Fitting Mixtures of Exponentials to Long-Tail Distributions to Analyze Network Performance Models

Traffic measurements from communication networks have shown that many quantities charecterizing network performance have long-tail probability distributions, i.e., with tails that decay more slowly than exponentially. File lengths, call holding times, scene lengths in MPEG video streams, and intervals between connection requests in Internet traffic all have been found to have long-tail distributions, being well described by distributions such as the Pareto and Weibull. It is known that long-tail distributions can have a dramatic effect upon performance, e.g., long-tail service-time distributions cause long-tail waiting-time distributions in queues, but it is often difficult to describe this effect in detail, because performance models with component long-tail distributions tend to be difficult to analyze. We address this problem by developing an algorithm for approximating a long-tail distribution by a hyperexponential distribution (a finite mixture of exponentials). We first prove that, in prinicple, it is possible to approximate distributions from a large class, including the Pareto and Weibull distributions, arbitrarily closely by hyperexponential distributions. Then we develop a specific fitting alogrithm. Our fitting algorithm is recursive over time scales, starting with the largest time scale. At each stage, an exponential component is fit in the largest remaining time scale iand then the fitted exponential component is subtracted from the distribution. Even though a mixture of exponentials has an exponential tail, it can match a long-tail distribution in the regions of primary interest when there are enough exponential components. When a good fit is achieved, the approximating hyperexponential distribution inherits many of the difficulties of the original long-tail distribution: e.g., it is still difficult to obtain reliable estimates from simulation experiments. However, some difficulties are avoided; e.g., it is possible to solve some queueing models that could not be solved before. We give examples showing that the fitting procedure is effective, both for directly matching a long-tail distribution and for predicting the performance in a queueing model with a long-tail service-time distribution.