Mathematical Foundations of Computer Networking (Addison-Wesley Professional Computing Series)
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“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.
transmitted exactly one, two, and three times? Solution: Assuming that the packet transmissions are independent events, we note that 0 the probability of success = p = 0.9. Therefore, p(1) = 0.1 * 0.9 = 0.9; p(2) = 1 2 0.1 * 0.9 = 0.09; p(3) = 0.1 * 0.9 = 0.009. Note the rapid decrease in the probability of more than two transmissions, even with a fairly high packet loss rate of 10%. Indeed, the expected number of transmissions is only 1/0.9 = 1.1 . 1.5.4 Poisson Distribution The Poisson
collecting samples, it is imperative to pay special attention to outliers, making certain that they are truly statistical aberrations before dismissing them or removing them from the data set. 2.2 Describing a Sample Parsimoniously After gathering a sample, the next step is to describe it parsimoniously, that is, with a few well-chosen statistics. These statistics can be thought to constitute a model of the sample: Each data item in the sample can be viewed as arising partly from this model and
count for each number of arrivals. What is the chisquared variate value for this reduced data set? Use this to determine whether the Poisson distribution is indeed a good distribution to describe the reduced data set. Number of Packet Arrivals 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Count 18 28 56 105 126 146 164 165 120 103 73 54 23 16 9 5 8. Independence, regression, and correlation A researcher measures the mean uplink bandwidth of ten desktop computers (in
expensive than Gaussian elimination, so Cramer’s theorem is useful mostly to give us insight into the nature of the solution. Cramer’s theorem states that if a system of n linear equations in n variables Ax = b has a nonzero coefficient determinant D = det A, the system has precisely one solution, given by xi = Di/D where Di is determinant of a matrix obtained by substituting b for the ith column in A. Thus, if we know the corresponding determinants, we can directly compute the xis by using this
3.23: DIAGONALIZATION 4 6 . Recall from Example 3.15 that it has two 12 4 1 1 eigenvalues, O = 4 r 6 2 , corresponding to the eigenvectors and . 2 – 2 Consider the matrix A = Chapter 3 13 8 This allows us to write out the matrix X as –1 find that X 1 1 2– 2 Linear Algebra . From Equation 3.13, we 1 = ------------------------------ – 2 – 1 . Therefore, we diagonalize A as – 2 – 2 – 2 1 1 -------------- – 2 – 1 4 6 – 2 2 – 2 1 12 4 1 1 1 = ------2 2– 2 2 1 23 2 –1 6 2 0 = 4+6