Self-Similar Processes in Telecommunications

About the authors.

Acknowledgements.

1 Principal Concepts of Fractal Theory and Self-Similar Processes.

1.1 Fractals and Multifractals.

1.1.1 Fractal Dimension of a Set.

1.1.2 Multifractals.

1.1.3 Fractal Dimension D0 and Informational Dimension D1.

1.1.4 Legendre Transform.

1.2 Self-Similar Processes.

1.2.1 Definitions and Properties of Self-Similar Processes.

1.2.2 Multifractal Processes.

1.2.3 Long-Range and Short-Range Dependence.

1.2.4 Slowly Decaying Variance.

1.3 ‘Heavy Tails’.

1.3.1 Distribution with ‘Heavy Tails’ (DHT).

1.3.2 ‘Heavy Tails’ Estimation.

1.4 Hurst Exponent Estimation.

1.4.1 Time Domain Methods of Hurst Exponent Estimation.

1.4.2 Frequency Domain Methods of Hurst Exponent.

Estimation.

1.5 Hurst Exponent Estimation Problems.

1.5.1 Estimation Problems.

1.5.2 Nonstationarity Problems.

1.5.3 Computational Problems.

1.6 Self-Similarity Origins in Telecommunication Traffic.

1.6.1 User’s Behaviour.

1.6.2 Data Generation Data Structure and Its Search.

1.6.3 Traffic Aggregation.

1.6.4 Means of Network Control.

1.6.5 Control Mechanisms based on Feedback.

1.6.6 Network Development.

References.

2 Simulation Methods for Fractal Processes.

2.1 Fractional Brownian Motion.

2.1.1 RMD Algorithm for FBM Generation.

2.1.2 SRA Algorithm for FBM Generation.

2.2 Fractional Gaussian Noise.

2.2.1 FFT Algorithm for FGN Synthesis.

2.2.2 Advantages and Shortcomings of FBM/FGN Models.

in Network Applications.

2.3 Regression Models of Traffic.

2.3.1 Linear Autoregressive (AR) Processes.

2.3.2 Processes of Moving Average (MA).

2.3.3 Autoregressive Models of Moving Average, ARMAðp; qÞ.

2.3.4 Fractional Autoregressive Integrated Moving Average.

(FARIMA) Process.

2.3.5 Parametric Estimation Methods.

2.3.6 FARIMAðp,d,qÞ Process Synthesis.

2.4 Fractal Point Process.

2.4.1 Statistical Characteristics of the Point Process.

2.4.2 Fractal Structure of FPP.

2.4.3 Methods of FPP Formation.

2.5 Fractional Levy Motion and its Application to Network.

Traffic Modelling.

2.5.1 Fractional Levy Motion and Its Properties.

2.5.2 Algorithm of Fractional Levy Motion Modelling.

2.5.3 Fractal Traffic Formation Based on FLM.

2.6 Models of Multifractal Network Traffic.

2.6.1 Multiplicative Cascades.

2.6.2 Modified Estimation Method of Multifractal Functions.

2.6.3 Generation of Traffic the Multifractal Model.

2.7 LRD Traffic Modelling with the Help of Wavelets.

2.8 M/G/1Model.

2.8.1 M/G/1Model and Pareto Distribution.

2.8.2 M/G/1Model and Log-Normal Distribution.

References.

3 Self-Similarity of Real Time Traffic.

3.1 Self-Similarity of Real Time Traffic Preliminaries.

3.2 Statistical Characteristics of Telecommunication Real Time Traffic.

3.2.1 Measurement Organization.

3.2.2 Pattern of TN Traffic.

3.3 Voice Traffic Characteristics.

3.3.1 Voice Traffic Characteristics at the Call Layer.

3.3.2 Voice Traffic Characteristics at the Packet Layer.

3.4 Multifractal Analysis of Voice Traffic.

3.4.1 Basics.

3.4.2 Algorithm for the Partition Function SmðqÞ Calculation.

3.4.3 Multifractal Properties of Multiplexed Voice Traffic.

3.4.4 Multifractal Properties of Two-Component Voice Traffic.

3.5 Mathematical Models of VoIP Traffic.

3.5.1 Problem Statement.

3.5.2 Voice Traffic Models at the Call Layer.

3.5.3 Estimation of Semi-Markovian Model Parameters and the Modelling.

Results of the Voice Traffic at the Call Layer.

3.5.4 Mathematical Models of Voice Traffic at the Packets Layer.

3.6 Simulation of the Voice Traffic.

3.6.1 Simulation Structure.

3.6.2 Parameters Choice of Pareto Distributions for Voice.

Traffic Source in ns2.

3.6.3 Results of Separate Sources Modelling.

3.6.4 Results of Traffic Multiplexing for the Separate.

ON/OFF Sources.

3.7 Long-Range Dependence for the VBR-Video.

3.7.1 Distinguished Characteristics of Video Traffic.

3.7.2 Video Conferences.

3.7.3 Video Broadcasting.

3.7.4 MPEG Video Traffic.

3.7.5 Nonstationarity of VBR Video Traffic.

3.8 Self-Similarity Analysis of Video Traffic.

3.8.1 Video Broadcasting Wavelet Analysis.

3.8.2 Numerical Results.

3.8.3 Multifractal Analysis.

3.9 Models and Modelling of Video Sequences.

3.9.1 Nonstationarity Types for VBR Video Traffic.

3.9.2 Model of the Video Traffic Scene Changing Based on the.

Shifting Level Process.

3.9.3 Video Traffic Models in the Limits of the Separate Scene.

3.9.4 Fractal Autoregressive Models of p-Order.

3.9.5 MPEG Data Modelling Using I, P and B Frames Statistics.

3.9.6 ON/OFF Model of the Video Sequences.

3.9.7 Self-Similar Norros Model.

3.9.8 Hurst Exponent Dependence on N.

References.

4 Self-Similarity of Telecommunication Networks Traffic.

4.1 Problem Statement.

4.2 Self-Similarity and ‘Heavy Tails’ in Lan Traffic.

4.2.1 Experimental Investigations of Ethernet Traffic Self-Similar.

Structure.

4.2.2 Estimation of Testing Results.

4.3 Self-Similarity of WAN Traffic.

4.3.1 WAN Traffic at the Application Level.

4.3.2 Some Limiting Results for Aggregated WAN Traffic.

4.3.3 The Statistical Analysis of WAN Traffic at the.

Application Level.

4.3.4 Multifractal Analysis of WAN Traffic.

4.4 Self-Similarity of Internet Traffic.

4.4.1 Results of Experimental Studies.

4.4.2 Stationarity Analysis of IP Traffic.

4.4.3 Nonstationarity of Internet Traffic.

4.4.4 Scaling Analysis.

4.5 Multilevel ON/OFF Model of Internet Traffic.

4.5.1 Problem Statement.

4.5.2 Estimation of Parameters and Model Parameterization.

4.5.3 Parallel Buffer Structure for Active Queue Control.

References.

5 Queuing and Performance Evaluation of Telecommunication.

Networks under Traffic Self-Similarity Conditions.

5.1 Traffic Fractality Influence Estimate on Telecommunication.

Networks Queuing.

5.1.1 Monofractal Traffic.

5.1.2 Communication System Model and the Packet Loss Probability.

Estimate for the Asymptotic Self-Similar Traffic Described by.

Pareto Distribution.

5.1.3 Queuing Model with Fractal Levy Motion.

5.1.4 Estimate of the Effect of Traffic Multifractality Effect on Queuing.

5.2 Estimate of Voice Traffic Self-Similarity Effects on the iP Networks.

Input Parameter Optimization.

5.2.1 Problem Statement.

5.2.2 Simulation Structure.

5.2.3 Estimate of the Traffic Self-Similarity Influence on QoS.

5.2.4 TN input Parameter Optimization for Given QoS Characteristics.

5.2.5 Conclusions.

5.3 Telecomminication Network Parameters Optimization Using the Tikhonov.

Regularization Approach.

5.3.1 Problem Statement.

5.3.2 Telecommunication Network Parameter Optimization on the Basis of.

the Minimization of the Discrepancy Functional of QoS Parameters.

5.3.3 Optimization Results.

5.3.4 TN Parameter Optimization on the Basis of Tikhonov.

Functional Minimization.

5.3.5 Regularization Results.

5.3.6 Conclusions.

5.4 Estimation of the Voice Traffic Self-Similarity Influence on QoS.

with Frame Relay Networks.

5.4.1 Pocket Delay at Transmission through the Frame Relay Network.

5.4.2 Frame Relay Router Modelling.

5.4.3 Simulation Results.

5.5 Bandwidth Prediction in Telecommunication Networks.

5.6 Congestion Control of Self-Similar Traffic.

5.6.1 Unimodal Ratio Loading/Productivity.

5.6.2 Selecting Aggressiveness Control (SAC) Scheme.

References.

Appendix A List of Symbols.

Appendix B List of Acronyms.

Index.