By Fayez Gebali (auth.)

This textbook offers the mathematical conception and methods valuable for studying and modeling high-performance international networks, reminiscent of the net. the 3 major construction 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 presents the elemental recommendations for modeling and examining those final parts. issues coated contain, yet are usually not restricted to: Markov chains and queuing research, site visitors modeling, interconnection networks and turn architectures and buffering strategies.

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**Example text**

We have n D 50 and k D 4: ! 8 Probability We define probability using the relative-frequency approach. Suppose we perform an experiment like the tossing of a coin for N times. We define event A is when the coin lands head up. Define NA as the number of times that event A occurs when the coin tossing experiment is repeated N times. 1 NA N This equation defines the relative frequency that event A happens. 9 Axioms of Probability We defined our sample space S as the set of all possible outcomes of an experiment.

According to references [2–5], a random variable is simply a numerical description of the outcome of a random experiment. We are free to choose the function that maps or assigns a numerical value to each outcome depending on the situation at hand. Later we shall see that the choice of this function is rather obvious in most situations. 3 graphically shows the steps leading to assigning a numerical value to the outcome of a random experiment. First we run the experiment then we observe the resulting outcome.

6. 95. Find the joint pmf of X and Y . 1; 1/ D 0:1 0:95 D 0:095 Note that the sum of all the probabilities must add up to 1. 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.