have had much attention, but a thorough comparison of these models lacks. In this thesis both models are compared on several aspects, both in a theoretical as well as in an empirical setting. Moreover, they are compared with physical process models, which can capture structural insights into the underlying process. For the latter a new framework is developed to draw inference. Next, models for imperfect maintenance are investigated. Finally, a review is given for systems consisting of multiple components

Automated Response Surface Methodology for Stochastic Optimization Models with Unknown Variance

Response Surface Methodology (RSM) is a tool that was introduced in the early 50´s by Box and Wilson (1951). It is a collection of mathematical and statistical techniques useful for the approximation and optimization of stochastic models. Applications of RSM can be found in e.g. chemical, engineering and clinical sciences. In this paper we are interested in finding the best settings for an automated RSM procedure when there is very little information about the stochastic objective function. We will present a framework of the RSM procedures for finding optimal solutions in the presence of noise. We emphasize the use of both stopping rules and restart procedures. Good stopping rules recognize when no further improvement is being made. Restarts are used to escape from non-optimal regions of the domain. We compare different versions of the RSM algorithms on a number of test functions, including a simulation model for cancer screening. The results show that co! nsiderable improvement is possible over the proposed settings in the existing literature.response surface methodology;simulation optimization

Automated Response Surface Methodology for Stochastic Optimization Models with Unknown Variance

Response Surface Methodology (RSM) is a tool that was introduced in the early 50´s by Box and Wilson (1951). It is a collection of mathematical and statistical techniques useful for the approximation and optimization of stochastic models. Applications of RSM can be found in e.g. chemical, engineering and clinical sciences. In this paper we are interested in finding the best settings for an automated RSM procedure when there is very little information about the stochastic objective function. We will present a framework of the RSM procedures for finding optimal solutions in the presence of noise. We emphasize the use of both stopping rules and restart procedures. Good stopping rules recognize when no further improvement is being made. Restarts are used to escape from non-optimal regions of the domain. We compare different versions of the RSM algorithms on a number of test functions, including a simulation model for cancer screening. The results show that co! nsiderable improvement is possible over the proposed settings in the existing literature

A general framework for statistical inference on discrete event systems.

We present a framework for statistical analysis of discrete event systems which combines tools such as simulation of marked point processes, likelihood methods, kernel density estimation and stochastic approximation to enable statistical analysis of the discrete event system, even if conventional approaches fail due to the mathematical intractability of the model. The approach is illustrated with an application to modelling and estimating corrosion of steel gates in the Dutch Haringvliet storm surge barrier

Optimal maintenance of multi-component systems: a review

In this article we give an overview of the literature on multi-component maintenance optimization. We focus on work appearing since the 1991 survey "A survey of maintenance models for multi-unit systems" by Cho and Parlar. This paper builds forth on the review article by Dekker et al. (1996), which focusses on economic dependence, and the survey of maintenance policies by Wang (2002), in which some group maintenance and some opportunistic maintenance policies are considered. Our classification scheme is primarily based on the dependence between components (stochastic, structural or economic). Next, we also classify the papers on the basis of the planning aspect (short-term vs long-term), the grouping of maintenance activities (either grouping preventive or corrective maintenance, or opportunistic grouping) and the optimization approach used (heuristic, policy classes or exact algorithms). Finally, we pay attention to the applications of the models