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

2. Introduction – Motivation for Part 3 • 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 Chapter 7 Overview of Probability

3. Chapters 7 Overview of Probability and Stochastic Processes

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

5. Pr[AB] Pr[AB] = Pr[B] Conditional Probability • 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 • A and B are independent events if Pr[AB] = Pr[A]Pr[B] Chapter 7 Overview of Probability

6. Total Probability • Given a set of mutually exclusive events E1, E2, …, En covering all possible outcomes, and • Given an arbitrary event A, then: Pr[A] =  Pr[AEi]Pr[Ei] n i = 1 Chapter 7 Overview of Probability

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

8. Pr[WB|Blue] Pr[Blue] Pr[Blue|WB] = Pr[WB|Blue] Pr[Blue] + Pr[WB|Yellow] Pr[Yellow] (0.8)(0.15) = = 0.41 (0.8)(0.15) + (0.2)(0.85) Bayes’s Theorem Example – “The Juror’s Fallacy” • Hit & run accident involving a taxi • 85% of taxis are yellow, 15% are 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 • What is the probability that the cab was blue? Chapter 7 Overview of Probability

9. Bayes’s Theorem Example • Network injects errors (flips bits) • Assume Pr[S1] = Pr[S0] = p = 0.5 • Assume Pr[R1] = Pr[R0] = (1-p) = 0.5 • Given error injection, such that Pr[R0S1] =pa and Pr[R1S0] =pb, then : Pr[S1R0] = Error Injection Receiver R Sender S pa p Pr[R0S1] Pr[S1] = Pr[R0S1] Pr[S1] + Pr[R0S0] Pr[S0] pa p + (1-pb)(1-p) Chapter 7 Overview of Probability

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

11. Random Variables • Continuous random variables can be described by either a distribution function or a density function • Discrete random variables are described by a probability function Px(k) = Pr[X=k] • Random variable characteristics: • Mean value: E[X] • Second moment: E[X2] • Variance: Var[X] = E[X2] - E[X]2 • Standard deviation:X = Var[X] Chapter 7 Overview of Probability

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

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

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

15. Probability Distributions – Relevance to Networks 2 • Service times of queues (ttrans) 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 Chapter 7 Overview of Probability