Topic
Fractional Poisson process
About: Fractional Poisson process is a research topic. Over the lifetime, 868 publications have been published within this topic receiving 17858 citations.
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TL;DR: This paper presents a proof of this result under one basic assumption: the process being observed cannot anticipate the future jumps of the Poisson process.
Abstract: In many stochastic models, particularly in queueing theory, Poisson arrivals both observe (see) a stochastic process and interact with it. In particular cases and/or under restrictive assumptions it has been shown that the fraction of arrivals that see the process in some state is equal to the fraction of time the process is in that state. In this paper, we present a proof of this result under one basic assumption: the process being observed cannot anticipate the future jumps of the Poisson process.
1,226 citations
TL;DR: In this article, a simple and relatively efficient method for simulating one-dimensional and two-dimensional nonhomogeneous Poisson processes is presented, which is applicable for any rate function and is based on controlled deletion of points in a Poisson process whose rate function dominates the given rate function.
Abstract: : A simple and relatively efficient method for simulating one- dimensional and two-dimensional nonhomogeneous Poisson processes is presented. The method is applicable for any rate function and is based on controlled deletion of points in a Poisson process whose rate function dominates the given rate function. In its simplest implementation, the method obviates the need for numerical integration of the rate function, for ordering of points, and for generation of Poisson variates.
890 citations
Book•
01 May 1997
TL;DR: In this paper, the Mixed Poisson Distributions (MPD) is defined as a mixture of Cox Processes, Gauss-Poisson Processes and Mixed Renewal Processes.
Abstract: Preface Introduction The Mixed Poisson Distributions Some Basic Concepts The Mixed Poisson Process Some Related Processes Cox Processes Gauss-Poisson Processes Mixed Renewal Processes Characterization of Mixed Poisson Processes Reliability Properties of Mixed Poisson Processes Characterization within Birth Processes Characterization within Stationary Point Processes Characterization within General Point Processes Compound Mixed Poisson Distributions Compound Distributions Exponential Bounds Asymptotic Behaviour Recursive Evaluation The Risk Business The Claim Process Ruin Probabilities
360 citations
TL;DR: The Chen-Stein method of Poisson approximation is a powerful tool for computing an error bound when approximating probabilities using the Poisson distribution as discussed by the authors, in many cases, this bound may be given in terms of first and second moments alone.
Abstract: The Chen-Stein method of Poisson approximation is a powerful tool for computing an error bound when approximating probabilities using the Poisson distribution. In many cases, this bound may be given in terms of first and second moments alone. We present a background of the method and state some fundamental Poisson approximation theorems. The body of this paper is an illustration, through varied examples, of the wide applicability and utility of the Chen-Stein method. These examples include birthday coincidences, head runs in coin tosses, random graphs, maxima of normal variates and random permutations and mappings. We conclude with an application to molecular biology. The variety of examples presented here does not exhaust the range of possible applications of the Chen-Stein method.
333 citations
TL;DR: In this article, a fractional non-Markov Poisson stochastic process has been developed based on fractional generalization of the Kolmogorov-Feller equation.
302 citations