part three : chapters 7-9

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Part ThreePart Three: : Chapters 7-9Chapters 7-9

Performance Performance Modeling and Modeling and EstimationEstimation

Chapter 7 Overview of Probability2

Introduction – Motivation Introduction – Motivation for Part 3for Part 3 Provide a Provide a briefbrief review of topics that will review of topics that will

help us: help us: – Statistically characterize network traffic flowStatistically characterize network traffic flow– Model and estimate performance Model and estimate performance

parametersparameters Set stage for discussion of traffic Set stage for discussion of traffic

management and routing later in the management and routing later in the coursecourse

NOTNOT a condensed class in probability a condensed class in probability theorytheory

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Chapters 7Chapters 7

Overview ofOverview ofProbability and Probability and Stochastic Stochastic ProcessesProcesses


Probability DefinitionsProbability DefinitionsRelative Frequency Definition:Relative Frequency Definition:

Pr[A] =Pr[A] =

where where nn is the number of trials, and is the number of trials, and nnAA the the number of times event number of times event AA occurred occurred

Classical Definition:Classical Definition:

Pr[A] =Pr[A] =

where where NN is the number of equally likely is the number of equally likely outcomes and outcomes and NNAA is the number of outcomes is the number of outcomes in which event in which event AA occurs occurs

limlimn -> n ->

nnAA

nn

NNAA

NN


Conditional ProbabilityConditional Probability

The conditional probability of an The conditional probability of an event A, given that event B has event A, given that event B has occurred is:occurred is:

Where Where Pr[APr[AB]B] encompasses all encompasses all possible outcomes that satisfy both possible outcomes that satisfy both conditionsconditions

A and B are A and B are independent independent events if events if Pr[APr[AB] = Pr[A]Pr[B]B] = Pr[A]Pr[B]

Pr[APr[AB] = B] =

Pr[B]Pr[B]

Pr[APr[AB]B]


Total ProbabilityTotal ProbabilityGiven a set of Given a set of

mutually exclusive mutually exclusive eventsevents E E11, E, E22, …, E, …, En n

covering all possible covering all possible outcomes, andoutcomes, and

Given an Given an arbitrary arbitrary eventevent A, then: A, then:

Pr[A] = Pr[A] = Pr[A Pr[AEEii]Pr[E]Pr[Eii]]

nn

i = 1i = 1


Bayes’s Theorem Bayes’s Theorem ““Posterior odds” – the Posterior odds” – the

probability that an probability that an event really occurred, event really occurred, given evidence in given evidence in favor of it:favor of it:

Pr[EPr[EiiA] =A] =

Pr[APr[AEEii] Pr[E] Pr[Eii]]Pr[A]Pr[A] ==

Pr[APr[AEEii] Pr[E] Pr[Eii]]nn

i = 1i = 1 Pr[APr[AEEii]Pr[E]Pr[Eii]]


Bayes’s Theorem Example Bayes’s Theorem Example – “The Juror’s Fallacy”– “The Juror’s Fallacy” Hit & run accident involving a taxiHit & run accident involving a taxi 85% of taxis are yellow, 15% are blue85% of taxis are yellow, 15% are blue Eyewitness reported that the taxi Eyewitness reported that the taxi

involved in the accident was blueinvolved in the accident was blue Data shows that eyewitnesses are Data shows that eyewitnesses are

correct on car color 80% of the timecorrect on car color 80% of the time What is the probability that the cab was What is the probability that the cab was

blue?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


Bayes’s Theorem ExampleBayes’s Theorem Example

Network injects errors (flips bits)Network injects errors (flips bits) Assume Pr[S1] = Pr[S0] = p = Assume Pr[S1] = Pr[S0] = p =

0.50.5 Assume Pr[R1] = Pr[R0] = (1-p) Assume Pr[R1] = Pr[R0] = (1-p)

= 0.5= 0.5 Given error injection, such that Given error injection, such that

Pr[R0Pr[R0S1] =pS1] =pa a and Pr[R1and Pr[R1S0] =pS0] =pbb, , then :then :

Pr[S1Pr[S1R0] =R0] =

Pr[R0Pr[R0S1] Pr[S1]S1] Pr[S1]

Pr[R0Pr[R0S1] Pr[S1] + Pr[R0S1] Pr[S1] + Pr[R0S0] Pr[S0]S0] Pr[S0] ppa a pp

ppa a p + (1-pp + (1-pbb)(1-p))(1-p)==

Sender SSender S Receiver RReceiver R

Error InjectionError Injection


Random VariablesRandom Variables Association of real numbers with Association of real numbers with

events, e.g. assigning a value to each events, e.g. assigning a value to each outcome of an experimentoutcome of an experiment

A A random variablerandom variable XX is a is a functionfunction that that assigns a real number (probability) to assigns a real number (probability) to every outcome in a sample space, and every outcome in a sample space, and satisfies the following conditions:satisfies the following conditions:

1.1. the set {X the set {X x} is an event for every x x} is an event for every x2.2. Pr[X= Pr[X= ] = Pr[X = -] = Pr[X = -] = 0] = 0

Simply put: an RV maps an event Simply put: an RV maps an event space into the domain of positive real space into the domain of positive real numbers.numbers.

A random variable can be A random variable can be continuouscontinuous or or discretediscrete


Random VariablesRandom Variables Continuous random variables can Continuous random variables can

be described by either a be described by either a distribution distribution functionfunction or a or a density functiondensity function

Discrete random variables are Discrete random variables are described by a probability function described by a probability function PPxx(k) = Pr[X=k](k) = Pr[X=k]

Random variable characteristics:Random variable characteristics:– Mean value:Mean value: E[X] E[X]– Second moment:Second moment: E[X E[X22]]– Variance:Variance: Var[X] = E[X Var[X] = E[X22] - E[X]] - E[X]22

– Standard deviation:Standard deviation: XX = Var[X] = Var[X]


Probability DistributionsProbability Distributions

F(x) = Pr[XF(x) = Pr[Xx] = 1 – ex] = 1 – e--xx

Exponential DistributionExponential Distribution Exponential DensityExponential Density

E[X] = X = 1/

f(x) = F(x) = f(x) = F(x) = e e --

xxdd

dxdx



F(x) = Pr[XF(x) = Pr[Xx] = 1 – ex] = 1 – e--xx

Exponential DistributionExponential Distribution Exponential DensityExponential Density

f(x) = F(x) = f(x) = F(x) = e e --

xxdd

dxdx

http://upload.wikimedia.org/wikipedia/commons/7/77/Exponential_distribution_cdf.png

http://upload.wikimedia.org/wikipedia/commons/b/b1/Exponential_distribution_pdf.png



Poisson DistributionPoisson Distribution Normal DensityNormal Density

Pr[X=k] = ePr[X=k] = e-- f(x) =f(x) =kk

k!k!ee-(x--(x-))22/2/222

22

E[X] = Var[X] =


Probability Distributions – Probability Distributions – Relevance to Networks 2Relevance to Networks 2 Service times of queues (tService times of queues (ttranstrans) in ) in

packet switching routers can be packet switching routers can be effectively modeled as effectively modeled as exponentialexponential

Arrival pattern of packets at a Arrival pattern of packets at a router is often router is often Poisson Poisson in naturein nature

and, arrival interval is exponential (and, arrival interval is exponential (why?why?)) Central Limit TheoremCentral Limit Theorem: the : the

distribution of a very large number distribution of a very large number of independent RVs is of independent RVs is approximately approximately normalnormal, , independent of individual independent of individual distributionsdistributions

part three : chapters 7-9

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