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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 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN CRC Press is an imprint of Taylor & Francis Group, LLC © 2022 Ming Li Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www. copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. 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