Fractional Ornstein-Uhlenbeck L´ evy Processes and the Telecom Process: Upstairs and Downstairs Robert L. Wolpert ∗ Murad S. Taqqu † February 15, 2005 Revision :3.3 Abstract We model the workload of a network device responding to a random flux of work requests with various intensities and durations in two ways, a conventional univariate stochastic integral approach (“downstairs”) and a higher-dimensional random field approach (“upstairs”). The models feature Gaussian, stable, Poisson and, more generally, infinitely divisible distributions reflecting the aggregate work requests from independent sources. We focus on the fractional Ornstein-Uhlenbeck L´ evy process and the Telecom process which is the limit of renewal reward processes where both the interrenewal times and the rewards are heavy-tailed. We show that the Telecom process can be interpreted as the workload of a network responding to job requests with stable infinite variance intensities and durations and that fractional Brownian motion can be interpreted in the same way but with finite variance intensities. This explains the ubiquitous presence of fractional Brownian motion in network traffic. Key words: Fractional Brownian motion, Fractional L´ evy motion, Fractional stable motion, In- finitely divisible distributions, L´ evy processes, Moving averages, Stable processes. AMS 2000 subject classifications: Primary 90B15, secondary 60G18, 60H05, 60G57. ∗ Robert L. Wolpert is Professor of Statistics and Decision Sciences, and Professor of the Environment and Earth Sciences, at Duke University in Durham, NC 27708, USA. † Murad S. Taqqu is Professor of Mathematics and Statistics at Boston University in Boston, MA 02459, USA. c  2004–2005 Robert L Wolpert, all rights reserved. 1