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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. 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. Probability Joint and Conditional Probability. Random Variables. Moments and Moment Generating Functions. Standard Discrete Distributions. Standard Continuous Distributions. Useful Theorems. Jointly Distributed Random Variables. Bayesian Networks. Further Reading. Exercises. Statistics Sampling a Population. Describing a Sample Parsimoniously Inferring Population Parameters from Sample Parameters. Testing Hypotheses about Outcomes of Experiments. Independence and Dependence: Regression and Correlation. Comparing Multiple Outcomes Simultaneously: Analysis of Variance. Design of Experiments. Dealing with Large Data Sets. Common Mistakes in Statistical Analysis. Further Reading. Exercises. Linear Algebra Vectors and Matrices. Vector and Matrix Algebra. Linear Combinations, Independence, Basis, and Dimension. Using Matrix Algebra to Solve Linear Equations. Linear Transformations, Eigenvalues, and Eigenvectors. Stochastic Matrices. Exercises. Optimization System Modeling and Optimization. Introduction to Optimization. Optimizing Linear Systems. Integer Linear Programming. Dynamic Programming. Nonlinear Constrained Optimization. Heuristic Nonlinear Optimization. Exercises. Signals, Systems, and Transforms Background. Signals. Systems. Analysis of a Linear Time-Invariant System. Transforms. The Fourier Series. The Fourier Transform and Its Properties. The Laplace Transform. The Discrete Fourier Transform and Fast Fourier Transform. The Z Transform. Further Reading. Exercises. Stochastic Processes and Queueing Theory Overview. Stochastic Processes. Continuous-Time Markov Chains. Birth-Death Processes. The M/M/1 Queue. Two Variations on the M/M/1 Queue. Other Queueing Systems. Further Reading. Exercises. Game Theory Concepts and Terminology. Solving a Game. Mechanism Design. Limitations of Game Theory. Further Reading. Exercises. Elements of Control Theory Overview of a Controlled System. Modeling a System. A First-Order System. A Second-Order System. Basics of Feedback Control. PID Control. Advanced Control Concepts. Stability. State Space–Based Modeling and Control. Digital Control. Partial Fraction Expansion. Further Reading. Exercises. Information Theory A Mathematical Model for Communication. From Messages to Symbols. Source Coding. The Capacity of a Communication Channel. The Gaussian Channel. Further Reading. Exercises. Solutions to Exercises