Gaussian and multifractal processes in teletraffic theory: Dissertation

Petteri Mannersalo

Research output: ThesisDissertationCollection of Articles

Abstract

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.
Original languageEnglish
QualificationDoctor Degree
Awarding Institution
  • Aalto University
Supervisors/Advisors
  • Virtamo, Jorma, Supervisor, External person
Award date25 Apr 2003
Place of PublicationEspoo
Publisher
Print ISBNs951-38-6036-1
Electronic ISBNs951-38-6037-X
Publication statusPublished - 2003
MoE publication typeG5 Doctoral dissertation (article)

Fingerprint

Time Scales
Multifractal Spectrum
Gaussian Process
Increment
Trace
Traffic
Processor Sharing
Queue Length Distribution
Variance Function
Long-range Dependence
Traffic Model
Constant function
Nondegeneracy
Moving Average
Stationarity
Self-similarity
Scaling Laws
Queueing System
Probable
Burst

Keywords

  • Gaussian processes
  • multifractals
  • queueing systems
  • performance analysis
  • traffic modeling

Cite this

Mannersalo, P. (2003). Gaussian and multifractal processes in teletraffic theory: Dissertation. Espoo: VTT Technical Research Centre of Finland.
Mannersalo, Petteri. / Gaussian and multifractal processes in teletraffic theory : Dissertation. Espoo : VTT Technical Research Centre of Finland, 2003. 47 p.
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author = "Petteri Mannersalo",
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publisher = "VTT Technical Research Centre of Finland",
number = "491",
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Mannersalo, P 2003, 'Gaussian and multifractal processes in teletraffic theory: Dissertation', Doctor Degree, Aalto University, Espoo.

Gaussian and multifractal processes in teletraffic theory : Dissertation. / Mannersalo, Petteri.

Espoo : VTT Technical Research Centre of Finland, 2003. 47 p.

Research output: ThesisDissertationCollection of Articles

TY - THES

T1 - Gaussian and multifractal processes in teletraffic theory

T2 - Dissertation

AU - Mannersalo, Petteri

PY - 2003

Y1 - 2003

N2 - 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.

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KW - Gaussian processes

KW - multifractals

KW - queueing systems

KW - performance analysis

KW - traffic modeling

M3 - Dissertation

SN - 951-38-6036-1

T3 - VTT Publications

PB - VTT Technical Research Centre of Finland

CY - Espoo

ER -

Mannersalo P. Gaussian and multifractal processes in teletraffic theory: Dissertation. Espoo: VTT Technical Research Centre of Finland, 2003. 47 p.