Overview
Two National Science Foundation (NSF) grants were awarded to George Mason University, Noblis and the University of Windsor to research the feasibility of approaches to modeling queues with heavy-tailed interarrival and service distributions: an NSF Small Grant for Exploratory Research (SGER), awarded April 2000 - April 2001, and an NSF Division of Design, Manufacture and Industrial Innovation three year grant (which was extended to a fourth year), awarded effective September 1, 2002.
This research is concerned with developing procedures for modeling queues with heavy-tailed distributions for interarrival and/or service times. These types of probability distributions decay much more slowly than exponential. Distributions of this type render queueing analyses very difficult, in that the Laplace-Stieltjes transforms (LSTs) of interarrival and/or service times, which play such a crucial role in analytical queueing theory, often do not have closed form. The approaches proposed herein avoid the problems and pitfalls of finding approximating distributions by using the actual heavy-tailed distributions themselves.
The procedures developed under this research grant fall into three areas. One method approximates the LSTs needed to produce the output measures (waiting-time and system-size distributions) of interest by directly approximating the LSTs using a discretized version of the heavy-tailed distribution itself (transform approximation method [TAM]). Another method avoids using the LST directly by solving an integral equation involving the complementary cdf of the heavy-tailed distribution (level crossing [LC] method). While discrete-event simulation is an alternative to analytical queueing analyses, this also has its limitations; and was examined in this research. Simulation has difficulty when modeling certain of the heavy-tailed distributions, especially with large coefficients of variation (standard deviation divided by mean). The use of quantile estimators with discrete-event simulation was also researched.
