Fractional Ornstein-Uhlenbeck Lévy Processes and
the Telecom Process: Upstairs and Downstairs

Robert L. Wolpert1 and Murad S. Taqqu2

Duke University Institute of Statistics and Decision Sciences1 and
Boston University Department of Mathematics2

Revised February 2005

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 can be interpreted in the same way but with finite variance intensities. This explains the ubiquitous presence of fractional Brownian motion in network traffic.

Keywords: Fractional Brownian motion, Fractional Lévy motion, Fractional Stable motion, Infinitely divisible distributions, Lévy processes, Moving averages, Nonparametric Bayesian analysis, Stable processes.

Cite as: 

@Article{Wolp:Taqq:2005,
      Author = "Robert L. Wolpert and Murad Taqqu",
       Title = "Fractional {O}rnstein-{U}hlenbeck {L\'e}vy Processes
                and the {T}elecom Process: {U}pstairs and Downstairs",
     Journal = "Signal Processing",
      Volume = 85,
      Number = 8,
       Month = aug,
       Pages = "1523--1545",
        Year = 2005,
}

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