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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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TL;DR: In this paper, the authors study random variables related to a shock reliability model and obtain properties of the distribution function of the random variables involved and obtain their limit behavior when k tends to infinity or when the probability of entering a critical set tends to zero.
Abstract: In this paper we study random variables related to a shock reliability model. Our models can be used to study systems that fail when k consecutive shocks with critical magnitude (e.g. above or below a certain critical level) occur. We obtain properties of the distribution function of the random variables involved and we obtain their limit behaviour when k tends to infinity or when the probability of entering a critical set tends to zero. This model generalises the Poisson shock model.
149 citations
TL;DR: In this article, the authors consider an insurance portfolio situation in which there is possible dependence between the waiting time for a claim and its actual size and obtain explicit exponential estimates for infinite and finite-time ruin probabilities in the case of light-tailed claim sizes.
Abstract: We consider an insurance portfolio situation in which there is possible dependence between the waiting time for a claim and its actual size. By employing the underlying random walk structure we obtain explicit exponential estimates for infinite- and finite-time ruin probabilities in the case of light-tailed claim sizes. The results are illustrated in several examples, worked out for specific dependence structures.
147 citations
TL;DR: A model of the telephone network in which there are three distinct channels is applied to study the effect that interleaving (time division multiplexing) has on the effectiveness of error-correcting codes.
Abstract: The error statistics from data-transmission field tests on the telephone network may be compactly represented by about one dozen parameters. These relate to a model of the telephone network in which there are three distinct channels. The errors in binary data on each channel are produced by a renewal process in which a bit-error is a renewal event. The mixture of three such channels allows a close fit to the error statistics for a large range of block lengths. It is not implied that the errors on the telephone network are actually produced by such processes, but merely that they may be conveniently and compactly represented by them. Use of this model simplifies the analysis of error-control systems and the determination of error rates for error-control codes. In this paper the model is applied to study the effect that interleaving (time division multiplexing) has on the effectiveness of error-correcting codes.
141 citations
TL;DR: In this paper, the renewal theorems for non-discrete non-negative random variables with mean k > o were studied. But the main emphasis was on processes which are not discrete.
Abstract: A sequence of non-negative random variables {Xi} is called a renewal process, and if the Xi may only take values on some sequence it is termed a discrete renewal process. The greatest k such that X1 + X2 + … + Xk ≤ x(> o) is a random variable N(x) and theorems concerning N(x) are renewal theorems. This paper is concerned with the proofs of a number of renewal theorems, the main emphasis being on processes which are not discrete. It is assumed throughout that the {Xi} are independent and identically distributed.If H(x) = Ɛ{N(x)} and K(x) is the distribution function of any non-negative random variable with mean K > o, then it is shown that for the non-discrete processwhere Ɛ{Xi} need not be finite; a similar result is proved for the discrete process. This general renewal theorem leads to a number of new results concerning the non-discrete process, including a discussion of the stationary “age-distribution” of “renewals” and a discussion of the variance of N(x). Lastly, conditions are established under whichThese new conditions are much weaker than those of previous theorems by Feller, Tacklind, and Cox and Smith.
139 citations
TL;DR: In this article, a treatment in terms of the theory of probability, which uses the modern developments of this theory, will shed new light on the subject, and the purpose of this treatment is to present such a treatment.
Abstract: Renewal theory is ordinarily reduced to the theory of certain types of integral equations. Since the basis for the integral equations is a simple probability process, however, it is to be expected that a treatment in terms of the theory of probability, which uses the modern developments of this theory, will shed new light on the subject. The purpose of this paper is to present such a treatment. Renewal theory, in the simplest case, deals with the following situation. A population of individuals is at hand; when any individual dies he is immediately replaced by a newly born individual. The problem is to investigate the development of the population, particularly the age distribution. Since the individuals are supposed to live and die independently of each other it is sufficient for most purposes to consider a population consisting at any time of only a single individual. At birth, the future lifetime of the individual is a chance variable with distribution function F(x) (the probability of death before age x). At any age x, the lifetime remaining then has distribution function
139 citations