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Renewal theory
About: Renewal theory is a research topic. Over the lifetime, 2381 publications have been published within this topic receiving 54908 citations.
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22 Mar 2017
TL;DR: In this article, a single unit repairable model with working and repair time omission under an alternative renewal process was studied, where the working time is shorter than threshold τ 1 and the repair time is omitted.
Abstract: In this article, we study a single-unit repairable model with working and repair time omission under an alternative renewal process. As the working time is shorter than threshold τ1, we regard some...
9 citations
01 Apr 1965
TL;DR: In this paper, it was shown that the product of two renewal densities is not a renewal density function, and that in some cases it is true, while in others it is not always true.
Abstract: Renewal processes in discrete time (or as they are commonly termed, recurrent events) are appropriately described by renewal sequences { u n } which are generated by discrete distributions , according to the equation Any two renewal sequences { u ′ n }, { u ″ n } define another renewal sequence { u n } by means of their term-by-term product { u n } = { u ′ n u ″ n }, for the joint occurrence of two independent recurrent events ℰ′ and ℰ″ is also a recurrent event. Considering a renewal process in continuous time for which we shall suppose a frequency function f ( x ) of the lifetime distribution exists, so that a renewal density exists, the analogous property would be that for two renewal density functions h 1 ( x ) and h 2 ( x ), the function h ( x ) = h 1 ( x ) h 2 ( x ) is a renewal density function. A little intuitive reflexion shows that while h ( x ) dx has a probability density interpretation, this is not in general true of h 1 ( x ) h 2 ( x ) dx . It is not surprising therefore to find in example 1 a case where the product of two renewal densities is not a renewal density. Example 2, on the other hand, shows that in some cases it is true, and taken together with example 1, there is suggested the problem of characterizing the class of renewal densities h ( x ) for which α h ( x ) is a renewal density for all finite positive α and not merely α in 0 A of renewal densities for which h 1 ( x ) and imply that .
9 citations
TL;DR: In this paper, the authors consider a renewal process observed over a fixed length of time starting from a random time point and only the times of renewals that occur within the observation window are recorded.
Abstract: Suppose we have a renewal process observed over a fixed length of time starting from a random time point and only the times of renewals that occur within the observation window are recorded. Assuming a parametric model for the renewal time distribution with parameter θ, we obtain the likelihood of the observed data and describe the exact and asymptotic behavior of the Fisher information (FI) on θ contained in this window censored renewal process. We illustrate our results with exponential, gamma, and Weibull models for the renewal distribution. We use the FI matrix to determine optimal window length for designing experiments with recurring events when the total time of observation is fixed. Our results are useful in estimating the standard errors of the maximum likelihood estimators and in determining the sample size and duration of clinical trials that involve recurring events associated with diseases such as lupus.
9 citations
TL;DR: In this article, the authors considered supercritical linear birth-and-death processes under the influence of disasters that arrive as a renewal process independently of the population size and the novelty of this paper lies in assuming that the killing probability in a disaster is a function of the time that has elapsed since the last disaster.
9 citations
TL;DR: A general framework for single component replacement models is discussed, based on the regenerative structure of these models and by using results from renewal theory a unified presentation of the discounted and average finite and infinite horizon cost models is given.
Abstract: In this paper we discuss a general framework for single component replacement models. This framework is based on the regenerative structure of these models and by using results from renewal theory a unified presentation of the discounted and average finite and infinite horizon cost models is given. Finally, some well-known replacement models are discussed, and making use of the previous results an easy derivation of their cost functions is presented.
9 citations