In this report we study the supremum distribution of a general class of Gaussian processes {Xt : t 2 0) having stationary increments. This distribution is directly related to the steady state queue length distribution of a queueing system, and hence its study is also important for various applications including communication network analysis. Our study is based on Extreme Value Theory and we show that log P({s~p~X:>t ~> x)) + asymptotically grows at most (on the order of) log x, where m, corresponds to the reciprocal of the maximum (normalized) variance of Xt. This result is considerably stronger than the existing results in the literature based on Large Deviation Theory. We further show that this improvement can be critical in characterizing the asymptotic behavior of P ( { s ~ p ~X>t >~ x)). The types of Gaussian processes that our results cover also include a large class of processes that exhibit self-similarity and other types of long-range dependence.

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