The Importance of Power-Tail Distributions for Modeling Queueing Systems

Abstract

Power-tail distributions are those for which the reliability function is of the form x−α for large x. Although they look well behaved, they have the singular property that E(Xℓ) = ∞ for all ℓ ≥ α. Thus it is possible to have a distribution with an infinite variance, or even an infinite mean. As pathological as these distributions seem to be, they occur everywhere in nature, from the CPU time used by jobs on main-frame computers to sizes of files stored on discs, earthquakes, or even health insurance claims. Recently, traffic on the “electronic super highway” was revealed to be of this type, too. In this paper we first describe these distributions in detail and show their suitability to model self-similar behavior, e.g., of the traffic stated above. Then we show how these distributions can occur in computer system environments and develop a so-called truncated analytical model that in the limit is power-tail. We study and compare the effects on system performance of a GI/M/1 model both for the truncated and the limit case, and demonstrate the usefulness of these approaches particularly for Markov modeling with LAQT (Linear Algebraic Approach to Queueing Theory, Lipsky 1992) techniques.

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