Renewal theory

About: Renewal theory is a research topic. Over the lifetime, 2381 publications have been published within this topic receiving 54908 citations.

Repairable Systems Reliability

Abstract: Nonrepairable systems fail just once, so models for nonrepairable systems must account for random lifetimes. It is often reasonable to assume that different units have random lifetimes that are independent and follow the same distribution, leading to the usual i.i.d. (independent and identically distributed) assumption. By contrast, models for repairable systems must account for successive failures in time. For a given system, these times between failure are often not independent and not identically distributed. Various assumptions about the failure process lead to different models for repairable systems. In this article, we present the nonhomogeneous Poisson process (NHPP), the renewal process, the piecewise exponential model, and the modulated power law process (a compromise between renewal process and NHPP). Inference for a single system as well as for multiple copies of a system is discussed. We also discuss briefly the use of covariates or concomitant variables. Keywords: Poisson process; power law process; renewal process; piecewise exponential; multiple systems

Conditional, Non-Homogeneous and Doubly Stochastic Compound Poisson Processes with Stochastic Discounted Claims

Abstract: In this paper, we study the conditional, non-homogeneous and doubly stochastic compound Poisson process with stochastic discounted claims. We derive the moment generating functions of these risk processes and find their inverses, numerically or analytically, by using their corresponding characteristic functions. We then compare their distributions and some risk measures as the VaR and TVaR, and we examine the case where there is a possible dependence between the occurrence time and the severity of the claim.

Stochastic Approach to Planning of Spares for Complex Deteriorating Industrial System

Abstract: An easily implementable stochastic model for planning the inventory of spare components needed for corrective replacements of system components is presented. The probability of spare shortage in a given time interval is chosen as the decision criterion. The model is based on the assumption that the system contains a great number of identical independent components subject to wear-out. It can be used to determine the minimal number of spare components needed at the beginning of the planning interval to fulfil the requirement for an acceptable shortage probability during this interval. Besides, the model enables calculation of the probability of spare shortage during a given time interval considering the existing inventory level at the beginning of this interval. The model contains a few constant parameters that can be estimated from the component field data. It also includes two time dependent parameters which are calculated using the renewal function from the renewal theory. A discrete approximati...

A hidden renewal model for monitoring aquatic systems biosensors

Abstract: This article proposes a method to model signals of oysters' openings over time using a four-state renewal process. Two of them are of particular interest and correspond to instants when the animals are open or closed. An estimator of the cumulative jump rate of the renewal process is provided. It relies on observations of the jumps between the four states. Here these measures are not available but the observed signal is assumed to take ranges of real values according to this underlying process. A procedure to estimate a probability density function that summarizes the information of the signal is explained. This leads to estimation of the hidden renewal process and of its cumulative jump rate for each oyster. We propose to classify these estimated functions for a group of oysters in order to discriminate these animals according to their health status. Such a diagnosis is essential when using these animals as biosensors for water quality assessment.