Exact Asymptotic Queue Length Distribution for Fractional Brownian Traac*
نویسنده
چکیده
For a storage model with a fractional Brownian input process, an exact asymptotic expression for the queue length distribution is obtained, using a few plausible assumptions. The probability that the queue length is greater than x is shown to be asymptotically proportional to a Weibullian function multiplied by x ? : For a fractional Brownian input process with Hurst parameter H; the exponent is shown to be equal to (1 ? H)(2H ? 1)=H: The proportionality constant in front of the asymptotic form of the queue length distribution can be evaulated by solving a related random walk problem, and is computed numerically for various values of H: 1.Introduction Measurements on data traac in the last few years in various environments have revealed that data traac has long range dependence, and is in fact statistically fractal or self-similar in nature over a wide range of timescales ..1] This creates serious diiculties for traditional traac models, which almost all assume that the traac is Markovian in nature. Qualitatively, it is not diicult to see that long-range dependence in the input traac will lead to larger delays in processing the traac through a server. If the input rate at any time is even slightly above the server speed, and the increased traac rate persists for a long time (as it often does when the input has long range dependence), there will be serious backups towards the end of this`hyperactive' period. Quantitatively, Norross2] has considered a simple model where the input traac is described by fractional Brownian motion (FBM), in which case the traac rate has long-range dependence.x For this model, Norross2] has obtained a lower bound for the queue length distribution when the traac * Submitted to Advances in Performance Analysis x Hereafter in this paper we shall occasionally refer to a FBM process as having long-range dependence, by which of course we mean the increments to the FBM process.
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