Fractal Teletraffic Modeling and Delay Bounds in Computer Communications

Description

By deploying time series analysis, Fourier transform, functional analysis, min-plus convolution, and fractional order systems and noise, this book proposes fractal traffic modeling and computations of delay bounds, aiming to improve the quality of service in computer communication networks. As opposed to traditional studies of teletraffic delay bounds, the author proposes a novel fractional noise, the generalized fractional Gaussian noise (gfGn) approach, and introduces a new fractional noise, generalized Cauchy (GC) process for traffic modeling. Researchers and graduates in computer science, applied statistics, and applied mathematics will find this book beneficial. Ming Li, PhD, is a professor at Ocean College, Zhejiang University, and the East China Normal University. He has been an active contributor for many years to the fields of computer communications, applied mathematics and statistics, particularly network traffic modeling, fractal time series, and fractional oscillations. He has authored more than 200 articles and 5 monographs on the subjects. He was identified as the Most Cited Chinese Researcher by Elsevier in 2014–2020. Professor Li was recognized as a top 100,000 scholar in all fields in 2019–2020 and a top 2% scholar in the field of Numerical and Computational Mathematics in 2021 by Prof. John P. A. Ioannidis, Stanford University. Cover Half Title Title Page Copyright Page Dedication Table of Contents Preface Acknowledgments Author CHAPTER 1 Time Series 1.1 RANDOM PROCESSES 1.2 ERGODICITY 1.3 PROBABILITY DENSITY FUNCTION 1.3.1 PDF of Brownian Motion 1.3.2 Gaussian Distribution 1.3.3 Poisson Distribution 1.3.4 Cauchy Distribution 1.4 CORRELATION FUNCTIONS 1.4.1 Auto-Correlation Functions 1.4.2 Auto-Covariance Functions 1.4.3 Cross-Correlation Functions 1.4.4 Cross-Covariance Functions 1.5 POWER SPECTRA 1.5.1 Power Auto-Spectrum Density Functions 1.5.2 Power Cross-Spectrum Density Functions 1.6 WHITE NOISE 1.7 RANDOM FUNCTIONS OF INTEREST IN TRAFFIC THEORY REFERENCES CHAPTER 2 Fourier Transform 2.1 BASIC IN FOURIER TRANSFORM 2.2 DELTA FUNCTION 2.3 NASCENT DELTA FUNCTIONS 2.4 BASIC PROPERTIES OF FOURIER TRANSFORM 2.5 CONVOLUTION REFERENCES CHAPTER 3 Applied Functional 3.1 LINEAR SPACES 3.1.1 Notion of Linear Spaces 3.1.2 Isomorphism of Linear Spaces 3.1.3 Subspaces and Affine Manifold 3.1.4 Convex Sets 3.2 METRIC SPACES 3.2.1 Concept of Metric Spaces 3.2.2 Limit in Metric Spaces 3.2.3 Balls Viewed from Metric Spaces 3.2.4 Postscript 3.3 LINEAR NORMED SPACES 3.3.1 Notion of Norm and Normed Spaces 3.3.2 Norms 3.3.3 Equivalence of Norms 3.3.4 Equivalence of Linear Normed Spaces 3.4 BANACH SPACES 3.4.1 Concept of Banach Spaces 3.4.2 Completeness 3.4.3 Series in Banach Spaces 3.4.4 Separable Banach Spaces and Completion of Spaces 3.5 HILBERT SPACES 3.5.1 Inner Product Spaces 3.5.1.1 Concept of Inner Product Spaces 3.5.1.2 Orthogonality 3.5.1.3 Continuity 3.5.2 Hilbert Spaces 3.6 BOUNDED LINEAR OPERATORS REFERENCES CHAPTER 4 Min-Plus Convolution 4.1 CONVENTIONAL CONVOLUTION 4.2 MIN-PLUS CONVOLUTION 4.3 IDENTITY IN THE MIN-PLUS CONVOLUTION 4.4 PROBLEM STATEMENTS 4.5 EXISTENCE OF MIN-PLUS DE-CONVOLUTION 4.5.1 Preliminaries 4.5.2 Proof of Existence 4.6 THE CONDITION OF THE EXISTENCE OF MIN-PLUS DE-CONVOLUTION 4.7 REPRESENTATION OF THE IDENTITY IN MIN-PLUS CONVOLUTION REFERENCES CHAPTER 5 Noise and Systems of Fractional Order 5.1 DERIVATIVES AND INTEGRALS OF FRACTIONAL ORDER 5.2 MIKUSINSKI OPERATOR OF FRACTIONAL ORDER 5.3 FRACTIONAL DERIVATIVES: A CONVOLUTION VIEW 5.4 FRACTIONAL ORDER DELTA FUNCTION 5.5 LINEAR SYSTEMS DRIVEN BY FRACTIONAL NOISE 5.6 FRACTIONAL SYSTEMS DRIVEN BY NON-FRACTIONAL NOISE 5.7 FRACTIONAL SYSTEMS DRIVEN BY FRACTIONAL NOISE REFERENCES CHAPTER 6 Fractional Gaussian Noise and Traffic Modeling 6.1 FRACTIONAL GAUSSIAN NOISE 6.2 FRACTIONAL GAUSSIAN NOISE IN TRAFFIC MODELING 6.3 APPROXIMATION OF THE ACF OF FRACTIONAL GAUSSIAN NOISE 6.4 FRACTAL DIMENSION OF FRACTIONAL GAUSSIAN NOISE 6.5 PROBLEM STATEMENTS REFERENCES CHAPTER 7 Generalized Fractional Gaussian Noise and Traffic Modeling 7.1 GENERALIZED FRACTIONAL GAUSSIAN NOISE 7.2 TRAFFIC MODELING USING GENERALIZED FRACTIONAL GAUSSIAN NOISE 7.3 APPROXIMATION OF THE ACF OF GENERALIZED FRACTIONAL GAUSSIAN NOISE 7.4 FRACTAL DIMENSION OF GENERALIZED FRACTIONAL GAUSSIAN NOISE REFERENCES CHAPTER 8 Generalized Cauchy Process and Traffic Modeling 8.1 MEANING OF GENERALIZED CAUCHY PROCESS IN THE BOOK 8.2 HISTORICAL VIEW 8.3 GC PROCESS 8.4 TRAFFIC MODELING USING THE GC PROCESS REFERENCES CHAPTER 9 Traffic Bound of Generalized Cauchy Type 9.1 PROBLEM STATEMENTS AND RESEARCH AIM 9.2 UPPER BOUND OF THE GENERALIZED CAUCHY PROCESS 9.3 DISCUSSIONS REFERENCES CHAPTER 10 Fractal Traffic Delay Bounds 10.1 BACKGROUND 10.2 FRACTAL DELAY BOUNDS 10.2.1 Fractal Delay Bound 1 10.2.2 Fractal Delay Bound 2 10.2.3 Fractal Delay Bound 3 10.2.4 Fractal Delay Bound 4 10.3 DISCUSSION REFERENCES CHAPTER 11 Computations of Scale Factors 11.1 BACKGROUND 11.2 PROBLEM STATEMENT 11.3 RESEARCH THOUGHTS FOR PROBLEM SOLVING 11.3.1 Idea 1 11.3.2 Idea 2 11.4 RESULTS 11.4.1 Computation Formulas of r and a 11.4.2 Asymptotic Computation Formulas of r and a 11.5 CASE STUDY 11.5.1 Traffic Data 11.5.2 Computations of r[sub(0min)] and a[sup(-1)][sub(min)] of Traffic Traces 11.5.2.1 Computations of σ and ρ of Traffic Traces 11.5.2.2 Values of r[sub(0min)] and a[sup(-1)][sub(min)] of Traffic Traces 11.6 APPLICATIONS 11.6.1 Physical Meaning of Asymptotic Scale Factors 11.6.2 Applications 11.6.2.1 Approximations of Traffic Bound 11.6.2.2 Applications to Fractal Delay Bounds 11.7 CONCLUDING REMARKS REFERENCES CHAPTER 12 Postscript 12.1 LOCAL VERSUS GLOBAL OF FRACTAL TIME SERIES 12.2 LOCAL VERSUS GLOBAL OF TRAFFIC TIME SERIES 12.3 PROBLEM REFERENCES INDEX