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Zbl 1160.94370
Wolpert, Robert L.; Taqqu, Murad S.
Fractional Ornstein-Uhlenbeck Lévy processes and the telecom process: Upstairs and downstairs. (English)
[J] Signal Process. 85, No. 8, 1523-1545 (2005). ISSN 0165-1684

Summary: 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 Lévy 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 (fBM) can be interpreted in the same way but with finite variance intensities. This explains the ubiquitous presence of fBM in network traffic.

MSC 2000:: *94A12 Signal theory
90B15 Flows in networks with stochastic elements
60G18 Self-similar processes
60H05 Stochastic integrals
60G57 Random measures

Keywords: fractional Brownian motion; fractional Lévy motion; fractional stable motion; infinitely divisible distributions; Lévy processes; moving averages; stable processes

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