Gaussian and multifractal processes in teletraffic theory

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 longrange 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 nondegeneracy, establish a power law for the moments and obtain a formula for the multifractal spectrum.