By Fayez Gebali (auth.)

This textbook offers the mathematical thought and strategies precious for reading and modeling high-performance worldwide networks, akin to the net. 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 ebook offers the fundamental suggestions for modeling and interpreting those final elements. themes lined contain, yet should not constrained to: Markov chains and queuing research, site visitors modeling, interconnection networks and turn architectures and buffering strategies.

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**Example text**

3 Counting Sample Space Points In many discussions of probability, we need to determine the number of possible outcomes of a given experiment or trial. The multiplication principle, permutations, and combinations, to be discussed in the following sections, will prove useful. 4 The Multiplication Principle The fundamental principle of counting is useful in finding the number of points in the sample space. Suppose that the number of outcomes for doing experiment E1 is x and the number of outcomes for doing experiment E2 is y.

Consider a chance experiment where an event A occurs at a rate events/second. In a small time interval (t ) the probability that the event A occurs is p D t . We chose t so small so that event A occurs at most only once. For example, might indicate the average number of packets arriving at a link per unit time. In that case the variable t will indicate time. might also refer to the average number of bits in error for every 1,000 bits, say. In that case, the variable t would indicate the number of bits under consideration.

133) We need to find fX and EŒX . For that we use the fundamental law of probability. x/ D 0 for jxj > 1: Now we can write: ˇ ˇ ˇ d! 135) and: ˇ ˇ ˇ d! 138) due to the odd symmetry of the function being integrated. 27. 139) Derive Y as a function of X: This example is fundamentally important since it shows how we can find the transformation that allows us to obtain random numbers following a desired pdf given the random numbers for the uniform distribution. This point will be further discussed in the next section.