Computer communication and networks ppt

Network Layer Performance Modeling & Analysis and data communication and computer networks ppt
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Dr.ShivJindal,India,Teacher
Published Date:19-07-2017
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Computer Communication Networks (CCN) Network Layer Performance Modeling & Analysis 1Overview • Network Layer Performance Modeling & Analysis – Part I: Essentials of Probability – Part II: Inside a Router – Part III: Network Analysis Rensselaer Polytechnic Institute © Shivkumar Kalvanaraman & © Biplab Sikdar 2Network Layer Performance Modeling & Analysis: Part I Essential of Probability • Motivation • Basic Definitions • Modeling Experiments with Uncertainty • Random Variables: Geometric, Poisson, Exponential Rensselaer Polytechnic Institute © Shivkumar Kalvanaraman & © Biplab Sikdar 3Network Layer Performance Modeling & Analysis: Part I Essential of Probability • Read any of the probability references, e.g. Ross, Molloy, Papoulis, Stark and Wood • Check out the WWW version of the notes: http://networks.ecse.rpi.edu/vas tola/pslinks/perf/node1.html Rensselaer Polytechnic Institute © Shivkumar Kalvanaraman & © Biplab Sikdar 4Motivation for learning Probability in CCN Rensselaer Polytechnic Institute © Shivkumar Kalvanaraman & © Biplab Sikdar 5Basic Definitions • Think of Probability as modeling an experiment • The Set of all possible outcomes is the sample space: S • Classic “Experiment”: Tossing a die: S = 1,2,3,4,5,6 • Any subset A of S is an event: A = the outcome is even = 2,4,6 Rensselaer Polytechnic Institute © Shivkumar Kalvanaraman & © Biplab Sikdar 6Basic Operation of Events • For any two events A, B the following are also events: A A complement = all possible outcomes not in A AB  A union B = all outcomes in A or B or both AB  A intersect B = all outcomes in A and B • Note , the empty set. S   • If AB , then A and B are mutually exclusive. Rensselaer Polytechnic Institute © Shivkumar Kalvanaraman & © Biplab Sikdar 7Basic Operation of Events • Can take many unions: A A... A 12 n • Or even infinite unions:  A A... A 12 n n1 • Ditto for intersections Rensselaer Polytechnic Institute © Shivkumar Kalvanaraman & © Biplab Sikdar 8Probability of Events •P is the Probability Mass function if it maps each event A, into a real number P(A), and: i.) P(A) 0 for every event A S ii.) P(S) = 1 iii.)If A and B are mutually exclusive events then, P(AB) P(A) P(B) Rensselaer Polytechnic Institute © Shivkumar Kalvanaraman & © Biplab Sikdar 9Probability of Events …In fact for any sequence of pair- wise-mutually-exclusive events A ,A ,A ,... (i.e. A A 0 for any i j) we 1 2 3 ij have   P A P() A nn nn  11  Rensselaer Polytechnic Institute © Shivkumar Kalvanaraman & © Biplab Sikdar 10Other Properties • P(A) 1 P(A) • PA ( ) 1 P(AB) P(A) P(B) P(AB) • • A B P(A) P(B) Rensselaer Polytechnic Institute © Shivkumar Kalvanaraman & © Biplab Sikdar 11Conditional Probability P(A B) • = (conditional) probability that the outcome is in A given that we know the outcome in B P() AB P(A B) P(B) 0 PB () •Example: Toss one die. Pi ( 3 i is odd)= •Note that: P(AB) P(B)P(A B) P(A)P(B A) Rensselaer Polytechnic Institute © Shivkumar Kalvanaraman & © Biplab Sikdar 12Independence • Events A and B are independent if P(AB) = P(A)P(B). • Example: A card is selected at random from an ordinary deck of cards. A=event that the card is an ace. B=event that the card is a diamond. P() AB PA () PB () P(A)P(B) Rensselaer Polytechnic Institute © Shivkumar Kalvanaraman & © Biplab Sikdar 13Independence • Event A and B are independent if P(AB) = P(A) P(B). • Independence does NOT mean that A and B have “nothing to do with each other” or that A and B “having nothing in common”. Rensselaer Polytechnic Institute © Shivkumar Kalvanaraman & © Biplab Sikdar 14Independence • Best intuition on independence is: A and B are independent if and only if (equivalently, ) P(B A) P(B) P(A B) P(A) i.e. if and only if knowing that B is true doesn’t change the probability that A is true. •Note: If A and B are independent and mutually exclusive, then P(A)=0 or P(B) = 0. Rensselaer Polytechnic Institute © Shivkumar Kalvanaraman & © Biplab Sikdar 15Random Variables • A random variable X maps each outcome s in the sample space S to a real number X(s). • Example: A fair coin is tossed 3 times. S=(TTT),(TTH),(THT),(HTT),(HHT),(HTH),(TH H),(HHH) Rensselaer Polytechnic Institute © Shivkumar Kalvanaraman & © Biplab Sikdar 16Random Variables • Let X be the number of heads tossed in 3 tries. X(TTT)= X(HHT)= X(TTH)= X(HTH)= X(THT)= X(THH)= X(HTT)= X(HHH)= • So P(X=0)= P(X=1)= P(X=2)= P(X=3)= Rensselaer Polytechnic Institute © Shivkumar Kalvanaraman & © Biplab Sikdar 17Random Variable as a Measurement • Think of much more complicated experiments – A chemical reaction. – A laser emitting photons. – A packet arriving at a router. Rensselaer Polytechnic Institute © Shivkumar Kalvanaraman & © Biplab Sikdar 18Random Variable as a Measurement • We cannot give an exact description of a sample space in these cases, but we can still describe specific measurements on them – The temperature change produced. – The number of photons emitted in one millisecond. – The time of arrival of the packet. Rensselaer Polytechnic Institute © Shivkumar Kalvanaraman & © Biplab Sikdar 19Random Variable as a Measurement • Thus a random variable can be thought of as a measurement on an experiment Rensselaer Polytechnic Institute © Shivkumar Kalvanaraman & © Biplab Sikdar 20