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Mathematical Foundations of Computer Networking

Book Description

“To design future networks that are worthy of society’s trust, we must put the ‘discipline’ of computer networking on a much stronger foundation. This book rises above the considerable minutiae of today’s networking technologies to emphasize the long-standing mathematical underpinnings of the field.”

–Professor Jennifer Rexford, Department of Computer Science, Princeton University

“This book is exactly the one I have been waiting for the last couple of years. Recently, I decided most students were already very familiar with the way the net works but were not being taught the fundamentalsthe math. This book contains the knowledge for people who will create and understand future communications systems."

–Professor Jon Crowcroft, The Computer Laboratory, University of Cambridge

The Essential Mathematical Principles Required to Design, Implement, or Evaluate Advanced Computer Networks

Students, researchers, and professionals in computer networking require a firm conceptual understanding of its foundations. Mathematical Foundations of Computer Networking provides an intuitive yet rigorous introduction to these essential mathematical principles and techniques.

Assuming a basic grasp of calculus, this book offers sufficient detail to serve as the only reference many readers will need. Each concept is described in four ways: intuitively; using appropriate mathematical notation; with a numerical example carefully chosen for its relevance to networking; and with a numerical exercise for the reader.

The first part of the text presents basic concepts, and the second part introduces four theories in a progression that has been designed to gradually deepen readers’ understanding. Within each part, chapters are as self-contained as possible.

The first part covers probability; statistics; linear algebra; optimization; and signals, systems, and transforms. Topics range from Bayesian networks to hypothesis testing, and eigenvalue computation to Fourier transforms.

These preliminary chapters establish a basis for the four theories covered in the second part of the book: queueing theory, game theory, control theory, and information theory. The second part also demonstrates how mathematical concepts can be applied to issues such as contention for limited resources, and the optimization of network responsiveness, stability, and throughput.

Table of Contents

  1. Title Page
  2. Copyright Page
  3. Contents
  4. Preface
    1. Motivation
    2. Organization
    3. Using This Book
    4. Acknowledgments
  5. 1. Probability
    1. 1.1. Introduction
    2. 1.2. Joint and Conditional Probability
    3. 1.3. Random Variables
    4. 1.4. Moments and Moment Generating Functions
    5. 1.5. Standard Discrete Distributions
    6. 1.6. Standard Continuous Distributions
    7. 1.7. Useful Theorems
    8. 1.8. Jointly Distributed Random Variables
    9. 1.9. Further Reading
    10. 1.10. Exercises
  6. 2. Statistics
    1. 2.1. Sampling a Population
    2. 2.2. Describing a Sample Parsimoniously
    3. 2.3. Inferring Population Parameters from Sample Parameters
    4. 2.4. Testing Hypotheses about Outcomes of Experiments
    5. 2.5. Independence and Dependence: Regression and Correlation
    6. 2.6. Comparing Multiple Outcomes Simultaneously: Analysis of Variance
    7. 2.7. Design of Experiments
    8. 2.8. Dealing with Large Data Sets
    9. 2.9. Common Mistakes in Statistical Analysis
    10. 2.10. Further Reading
    11. 2.11. Exercises
  7. 3. Linear Algebra
    1. 3.1. Vectors and Matrices
    2. 3.2. Vector and Matrix Algebra
    3. 3.3. Linear Combinations, Independence, Basis, and Dimension
    4. 3.4. Using Matrix Algebra to Solve Linear Equations
    5. 3.5. Linear Transformations, Eigenvalues, and Eigenvectors
    6. 3.6. Stochastic Matrices
    7. 3.7. Exercises
  8. 4. Optimization
    1. 4.1. System Modeling and Optimization
    2. 4.2. Introduction to Optimization
    3. 4.3. Optimizing Linear Systems
    4. 4.4. Integer Linear Programming
    5. 4.5. Dynamic Programming
    6. 4.6. Nonlinear Constrained Optimization
    7. 4.7. Heuristic Nonlinear Optimization
    8. 4.8. Exercises
  9. 5. Signals, Systems, and Transforms
    1. 5.1. Background
    2. 5.2. Signals
    3. 5.3. Systems
    4. 5.4. Analysis of a Linear Time-Invariant System
    5. 5.5. Transforms
    6. 5.6. The Fourier Series
    7. 5.7. The Fourier Transform and Its Properties
    8. 5.8. The Laplace Transform
    9. 5.9. The Discrete Fourier Transform and Fast Fourier Transform
    10. 5.10. The Z Transform
    11. 5.11. Further Reading
    12. 5.12. Exercises
  10. 6. Stochastic Processes and Queueing Theory
    1. 6.1. Overview
    2. 6.2. Stochastic Processes
    3. 6.3. Continuous-Time Markov Chains
    4. 6.4. Birth-Death Processes
    5. 6.5. The M/M/1 Queue
    6. 6.6. Two Variations on the M/M/1 Queue
    7. 6.7. Other Queueing Systems
    8. 6.8. Further Reading
    9. 6.9. Exercises
  11. 7. Game Theory
    1. 7.1. Concepts and Terminology
    2. 7.2. Solving a Game
    3. 7.3. Mechanism Design
    4. 7.4. Limitations of Game Theory
    5. 7.5. Further Reading
    6. 7.6. Exercises
  12. 8. Elements of Control Theory
    1. 8.1. Overview of a Controlled System
    2. 8.2. Modeling a System
    3. 8.3. A First-Order System
    4. 8.4. A Second-Order System
    5. 8.5. Basics of Feedback Control
    6. 8.6. PID Control
    7. 8.7. Advanced Control Concepts
    8. 8.8. Stability
    9. 8.9. State Space–Based Modeling and Control
    10. 8.10. Digital Control
    11. 8.11. Partial Fraction Expansion
    12. 8.12. Further Reading
    13. 8.13. Exercises
  13. 9. Information Theory
    1. 9.1. Introduction
    2. 9.2. A Mathematical Model for Communication
    3. 9.3. From Messages to Symbols
    4. 9.4. Source Coding
    5. 9.5. The Capacity of a Communication Channel
    6. 9.6. The Gaussian Channel
    7. 9.7. Further Reading
    8. 9.8. Exercises
  14. Solutions to Exercises
    1. Chapter 1 Probability
    2. Chapter 2 Statistics
    3. Chapter 3 Linear Algebra
    4. Chapter 4 Optimization
    5. Chapter 5 Signals, Systems, and Transforms
    6. Chapter 6 Stochastic Processes and Queueing Theory
    7. Chapter 7 Game Theory
    8. Chapter 8 Elements of Control Theory
    9. Chapter 9 Information Theory
  15. Index