1. 4. Overview of Probability Network Performance and Quality of Service

2. Motivation Provide a brief review of topics that will help us:  Statistically characterize network traffic flow  Model and estimate performance parameters Set stage for discussion of traffic management and routing later in the course NOT a condensed class in probability theory RQ122

3. Definitions of Probability Theory Probability is concerned with assignment of numbers to events. Pr[A] of an event A is a number between 0 and 1 that corresponds to the likelihood that the event A will occur. There are a number definitions of probability, we will discuss only three 1. Classical Definition 2. Relative Frequency Definition 3. Axiomatic Definition RQ123

4. Classical Definition If a random experiment (process with an uncertain outcome) can result in N mutually exclusive and equally likely outcomes, and if N A is the number of outcomes in which event A occurs, then the probability of A is Pr[A] = RQ124 NANNAN

5. Classical Definition Example: If we roll a die …  There are 6 equally likely outcomes i.e. N=6  There are three outcomes that correspond to the event [even].  In this case, Pr[even] = = = 0.5 Example: If we roll two dice …  There are 36 equally likely outcomes (6x6)  The probability that the sum is 7 is. RQ125 NANNAN 3636 6 36

6. Classical Definition What if N is not finite?  In that case, the Classical definition is not applicable. What if the outcomes are not equally likely?  Again, the Classical definition of probability is not applicable. In such cases, how might we define the probability of an outcome that has event A? RQ126

7. Relative Frequency Definition If a random experiment is repeated a large number of times, say n times, under identical conditions and if an event A is observed to occur n A times, then the probability of A is Pr[A] = The foundation of this approach is that there is some Pr[A]. We cannot deduce it, as in Classical probability, but we can estimate it. RQ127 lim n   nAnnAn

8. Relative Frequency Definition Example:  one tosses a coin, which might or might not be fair, 100 times and observes heads on 52 of the tosses.  One’s estimate of the probability of a head is Pr[head] ≈ or 0.52 RQ128 52 10 0

9. Axiomatic Definition The axiomatic approach build up probability theory from a number of assumptions (axioms). From these axioms, laws of probability are derived that can be used for calculations. RQ129 1.0  Pr[A]  1 for each even A 2.Pr[  ] = 1 3.Pr[A  B] = Pr[A] + Pr[B] if A and B are mutually exclusive Common Axioms:

10. Axiomatic Definition RQ1210 1.Pr[A] = 1 - Pr[A] 2.Pr[A  B] = 0 (if A and B are mutually exclusive) 3.Pr[A  B] = Pr[A] + Pr[B] – Pr[A  B] 4.Pr[A  B  C] = Pr[A] + Pr[B] + Pr[C] – Pr[A  B] – Pr[A  C] – Pr[B  C] + Pr[A  B  C] Important Laws:

11. Axiomatic Definition Example: If we roll a die …  If we assume that each of the 6 outcomes are equally likely, probability of each will be ⅙.  Pr[even] = Pr[2] + Pr[4] + Pr[6] = ½  Pr[less than 3] = Pr[1] + Pr[2] = ⅓  Pr[{even} U {less than 3}] = Pr[even] + Pr[less than 3] – Pr[2] = ½ + ⅓ – ⅙ = ⅔ RQ1211

12. Conditional Probability The conditional probability of an event A, given that event B has occurred is: Pr[AB] ≅ Pr[A  B] ≅ Pr[A and B] A and B are independent events if Pr[A  B] = Pr[A]Pr[B] Pr[A  B] = Pr[B] Pr[AB] RQ1212

13. Conditional Probability Example: What is the probability of getting a sum of 8 on the roll of two dice if we know that the face of at least one die is an even number?  Let, A = [sum of 8], B = [at least 1 die even] Pr[A | B] = = = RQ1213 Pr[B] Pr[AB] 1/12 ¾ 1919

14. Total Probability Given a set of mutually exclusive events E 1, E 2, …, E n covering all possible outcomes, and Given an arbitrary event A, then: Pr[A] = ∑ Pr[A  E i ]Pr[E i ] n i = 1 RQ1214

15. Bayes’s Theorem “Posterior odds” – the probability that an event really occurred, given evidence in favor of it: Pr[E i  A] = Pr[A  E i ] Pr[E i ] Pr[A] = n i = 1  Pr[A  E i ]Pr[E i ] RQ1215

16. Bayes’s Theorem Example Hit & run accident involving a taxi 85% of taxis are yellow, 15% are black Eyewitness reported that the taxi involved in the accident was black Data shows that eyewitnesses are correct on car color 80% of the time What is the probability that the cab was black? Pr[Black|WB] = Pr[WB|Black] Pr[Black] Pr[WB|Black] Pr[Black] + Pr[WB|Yellow] Pr[Yellow] = (0.8)(0.15) (0.8)(0.15) + (0.2)(0.85) = 0.41 RQ1216

17. Network injects errors (flips bits) Assume Pr[S1] = p = Pr[S0] = 1-p = 0.5 Assume Pr[R1] = Pr[R0] = 0.5 Given error injection, such that Pr[R0  S1] =p a and Pr[R1  S0] =p b, then : Pr[S1  R0] = Pr[R0  S1] Pr[S1] Pr[R0  S1] Pr[S1] + Pr[R0  S0] Pr[S0] p a p p a p + (1-p b )(1-p) = Sender S Receiver R Error Injection Bayes’s Theorem Example RQ1217

18. Random Variables A random variable is a variable whose possible values are numerical outcomes of a random phenomenon.  As opposed to other variables, a random variable conceptually does not have a single, fixed value; rather, it can take on a set of possible different values (each with an associated probability). There are two types of random variables, discrete and continuous. RQ1218

19. Random Variables Examples: 1. Select a soccer player; X = the number of goals the player has scored during the season. The values of X are 0, 1, 2, 3,... 2. Survey a group of 10 soccer players; Y = the average number of goals scored by the players during the season. The values of Y are 0, 0.1, 0.2,....,1.0, 1.1, … RQ1219

20. Random Variables A discrete random variable can take on only specific, isolated numerical values.  e.g. number of packets dropped during transmission A continuous random variable is one which takes an infinite number of possible values.  e.g. delay experienced by packets during transmission RQ1220

21. Random Variables Discrete random variables are described by a probability function P x (k) = Pr[X=k] Continuous random variables can be described by either a distribution function or a density function. Random variable characteristics:  Mean value: E[X]  Second moment: E[X 2 ]  Variance: Var[X] = E[X 2 ] - E[X] 2  Standard deviation:  X = (Var[X]) ½ RQ1221

22. Cumulative Distribution Function The Cumulative Distribution Function (CDF) of a random variable maps a given value a to the probability of the variable taking a value less than or equal to a: F X (a) = Pr[X ≤ a] RQ1222

23. Probability Density Function The above derivative of the CDF F(x) is called the probability density function of x. Given a pdf f(x), the probability of x being in the interval (x 1, x 2 ) can also be computed by integration: RQ1223

24. Mean and Variance Mean or Expected Value Variance: RQ1224

25. Probability Distributions F(x) = Pr[X  x] = 1 – e - x Exponential Distribution Exponential Density E[X] =  X = 1/ f(x) = F(x) = e - xddx RQ1225

26. Probability Distributions F(x) = Pr[X  x] = 1 – e - x Exponential Distribution Exponential Density f(x) = F(x) = e - xddx RQ1226

27. Probability Distributions Poisson Distribution Normal Density Pr[X=k] = e - f(x) = k k! e -(x-  ) 2 /2  2  2  E[X] = Var[X] = RQ1227

28. Probability Distributions – Relevance to Networks 2 Service times of queues (t trans ) in packet switching routers can be effectively modeled as exponential Arrival pattern of packets at a router is often Poisson in nature and, arrival interval is exponential (why?) Central Limit Theorem: the distribution of a very large number of independent RVs is approximately normal, independent of individual distributions RQ1228