Maxim Finkelstein

On the reversed hazard rate

Abstract The reversed hazard rate defined as the ratio of the density to the distribution function had attracted the attention of researchers only relatively recently. Being in a certain sense a dual function to an ordinary hazard rate, it still bears some interesting features useful in reliability analysis. One of its most important properties is the connection with the mean waiting time studied in this paper. The application to ordering of random variables via the proportional reversed hazard rate model is also considered. Possible applications are discussed.

An optimal age-based group maintenance policy for multi-unit degrading systems

In order to share maintenance set-up costs and reduce system breakdown, the group maintenance policies are widely used for complex multi-unit systems. In this paper, an optimal age-based group maintenance policy is proposed for a multi-unit series system whose components are subject to different gradual degradation phenomena. When the degradation level of a component reaches a given critical size, it is replaced by a new one and the other components undergo a preventive maintenance (PM) action; otherwise, a planned group PM is performed for the whole system at operational age T>0. The problem is to determine an optimal group maintenance time T⁎ such that the system’s average long-run maintenance cost per unit time is minimized. The explicit expression of the objective function is derived and sufficient conditions for existence and uniqueness of the optimal solution are obtained. Finally, the proposed maintenance policy is applied to a group of wind turbine bearings and the results are compared with the case without planned maintenance (i.e., reactive response) and with an individual age-based maintenance policy.

An opportunistic condition-based maintenance policy for offshore wind turbine blades subjected to degradation and environmental shocks

Offshore wind turbine blades are subject to multiple types of internal and external damages. Internal damages (such as fatigue, wear and cracks) are generally caused by system degradation, whereas the external damages (such as icing, wind and wave shocks) result from harsh marine environments. In this paper, we investigate an optimal opportunistic condition-based maintenance (OCBM) policy for a multi-bladed offshore wind turbine system subjected to stress corrosion cracking (SCC) and environmental shocks. When the length of a crack in a blade reaches critical size D, the blade breaks and it has to be replaced by a new one. An environmental shock is minor with probability 1−p and catastrophic with probability p (0≤p≤1). A minor shock causes an instant drop in power output without resulting in any system failure, whereas a catastrophic shock stops the wind turbine and requires system replacement. When the length of a crack in one of the blades exceeds a threshold d ( 0). The problem is to simultaneously determine the optimal control parameters d⁎ and T⁎ such that the average long-run maintenance cost per blade per unit time is minimized. The explicit expression of the objective function is derived and under certain conditions, the existence and uniqueness of the optimal solution are shown for the infinite-horizon case. The proposed maintenance model is tested on a three-bladed wind turbine and its performance over the system life cycle is evaluated using a Monte-Carlo simulation technique.

On optimal upgrade level for used products under given cost structures

Abstract In spite of the growing share of the second-hand market, often customers of used products encounter the following three problems: (a) they are uncertain regarding the durability and performance of these products due to lack of information on the item’s past usage and maintenance history, (b) they are uncertain about the accurate pricing of warranties and the post-warranty repair costs, and (c) sometimes, right after the sale, used items may have high failure rate and could be harmful to their new owner. Due to these problems, the dealers are currently carrying out actions such as overhaul and upgrade of the used products before their release. Reliability improvement, which is closely related to the concept of warranty, for used products is a relatively new concept and has received very limited attention. This paper also develops a stochastic model which results in the derivation of the optimal expected upgrade level under given structures of the profit and failure rate functions. We provide a numerical study to illustrate our results.

On statistical and information-based virtual age of degrading systems

Abstract Two approaches to defining a virtual age of a degrading system are considered. The first one is based on the fact that deterioration depends on the environment. In a more severe environment deterioration is more intensive, which means that objects are aging faster and therefore, the corresponding virtual age is larger than the calendar age in a baseline environment. The second approach is based on considering an observed level of individual degradation and comparing it with some average, ‘population degradation’. The virtual age of the series system is defined. Several illustrative examples are considered.

Asymptotic behavior of a general class of mixture failure rates

Mixtures of increasing failure rate distributions can decrease, at least in some intervals of time. Usually this property is observed asymptotically, as t → ∞, which is due to the fact that a mixture failure rate is ‘bent down’, as the weakest populations are dying out first. We consider a survival model that generalizes additive hazards models, proportional hazards models, and accelerated life models very well known in reliability and survival analysis. We obtain new explicit asymptotic relations for a general setting and study specific cases. Under reasonable assumptions we prove that the asymptotic behavior of the mixture failure rate depends only on the behavior of the mixing distribution in the neighborhood of the left-hand endpoint of its support, and not on the whole mixing distribution.

Admissible mixing distributions for a general class of mixture survival models with known asymptotics

Statistical analysis of data on the longest living humans leaves room for speculation whether the human force of mortality is actually leveling off. Based on this uncertainty, we study a mixture failure model, introduced by Finkelstein and Esaulova (2006) that generalizes, among others, the proportional hazards and accelerated failure time models. In this paper we first, extend the Abelian theorem of these authors to mixing distributions, whose densities are functions of regular variation. In addition, taking into account the asymptotic behavior of the mixture hazard rate prescribed by this Abelian theorem, we prove three Tauberian-type theorems that describe the class of admissible mixing distributions. We illustrate our findings with examples of popular mixing distributions that are used to model unobserved heterogeneity.

Burn-in by environmental shocks for two ordered subpopulations

Burn-in is a widely used engineering method of elimination of defective items before they are shipped to customers or put into field operation. In conventional burn-in procedures, components or systems are subject to a period of simulated operation prior to actual usage. Then those which failed during this period are scrapped and discarded. In this paper, we assume that the population of items is composed of two ordered subpopulations and the elimination of weak items by using environmental shocks is considered. Optimal severity levels of these shocks that minimize the defined expected costs are investigated. Some illustrative examples are discussed.

Shocks in homogeneous and heterogeneous populations

A system subject to a point process of shocks is considered. Shocks occur in accordance with a nonhomogeneous Poisson process. Different criterions of system failures are discussed in a homogeneous case. Two natural settings are analyzed. Heterogeneity is modeled by an unobserved univariate random variable (frailty). It is shown that reliability (safety) analysis for a heterogeneous case can differ dramatically from that for a homogeneous setting. A shock burn-in procedure for a heterogeneous population is described. The corresponding bounds for the failure rates are obtained.

On terminating Poisson processes in some shock models

A system subject to a point process of shocks is considered. Shocks occur in accordance with the homogeneous Poisson process. Different criteria of system failure (termination) are discussed and the corresponding probabilities of failure (accident)-free performance are derived. The described analytical approach is based on deriving integral equations for each setting and solving these equations through the Laplace transform. Some approximations are analyzed and further generalizations and applications are discussed.