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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: Feller's paper as discussed by the authors is a rigorous treatment of renewal theory, and to assist the reader his principal results are summarized below in demographic form and notation, and they can be found in Table 1.
Abstract: Feller’s paper is a rigorous treatment of renewal theory, and to assist the reader his principal results are summarized below in demographic form and notation.
287 citations
Book•
01 Jan 1999
TL;DR: In this paper, the authors introduce Markov Chains, Martingales, Poisson Processes, and Renewal Theory, as well as Brownian Motion, and conclude that renewal theory can be viewed as a form of Markov chains.
Abstract: 1. Markov Chains 2. Martingales 3. Poisson Processes 4. Markov Chains 5. Renewal Theory 6. Brownian Motion
286 citations
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TL;DR: In this paper, renewal theory is applied to counting processes and the interval distribution is distorted by the presence of a dead time, and the resulting counting statistics are used to determine the resulting count statistics.
269 citations
TL;DR: In this paper, the authors introduced the geometric process which is a sequence of independent nonnegative random variables, such that the distribution function of a random variable X n is F (a n−1 −1 normalized x), wherea is a positive constant, and the explicit expressions of the long-run average costs per unit time under each replacement policy are calculated, and therefore the corresponding optimal replacement policies can be found analytically or numerically.
Abstract: In this paper, we introduce and study the geometric process which is a sequence of independent non-negative random variablesX
1,X
2,... such that the distribution function ofX
n isF (a
n−1
x), wherea is a positive constant. Ifa>1, then it is a decreasing geometric process, ifa<1, it is an increasing geometric process. Then, we consider a replacement model as follows: the successive survival times of the system after repair form a decreasing geometric process or a renewal process while the consecutive repair times of the system constitute an increasing geometric process or a renewal process. Besides the replacement policy based on the working age of the system, a new kind of replacement policy which is determined by the number of failures is considered. The explicit expressions of the long-run average costs per unit time under each replacement policy are then calculated, and therefore the corresponding optimal replacement policies can be found analytically or numerically.
266 citations
TL;DR: Numerical investigation of errors in the approximation and subsequent experience has shown that this method of generating overflow traffic is accurate and very useful in both simulations and analyses of traffic systems.
Abstract: Traffic overflowing a first-choice trunk group can be approximated accurately by a simple renewal process called an interrupted Poisson process–a Poisson process which is alternately turned on for an exponentially distributed time and then turned off for another (independent) exponentially distributed time. The approximation is obtained by matching either the first two or three moments of an interrupted Poisson process to those of an overflow process. Numerical investigation of errors in the approximation and subsequent experience has shown that this method of generating overflow traffic is accurate and very useful in both simulations and analyses of traffic systems.
264 citations