# Mathematical Foundations of Computer Networking,1st edition

• Srinivasan Keshav
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Mathematical techniques pervade current research in computer networking, yet are not taught to most computer science undergraduates. This self-contained, highly-accessible book bridges the gap, providing the mathematical grounding students and professionals need to successfully design or evaluate networking systems. The only book of its kind, it brings together information previously scattered amongst multiple texts. It first provides crucial background in basic mathematical tools, and then illuminates the specific theories that underlie computer networking. Coverage includes: * Basic probability * Statistics * Linear Algebra * Optimization * Signals, Systems, and Transforms, including Fourier series and transforms, Laplace transforms, DFT, FFT, and Z transforms * Queuing theory * Game Theory * Control theory * Information theory

Â·Â Â Â  Brings together the math background needed to understand the latest networking research, and design or evaluate real networking systems

Â·Â Â Â  Includes modular, easy-to-understand introductions to probability, statistics, linear algebra, optimization, signals, systems, and transforms

Â·Â Â Â  Demystifies modern queuing, game, control, and information theories

Â·Â Â Â Â Â There will also be a set of homework exercises, in addition to those in the text, available to instructors, with separate solutions, for five chapters.

Â·Â Â Â  Roughly 40 hours of video are already on YouTube in the University of Waterloo channel with an additional 45 hours available at the end of 2011 at:Â  http://www.youtube.com/playlist?list=PL5216DFFFAEB1A6BB

Preface xv

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Chapter 1: Probability 1

1.1 Introduction 1

1.2 Joint and Conditional Probability 7

1.3 Random Variables 14

1.4 Moments and Moment Generating Functions 21

1.5 Standard Discrete Distributions 25

1.6 Standard Continuous Distributions 29

1.7 Useful Theorems 35

1.8 Jointly Distributed Random Variables 42

1.8.1 Bayesian Networks 44

1.10 Exercises 47

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Chapter 2: Statistics 53

2.1 Sampling a Population 53

2.2 Describing a Sample Parsimoniously 57

2.3 Inferring Population Parameters from Sample Parameters 66

2.4 Testing Hypotheses about Outcomes of Experiments 70

2.5 Independence and Dependence: Regression and Correlation 86

2.6 Comparing Multiple Outcomes Simultaneously: Analysis of Variance 95

2.7 Design of Experiments 99

2.8 Dealing with Large Data Sets 100

2.9 Common Mistakes in Statistical Analysis 103

2.11 Exercises 105

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Chapter 3: Linear Algebra 109

3.1 Vectors and Matrices 109

3.2 Vector and Matrix Algebra 111

3.3 Linear Combinations, Independence, Basis, and Dimension 114

3.4 Using Matrix Algebra to Solve Linear Equations 117

3.5 Linear Transformations, Eigenvalues, and Eigenvectors 125

3.6 Stochastic Matrices 138

3.7 Exercises 143

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Chapter 4: Optimization 147

4.1 System Modeling and Optimization 147

4.2 Introduction to Optimization 149

4.3 Optimizing Linear Systems 152

4.4 Integer Linear Programming 157

4.5 Dynamic Programming 162

4.6 Nonlinear Constrained Optimization 164

4.7 Heuristic Nonlinear Optimization 167

4.8 Exercises 170

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Chapter 5: Signals, Systems, and Transforms 173

5.1 Background 173

5.2 Signals 185

5.3 Systems 188

5.4 Analysis of a Linear Time-Invariant System 189

5.5 Transforms 195

5.6 The Fourier Series 196

5.7 The Fourier Transform and Its Properties 200

5.8 The Laplace Transform 209

5.9 The Discrete Fourier Transform and Fast Fourier Transform 216

5.10 The Z Transform 226

5.12 Exercises 234

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Chapter 6: Stochastic Processes and Queueing Theory 237

6.1 Overview 237

6.2 Stochastic Processes 240

6.3 Continuous-Time Markov Chains 252

6.4 Birth-Death Processes 255

6.5 The M/M/1 Queue 262

6.6 Two Variations on the M/M/1 Queue 266

6.7 Other Queueing Systems 270

6.9 Exercises 272

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Chapter 7: Game Theory 277

7.1 Concepts and Terminology 278

7.2 Solving a Game 291

7.3 Mechanism Design 301

7.4 Limitations of Game Theory 314

7.6 Exercises 316

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Chapter 8: Elements of Control Theory 319

8.1 Overview of a Controlled System 320

8.2 Modeling a System 323

8.3 A First-Order System 329

8.4 A Second-Order System 331

8.5 Basics of Feedback Control 336

8.6 PID Control 341

8.8 Stability 350

8.9 State Spaceâ€“Based Modeling and Control 360

8.10 Digital Control 364

8.11 Partial Fraction Expansion 367

8.13 Exercises 370

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Chapter 9: Information Theory 373

9.1 Introduction 373

9.2 A Mathematical Model for Communication 374

9.3 From Messages to Symbols 378

9.4 Source Coding 379

9.5 The Capacity of a Communication Channel 386

9.6 The Gaussian Channel 399

9.8 Exercises 407

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Solutions to Exercises 411

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Index 457

Srinivasan Keshav is a Professor and a Canada Research Chair at the David R. Cheriton School of Computer Science, University of Waterloo, Ontario,Â Canada.

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