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Fractional Ornstein-Uhlenbeck Lévy processes and the telecom process: Upstairs and downstairs. (English) Zbl 1160.94370
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:
94A12Signal theory (characterization, reconstruction, filtering, etc.)
90B15Network models, stochastic (optimization)
60G18Self-similar processes
60H05Stochastic integrals
60G57Random measures