# Mathematical Foundations of Computer Networking (Addison-Wesley Professional Computing Series)

Language: English

Pages: 496

ISBN: 0321792106

Format: PDF / Kindle (mobi) / ePub

“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 fundamentals*–*the 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.

Therefore, we can precisely compute the power of a test only in the context of an alternative hypothesis about the state of the world. Unfortunately, in many cases, this is impossible to determine. Therefore, despite its intuitive merit, the power of a test is rarely computed. 86 Chapter 2 Statistics 2.5 Independence and Dependence: Regression and Correlation Thus far, we have studied primarily single variables in isolation. In this section, we study data sets with two variables. In this

carefully justifying why the sample is representative of the chosen underlying population. 2.9.2 Lack of Confidence Intervals in Comparing Results Comparing the performance of two systems simply by comparing the mean values of performance metrics is an all-too-common mistake. The fact that one mean is greater than another is not statistically meaningful and may lead to erroneous conclusions. The simple solution is to always compare confidence intervals rather than means, as described in Section

all vectors that can be expressed as a linear combination of the eigenvectors is called the eigenspace of the matrix. EXAMPLE 3.16: COMPLEX EIGENVALUES Let us compute eigenvalues for the matrix A = characteristic determinant to 0: –O 1 –1 –O 0 1 . We do so by setting the –1 0 = 0 This gives us the characteristic equation 2 O +1 = 0 so that O = r i . This is the spectrum of the matrix, and the spectral radius is 1. We find the eigenvector corresponding to O = i by setting 3.5 Linear

subproblem recursively, and then put the solutions together again to obtain the final answer. Of course, the recursion needs to end in a “small” problem that can be easily solved. Moreover, it is critical that the recursion does not result in a proliferation of subproblems. EXAMPLE 4.8: FIBONACCI COMPUTATION Although not an optimization problem, the Fibonacci sequence clearly demonstrates the meaning of substructure, composition, and the need to limit the number of subproblems. This sequence is

generalize the summation over the entire number line to get f xt = ¦ x W G t – W (EQ 5.10) W = –f This summation is called the convolution of x(t) with G (t) and is denoted 3 xt Gt . In general, the convolution of two discrete functions x(t) and y(t) is defined as f xt yt = ¦ x W y t – W (EQ 5.11) W = –f Note that each value of the convolution of x(t) and y(t) (i.e., at time t) is the result of adding all product pairs x W and y t – W . This is