Gaussian and multifractal processes in teletraffic theory

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Doctoral thesis (article-based)
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44, [109]
VTT publications, 491
In this thesis, we consider two classes of stochastic models which both capture some of the essential properties of teletraffic. Teletraffic has two time regimes where profoundly different behavior and characteristics are seen. When traffic traces are observed at coarse resolutions, properties like self-similarity and long-range dependence are visible. In small time-scales, traffic exhibits complex scaling laws with much more spiky bursts than in coarser resolutions. The main part of the thesis is devoted to a large time-scale analysis by considering Gaussian processes and queueing systems with Gaussian input. In order to understand the small time-scale dynamics, first steps are taken towards general multifractal models offering a suitable basis for short time-scale teletraffic modeling. The family of Gaussian processes with stationary increments serves as the traffic model for large time-scales. First, we introduce a fast and accurate simulation algorithm, which can be used to generate long approximate Gaussian traces. Moreover, the algorithm is also modified to run on-the-fly. Then approximate queue length distributions for ordinary, priority and generalized processor sharing queues are derived using a most probable path approach. Simulation studies show that the performance formulae appear to be quite accurate over the full range of buffer levels. Finally, we construct a semi-stationary predictor, which uses a constant variance function and mean rate estimation based on a moving average method. Moreover, we show that measuring the past of a process by geometrically increasing intervals is a good engineering solution and a much better way than equally spaced measurements. We introduce a family of multifractal processes which belongs to the framework of T-martingales and multiplicative chaos introduced by Kahane. The family has many desirable properties like stationarity of increments, concave multifractal spectra and simple construction. We derive, for example, conditions for non-degeneracy, establish a power law for the moments and obtain a formula for the multifractal spectrum.
Gaussian processes, multifractals, queueing systems, performance analysis, traffic modeling
Other note
  • Norros, I., Mannersalo, P. and Wang, J. Simulation of fractional Brownian motion with conditionalized random midpoint displacement. Advances in Performance Analysis, 1999. Vol. 2(1), pp. 77-101.
  • Addie, R., Mannersalo, P. and Norros, I. Performance formulae for queues with Gaussian input. In Proceedings of ITC 16. Edinburgh, UK, 1999. Pp. 1169-1178.
  • Addie, R., Mannersalo, P. and Norros, I. Most probable paths and performance formulae for buffers with Gaussian input traffic. European Transactions in Telecommunications, 2002. Vol. 13(3), pp. 183-196.
  • Mannersalo, P. and Norros, I. Approximate formulae for Gaussian priority queues. In Proceedings of ITC 17. Salvador, Brazil, 2001. Pp. 991-1002.
  • Mannersalo, P. and Norros, I. GPS schedulers and Gaussian traffic. In Proceedings of IEEE Infocom 2002. New York, USA, 2002. Pp. 1660-1667.
  • Mannersalo, P. and Norros, I. A most probable path approach to queueing systems with general Gaussian input. Computer Networks, 2002. Vol. 40(3), pp. 399-412.
  • Mannersalo, P. Some notes on prediction of teletraffic. In Proceedings of 15th ITC Specialist Seminar. Würzburg, Germany, 2002. Pp. 220-229.
  • Mannersalo, P., Norros, I. and Riedi, R. Multifractal products of stochastic processes: construction and some basic properties. Advances in Applied Probability, 2002. Vol. 34(4), pp. 888-903.
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