By Fayez Gebali (auth.)

This textbook offers the mathematical thought and methods valuable for studying and modeling high-performance worldwide networks, reminiscent of the web. the 3 major construction blocks of high-performance networks are hyperlinks, switching gear connecting the hyperlinks jointly and software program hired on the finish nodes and intermediate switches. This booklet presents the fundamental strategies for modeling and reading those final parts. issues coated comprise, yet usually are not restricted to: Markov chains and queuing research, site visitors modeling, interconnection networks and turn architectures and buffering strategies.

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30 Individual pmf from a Given Joint pmf Sometimes we want to study an individual random variable even though our random experiment generates multiple RVs. 23. Assume a random experiment that generates two random variables X and Y with the given joint pmf. 32 Correlation 35 Find the individual pmf for X and Y . Are X and Y independent RVs? 31 Expected Value The joint pmf helps us find the expected value of one of the random variables. 24. 23 above, find the expected values for the random variables X and Y .

112) y We say that the two random variables X and Y are orthogonal when rXY D 0. 116) We say that the two random variables X and Y are uncorrelated when cXY D 0. 118) We expect that the correlation coefficient would decrease as the value of n becomes large to indicate that the random process “forgets” its past values. 35 Transforming Random Variables Mathematical packages usually have functions for generating random numbers following the uniform and normal distributions only. However, when we are simulating communication systems, we need to generate network traffic that follows other types of distributions.

13 Probability Density Function 13 Fig. x/ will never be negative. It can, however, be zero or even bigger than 1. x/. 14 Probability Mass Function For the case of a discrete random variable the cdf is discontinuous in the shape of a staircase. Therefore, its slope will be zero everywhere except at the discontinuities where it will be infinite. The pdf in the discrete case is called the Probability Mass Function (pmf) [1]. x/. 29) where the expression on the right-hand side indicates the probability that the random variable X has the value x.