By Fayez Gebali (auth.)

This textbook provides the mathematical idea and strategies precious for reading and modeling high-performance international networks, reminiscent of the web. 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 e-book offers the elemental recommendations for modeling and examining those final elements. subject matters coated contain, yet will not be constrained to: Markov chains and queuing research, site visitors modeling, interconnection networks and turn architectures and buffering strategies.

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Obviously the random variable ‚ is continuous since the pointer could point at any angle. The range of values for Â is between 0ı and 360ı . Â / has the following extreme values: F‚ 0ı D p Â Ä 0ı D 0 F‚ 360ı D p Â Ä 360ı D 1 There is no preference for the pointer to settle at any angle in particular and the CDF will have the distribution shown in Fig. 4. 12 1 Probability Fig. 20). The CDF for a discrete random variable will be a staircase as illustrated in the following example. 13. Consider again the case of the spinning pointer experiment but define the discrete random variable Q which identifies the quadrant in which the pointer rests in.

Assigning a numerical value to the outcome of a random experiment allows us to develop uniform analysis for many types of experiments independent of the nature of their specific outcomes [2]. We denote a random variable by a capital letter (the name of the function) and any particular value of the random variable is denoted by a lowercase letter (the value of the function). Examples of random variables, and their numerical values, could be: 1. Number of arriving packets at a given time instance is an example of a discrete random variable N with possible values n D 0; 1; 2; .

7. Packets arriving at a terminal could be one of three possible service classes: class A, class B, or class C. Assume that we received 10 packets and we found out that there were 2 packets in class A, 5 in class B, and 3 in class C. How many possible service class arrival order could we have received? We are not interested here in the sequence of received packets. Instead, we are interested only in the arrival order of the service classes. We have n1 D 2, n2 D 5, and n3 D 3 such that n D 10. The number of service class patterns is: xD 10Š D 2; 520 2Š 5Š 3Š In other words, there are 10 possibilities for receiving 10 packets such that exactly two of them belonged to class A, five belonged to class B, and three belonged to class C.