Maximum Queue Length of a Fluid Model with a Gaussian Input
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Issue Date
2007-12-11Author
Jin, Yasong
Publisher
University of Kansas
Format
111 pages
Type
Dissertation
Degree Level
PH.D.
Discipline
Mathematics
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This item is protected by copyright and unless otherwise specified the copyright of this thesis/dissertation is held by the author.
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A fractional Brownian queueing model, that is, a fluid model with an input of a fractional Brownian motion, was proposed in the 1990s to capture the self-similarity and long-range dependence observed in Internet traffic. Since then, a Gaussian queueing model, which is a queueing model with an input of a continuous Gaussian process, has received much attention. In this dissertation, a Gaussian queueing model is discussed and the maximum queue length over a time interval [0, t] is analyzed. Under some mild assumptions, it is shown that a limit of the maximum queue length suitably normalized is determined by a suitable function of the asymptotic variance of the Gaussian input. Some Gaussian queueing models, such as a queue with an input of several independent fractional Brownian motions and a queue with an input of an integrated Ornstein-Uhlenbeck process, are discussed as examples. For a fractional Brownian queueing model, the main results extend some related known results in the literature. The results on the maximum queue length provide insights for the occurrence of large excursions, which are also called congestion events, in a queueing process. In the context of a fractional Brownian queueing model the temporal properties of congestion events, such as the duration and the inter-congestion event time, are analyzed. A new method based on a Poisson clumping approximation is proposed to evaluate these properties. By comparing with simulation results, it is illustrated that the proposed methodology produces satisfying results for estimating the temporal properties of congestion events in a fractional Brownian queueing model.
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