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One of the major challenges in designing modern communication networks is providing quality of service to the individual users. An important part of this design process is understanding statistical characteristics of network traf®c streams and their impact on network performance. Unlike the conventional voice traf®c, modern data traf®c exhibits an increased level of ``burstiness'' that spans over multiple time scales. It was observed that sample paths of these data sequences show evidence of self-similarity | Self-Similar Network Traffic and Performance Evaluation Edited by Kihong Park and Walter Willinger Copyright 2000 by John Wiley Sons Inc. Print ISBN 0-471-31974-0 Electronic ISBN 0-471-20644-X 10 ASYMPTOTIC ANALYSIS OF QUEUES WITH SUBEXPONENTIAL ARRIVAL PROCESSES P. R. Jelenkovic Department of Electrical Engineering Columbia University New York NY 10027 INTRODUCTION One of the major challenges in designing modem communication networks is providing quality of service to the individual users. An important part of this design process is understanding statistical characteristics of network traffic streams and their impact on network performance. Unlike the conventional voice traffic modern data traffic exhibits an increased level of burstiness that spans over multiple time scales. It was observed that sample paths of these data sequences show evidence of self-similarity. Their autocorrelation structure is characterized by long-range dependency and the empirical distributions are easily matched with subexponential and long-tailed distributions. Early discovery of the self-similar nature of Ethernet traffic was reported in Leland et al. 42 see also Leland et al. 43 . More recently Crovella 22 attributed the long-range dependency of Ethernet traffic to the longtailed file sizes that are transferred over the network. Long-range dependency of the variable bit rate video traffic was demonstrated by Beran et al. 9 . Long-tailed characteristics of the scene length distribution of MPEG video streams were explored in Heyman and Lakshman 30 and Jelenkovic et al. 37 . Practical importance novelty and the intriguing nature of these phenomena have attracted a great number of scientists to develop new traffic models and to understand the impact of these models on network performance hi this development there 249 250 ASYMPTOTIC ANALYSIS OF QUEUES WITH SUBEXPONENTIAL ARRIVALS have been two basic approaches self-similar fractal processes and fluid renewal models with long-tailed .