Fractal Teletraffic Modeling and Delay Bounds in Computer Communications

239 Pages·2022·5.123 MB·English

by  Ming Li

Fractal Teletraffic
Modeling and Delay
Bounds in Computer
Communications

Fractal Teletraffic
Modeling and Delay
Bounds in Computer
Communications
Ming Li
First edition published 2022
by CRC Press
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© 2022 Ming Li
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ISBN: 978-1-032-21286-9 (hbk)
ISBN: 978-1-032-21526-6 (pbk)
ISBN: 978-1-003-26880-2 (ebk)
DOI: 10.1201/9781003268802
Typeset in Minion
by codeMantra
To Dr. Yonglang Zhang and Joanna Jiayue Li —
for Making It Both Possible and Worthwhile

Contents
Preface, xiii
Acknowledgments, xv
Author, xvii
Chapter 1 ◾ T ime Series 1
1.1 RANDOM PROCESSES 1
1.2 ERGODICITY 3
1.3 PROBABILITY DENSITY FUNCTION 3
1.3.1 PDF of Brownian Motion 4
1.3.2 Gaussian Distribution 5
1.3.3 Poisson Distribution 5
1.3.4 Cauchy Distribution 6
1.4 CORRELATION FUNCTIONS 7
1.4.1 Auto-Correlation Functions 7
1.4.2 Auto-Covariance Functions 9
1.4.3 Cross-Correlation Functions 10
1.4.4 Cross-Covariance Functions 11
1.5 POWER SPECTRA 12
1.5.1 Power Auto-Spectrum Density Functions 12
1.5.2 Power Cross-Spectrum Density Functions 13
1.6 WHITE NOISE 14
vii
viii ◾ Contents
1.7 RANDOM FUNCTIONS OF INTEREST
IN TRAFFIC THEORY 14
REFERENCES 15
Chapter 2 ◾ F ourier Transform 17
2.1 BASIC IN FOURIER TRANSFORM 17
2.2 DELTA FUNCTION 21
2.3 NASCENT DELTA FUNCTIONS 29
2.4 BASIC PROPERTIES OF FOURIER TRANSFORM 30
2.5 CONVOLUTION 42
REFERENCES 47
Chapter 3 ◾ A pplied Functional 49
3.1 LINEAR SPACES 49
3.1.1 Notion of Linear Spaces 49
3.1.2 Isomorphism of Linear Spaces 50
3.1.3 Subspaces and Affine Manifold 50
3.1.4 Convex Sets 51
3.2 METRIC SPACES 52
3.2.1 Concept of Metric Spaces 52
3.2.2 Limit in Metric Spaces 54
3.2.3 Balls Viewed from Metric Spaces 54
3.2.4 Postscript 56
3.3 LINEAR NORMED SPACES 56
3.3.1 Notion of Norm and Normed Spaces 56
3.3.2 Norms 57
3.3.3 Equivalence of Norms 59
3.3.4 Equivalence of Linear Normed Spaces 61
3.4 BANACH SPACES 61
3.4.1 Concept of Banach Spaces 61
3.4.2 Completeness 62
Contents ◾ ix
3.4.3 Series in Banach Spaces 63
3.4.4 Separable Banach Spaces and Completion of Spaces 64
3.5 HILBERT SPACES 66
3.5.1 Inner Product Spaces 66
3.5.1.1 Concept of Inner Product Spaces 66
3.5.1.2 Orthogonality 68
3.5.1.3 Continuity 68
3.5.2 Hilbert Spaces 69
3.6 BOUNDED LINEAR OPERATORS 71
REFERENCES 73
Chapter 4 ◾ M in-Plus Convolution 75
4.1 CONVENTIONAL CONVOLUTION 75
4.2 MIN-PLUS CONVOLUTION 82
4.3 IDENTITY IN THE MIN-PLUS CONVOLUTION 85
4.4 PROBLEM STATEMENTS 85
4.5 EXISTENCE OF MIN-PLUS DE-CONVOLUTION 87
4.5.1 Preliminaries 87
4.5.2 Proof of Existence 89
4.6 THE CONDITION OF THE EXISTENCE OF MIN-PLUS
DE-CONVOLUTION 91
4.7 REPRESENTATION OF THE IDENTITY IN MIN-PLUS
CONVOLUTION 91
REFERENCES 93
Chapter 5 ◾ Noise and Systems of Fractional Order 95
5.1 DERIVATIVES AND INTEGRALS OF FRACTIONAL
ORDER 95
5.2 MIKUSINSKI OPERATOR OF FRACTIONAL ORDER 100
5.3 FRACTIONAL DERIVATIVES: A CONVOLUTION VIEW 101
5.4 FRACTIONAL ORDER DELTA FUNCTION 103
5.5 LINEAR SYSTEMS DRIVEN BY FRACTIONAL NOISE 104