1. 1 Part Three: Chapters 7-9 Performance Modeling and Estimation

2. Chapter 7 Overview of Probability 2 Introduction – Motivation for Part 3 Provide a brief review of topics that will help us: 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 Set stage for discussion of traffic management and routing later in the course NOT a condensed class in probability theory NOT a condensed class in probability theory

3. 3 Chapters 7 Overview of Probability and Stochastic Processes

4. Chapter 7 Overview of Probability 4 Probability – Axiomatic Definition 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 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] Common Axioms: Important Laws:

5. Chapter 7 Overview of Probability 5 Venn Diagrams Complementation

6. Chapter 7 Overview of Probability 6 Venn Diagrams Intersection

7. Chapter 7 Overview of Probability 7 Venn Diagrams Mutual Exclusivity

8. Chapter 7 Overview of Probability 8 Probability Definitions Relative Frequency Definition: Pr[A] = where n is the number of trials, and n A the number of times event A occurred Classical Definition: Pr[A] = where N is the number of equally likely outcomes and N A is the number of outcomes in which event A occurs lim n ->  nAnAnnnAnAnnn NANANNNANANNN

9. Chapter 7 Overview of Probability 9 Conditional Probability The conditional probability of an event A, given that event B has occurred is: The conditional probability of an event A, given that event B has occurred is: Where Pr[A  B] encompasses all possible outcomes that satisfy both conditions Where Pr[A  B] encompasses all possible outcomes that satisfy both conditions A and B are independent events if Pr[A  B] = Pr[A]Pr[B] A and B are independent events if Pr[A  B] = Pr[A]Pr[B] Pr[A  B] = Pr[B] Pr[A  B]

10. Chapter 7 Overview of Probability 10 Total Probability Given a set of mutually exclusive events E 1, E 2, …, E n covering all possible outcomes, and Given a set of mutually exclusive events E 1, E 2, …, E n covering all possible outcomes, and Given an arbitrary event A, then: Given an arbitrary event A, then: Pr[A] =  Pr[A  E i ]Pr[E i ] Pr[A] =  Pr[A  E i ]Pr[E i ] n i = 1

11. Chapter 7 Overview of Probability 11 Bayes’s Theorem “Posterior odds” – the probability that an event really occurred, given evidence in favor of it: “Posterior odds” – the probability that an event really occurred, given evidence in favor of it: Pr[E i  A] = Pr[E i  A] = Pr[A  E i ] Pr[E i ] Pr[A] = n i = 1  Pr[A  E i ]Pr[E i ]

12. Chapter 7 Overview of Probability 12 Bayes’s Theorem Example – “The Juror’s Fallacy” Hit & run accident involving a taxi Hit & run accident involving a taxi 85% of taxis are yellow, 15% are blue 85% of taxis are yellow, 15% are blue Eyewitness reported that the taxi involved in the accident was blue Eyewitness reported that the taxi involved in the accident was blue Data shows that eyewitnesses are correct on car color 80% of the time Data shows that eyewitnesses are correct on car color 80% of the time What is the probability that the cab was blue? What is the probability that the cab was blue? Pr[Blue|WB] = Pr[WB|Blue] Pr[Blue] Pr[WB|Blue] Pr[Blue] + Pr[WB|Yellow] Pr[Yellow] = (0.8)(0.15) (0.8)(0.15) + (0.2)(0.85) = 0.41

13. Chapter 7 Overview of Probability 13 Bayes’s Theorem Example Network injects errors (flips bits) Network injects errors (flips bits) Assume Pr[S1] = Pr[S0] = p = 0.5 Assume Pr[S1] = Pr[S0] = p = 0.5 Assume Pr[R1] = Pr[R0] = (1-p) = 0.5 Assume Pr[R1] = Pr[R0] = (1-p) = 0.5 Given error injection, such that Pr[R0  S1] =p a and Pr[R1  S0] =p b, then : Given error injection, such that Pr[R0  S1] =p a and Pr[R1  S0] =p b, then : Pr[S1  R0] = 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

14. Chapter 7 Overview of Probability 14 Random Variables Association of real numbers with events, e.g. assigning a value to each outcome of an experiment Association of real numbers with events, e.g. assigning a value to each outcome of an experiment A random variable X is a function that assigns a real number (probability) to every outcome in a sample space, and satisfies the following conditions: A random variable X is a function that assigns a real number (probability) to every outcome in a sample space, and satisfies the following conditions: 1.the set {X  x} is an event for every x 2.Pr[X=  ] = Pr[X = -  ] = 0 Simply put: an RV maps an event space into the domain of positive real numbers. Simply put: an RV maps an event space into the domain of positive real numbers. A random variable can be continuous or discrete A random variable can be continuous or discrete

15. Chapter 7 Overview of Probability 15 Random Variables Continuous random variables can be described by either a distribution function or a density function Continuous random variables can be described by either a distribution function or a density function Discrete random variables are described by a probability function P x (k) = Pr[X=k] Discrete random variables are described by a probability function P x (k) = Pr[X=k] Random variable characteristics: 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]

16. Chapter 7 Overview of Probability 16 Probability Distributions F(x) = Pr[X  x] = 1 – e - x Exponential Distribution Exponential Density E[X] =  X = 1/ f(x) = F(x) = e - x ddx

17. Chapter 7 Overview of Probability 17 Probability Distributions F(x) = Pr[X  x] = 1 – e - x Exponential Distribution Exponential Density f(x) = F(x) = e - x ddx

18. Chapter 7 Overview of Probability 18 Probability Distributions Poisson Distribution Normal Density Pr[X=k] = e - Pr[X=k] = e - f(x) = kk! e -(x-  ) 2 /2  2  2   2  E[X] = Var[X] =

19. Chapter 7 Overview of Probability 19 Probability Distributions – Relevance to Networks 2 Service times of queues (t trans ) in packet switching routers can be effectively modeled as exponential 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 Arrival pattern of packets at a router is often Poisson in nature and, arrival interval is exponential (why?) 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 Central Limit Theorem: the distribution of a very large number of independent RVs is approximately normal, independent of individual distributions

20. Chapter 7 Overview of Probability 20 Multiple Random Variables Independence: Independence: – F(x,y ) = F(x )F(y ), and f(x,y ) = f(x )f(y ) Covariance: Covariance: –Cov(X,Y ) = E[XY ] – E[X ]E[Y ] Correlation coefficient Correlation coefficient –r(X,Y ) = Cov(X,Y ) /  x  y positively correlated: r(X,Y) > 0 positively correlated: r(X,Y) > 0 negatively correlated: r(X,Y) < 0 negatively correlated: r(X,Y) < 0 uncorrelated: r(X,Y) = Cov(X,Y) = 0 uncorrelated: r(X,Y) = Cov(X,Y) = 0

21. Chapter 7 Overview of Probability 21 Stochastic (Random) Processes Family of Random Variables Family of Random Variables –{x(t), t  T }, indexed by parameter t over index set T –index set is typically taken as time dimension –continuous- or discrete-time, t –continuous- or discrete-value, x(t )

22. Chapter 7 Overview of Probability 22 Brownian Motion Processes Stochastic process that describes Random Movement of particles Stochastic process that describes Random Movement of particles –Let B(t) denote displacement in one dimension after time t –Let B(t) – B(s) denote net movement over time interval (s,t) –Then, B(t) – B(s) has normal distribution Brownian probability density function: Brownian probability density function: f B (x,t) = f B (x,t) = e -x 2 /2  2 t  2  t  2  t

23. Chapter 7 Overview of Probability 23 Poisson Counting Process Pr[N(t) = k] = e - t e - t ( t) k k!

24. Chapter 7 Overview of Probability 24 Poisson Increment Process X(t) = N(t + L) – N(t) L