By Fayez Gebali (auth.)

This textbook provides the mathematical conception and methods precious for examining and modeling high-performance worldwide networks, akin to the net. the 3 major development blocks of high-performance networks are hyperlinks, switching apparatus connecting the hyperlinks jointly and software program hired on the finish nodes and intermediate switches. This publication offers the elemental ideas for modeling and interpreting those final parts. issues lined comprise, yet aren't constrained to: Markov chains and queuing research, site visitors modeling, interconnection networks and turn architectures and buffering strategies.

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6 Continuous and discrete random variables. (a) pdf for the continuous case. 15 Expected Value and Variance The pdf and pmf we obtained above help us find the expected value E ŒX of a random variable X . 33) i The expectation is sometimes referred to as the first moment of the random variable. 34) The mean is not exactly equal to the expected value since m changes its value depending on how many samples we take. However, as n ! 1, the two quantities become equal [1]. 35) The variance describes how much of the mass of the distribution is close to the expected value.

The atmospheric pressure at a given location is an example of a random variable P . This random variable could be discrete or continuous depending on the accuracy of the barometer (analog or digital). x/ and is defined as the probability that the random variable is less than or equal to x. 14) The subscript X denotes the random variable associated with the function while the argument x denotes a numerical value. x/ when we are dealing with a single random variable and there is no chance of confusion.

21 Rayleigh Distribution The Rayleigh distribution is used to model the phenomenon of fading in a wireless communication channel. 54) where a is the shape parameter. 12a shows the Rayleigh pdf distribution for three different values of the shape parameter a. 2a2 / This can be proven using the formulas in Appendix A. 2 0 0 0 2 4 6 8 10 0 x 2 4 6 8 10 x Fig. 12 The Rayleigh distribution for three different values of the shape parameter a. (a) Probability density function (pdf). 22 Common Discrete Distributions We discuss in the following sections some discrete random variables that are useful for network simulations.