Probability Review

Thinh Nguyen

Probability Theory Review

Sample space Bayes’ Rule Independence Expectation Distributions

Sample Space - Events

Sample Point The outcome of a random experiment

Sample Space S The set of all possible outcomes Discrete and Continuous

Events A set of outcomes, thus a subset of S Certain, Impossible and Elementary

Set Operations Union Intersection Complement

Properties Commutation

Associativity

Distribution

De Morgan’s Rule

A BA B

A B

CA

CA

A B B A

A B C A B C

A B C A B A C

C C CA B A B

S

A B

Axioms and Corollaries

Axioms If

If A1, A2, … are pairwise exclusive

Corollaries

A B

P A B P A P B

11

k kkk

P A P A

0 P A

1P S 1CP A P A

1P A 0P

P A B

P A P B P A B

Conditional Probability

Conditional Probability of event A given that event B has occurred

If B1, B2,…,Bn a partition of S, then

(Law of Total Probability)

A B

CA

S

A B

|

P A BP A B

P B

B1

B3

B2

A

1 1| ...

| j j

P A P A B P B

P A B P B

Bayes’ Rule

If B1, …, Bn a partition of S then

1

|

|

|

j

j

j j

n

k kk

P A BP B A

P A

P A B P B

P A B P B

likelihood priorposterior

evidence

Event Independence

Events A and B are independent if

If two events have non-zero probability and are mutually exclusive, then they cannot be independent

P A B P A P B

Random Variables

Random Variables

The Notion of a Random Variable The outcome is not always

a number Assign a numerical value to

the outcome of the experiment

Definition A function X which assigns

a real number X(ζ) to each outcome ζ in the sample space of a random experiment

S

x

Sx

ζ

X(ζ) = x

Cumulative Distribution Function Defined as the probability

of the event {X≤x}

Properties XF x P X x

0 1XF x

lim 1Xx

F x

lim 0Xx

F x

if then X Xa b F a F a

X XP a X b F b F a

1 XP X x F x

x

2

1

Fx(x)

¼

½

¾

10 3

1

Fx(x)

x

Types of Random Variables

Continuous Probability Density

Function

Discrete Probability Mass

Function

X k kP x P X x

X X k kk

F x P x u x x

XX

dF xf x

dx

x

X XF x f t dt

Probability Density Function

The pdf is computed from

Properties

For discrete r.v.

dx

fX(x)

XX

dF xf x

dx

b

XaP a X b f x dx

x

X XF x f t dt

1 Xf t dt

fX(x)

XP x X x dx f x dx

x

X X k kk

f x P x x x

Expected Value and Variance

The expected value or mean of X is

Properties

The variance of X is

The standard deviation of X is

Properties

XE X tf t dt

k X kk

E X x P x

E c c

E cX cE X

E X c E X c

22Var X E X E X

Std X Var X

0Var c

2Var cX c Var X

Var X c Var X

Queuing Theory

Example

Send a file over the internetSend a file over the internet

packet link

buffer

Modemcard

(fixed rate)

Delay Models

time

place

A

B

Cp

rop

agat

ion

tran

smis

sion

Com

pu

tati

on(Q

ueu

ing)

Queue Model

Practical Example

Multiserver queue

Multiple Single-server queues

Standard Deviation impact

Queueing Time

Queuing Theory

The theoretical study of waiting lines, expressed in mathematical terms

input output

queue

server

Delay= queue time +service time

The ProblemGiven One or more servers that render the service A (possibly infinite) pool of customers Some description of the arrival and service

processes.

Describe the dynamics of the system Evaluate its Performance

If there is more than one queue for the server(s), there may also

be some policy regarding queue changes for the customers.

Common Assumptions

The queue is FCFS (FIFO). We look at steady state : after the system has

started up and things have settled down.

State=a vector indicating the total # of customers in each queue at a particular time instant

(all the information necessary to completely describe the system)

Notation for queuing systems

A = the interarrival tim e distribution

B = the serv ice tim e distribution

c = the num ber of servers

d = the queue size lim it

A /B /c/d

M for Markovian (exponential) distribution

D for Deterministic distribution

G for General (arbitrary) distribution

:Where A and B can be

omitted if infiniteomitted if infinite

The M/M/1 System

Poisson Process

output

queue

Exponential server

Arrivals follow a Poisson process

a(t) = # of arrivals in time interval [0,t] = mean arrival rate t = k ; k = 0,1,…. ; 0

Pr(exactly 1 arrival in [t,t+]) = Pr(no arrivals in [t,t+]) = 1-Pr(more than 1 arrival in [t,t+]) = 0

Pr(a(t) = n) = e- t ( t)n/n!

Readily amenable for analysisReadily amenable for analysisReasonable for a wide variety of situations Reasonable for a wide variety of situations

Model for Interarrivals and Service times Customers arrive at times t0 < t1 < .... - Poisson distributed The differences between consecutive arrivals are the

interarrival times : n = tn - t n-1

n in Poisson process with mean arrival rate , are exponentially distributed,

Pr(n t) = 1 - e- t

Service times are exponentially distributed, with mean service rate :

Pr(Sn s) = 1 - e-s

System Features

Service times are independent service times are independent of the arrivals

Both inter-arrival and service times are memoryless Pr(Tn > t0+t | Tn> t0) = Pr(Tn t)

future events depend only on the present state

This is a Markovian System

Exponential Distribution

given an arrival at time x

|

Same as probability starting at time = 0

P x t xP x x t

P x

P x t P x

P x

e e

P x

e ee

e ee

x t x

x t x

x

x t

x

(( ) )( ( ))

( )

( ( )) ( )

( )

( ) ( )

( )

( )( )

( )

( )

1 1

1

1 11

Markov Models

BufferOccupancy

t t

• n+1

• n

• n-1

• n

departure

arrival

Probability of being in state n

P t t P t t t t t

P t t t

P t t t

t

P t t P tdP t

dtt

n n

n

n

n nn

( ) ( )[( )( ) ]

( )[( )( )]

( )[( )( )]

,

( ) ( )( )

1 1

1

1

0

1

1

as Taylor series

Steady State Analysis

Substituting for

Steady state

P0

P t t

P P P

P

n

n n n

( )

( )

1 1

1

Markov Chains

0 1 ... n-1 n n+1

Rate leaving n = Rate arriving n = Steady State State 0

PP P

P P PP P

n

n n

n n n

( )

( )

1 1

1 1

0 1

Substituting Utilization

P P P

P P P

P P P P P P

1 0 0

2 1 0

2 1 1 0 1 01

( )

( )

Substituting P1

P P P

P P P P

P Pnn

2 0 0

20 0 0

20

0

1

( )

• Higher states have decreasing probability• Higher utilization causes higher probability of higher states

What about P0

P P PP

PP

P

nn

n

n

n

n

nn

0 00 0

0

0

00

11

11

1

1

( )

Queue determined by

E(n), Average Queue Size

-1=

)1()1()(000

n

n

n

n

nn nnnPnEq

Selecting Buffers

E(N) 1/3 .251 .53 .759 .9

For large utilization, buffers grow exponentially

Throughput

Throughput=utilization/service time = /Ts

For =.5 and Ts=1ms

Throughput is 500 packets/sec

Intuition on Little’s Law

If a typical customer spends T time units, on the overage, in the system, then the number of customers left behind by that typical customer is equal to

qTq

Applying Little’s Law

)1()1(

/

)1()( so

1

1

)1(

1

)1(

)()(

or or )()(

Delay Average M/M/1

ss

qw

TTET

nETE

TqTwTEnE

Probability of Overflow

P n N pnn N

n

n N

N( ) ( )

1 1

11

Buffer with N Packets

1 with )1(1

)1(

1

)1( and

1

1

1

11

1+N1

110

1

00

00

NN

N

N

N

n

nN

NN

n

nN

nn

p

pp

ppp

Example Given

Arrival rate of 1000 packets/sec Service rate of 1100 packets/sec

Find Utilization

Probability of having 4 packets in the queue

1000

11000 91.

P

P P P P4

4

1 2 3 5

1 062

082 075 068 056

( ) .

. , . , . , .

Example

04,.05,.05,.06,.07,.07,.08,.09,.09,.10,.11.

041

1

yprobabilit loss cell buffers fixed 12With

28.)12(

buffers infiniteWith

99.91

)(

112

12

12

112

nP

.)(

P

nP

nE

Application to Statistcal Multiplexing Consider one transmission

line with rate R. Time-division Multiplexing

Divide the capacity of the transmitter into N channels, each with rate R/N.

Statistical Multiplexing Buffering the packets

coming from N streams into a single buffer and transmitting them one at a time.

R/N

R/N

R/N

R

1T

N

T

NNT

1

'

Network of M/M/1 Queues

11 2

334

2

211 3212 313

ii

iiL

321 LLLL 321

J

i ii

iT1

1

M/G/1 Queue

Q S0

S

2

2SSQ

Assume that every customer in the queue pays at rate R when his or her remaining service time is equal to R.

Time Queuing:

Time Service :

Q

S

Total cost paid by a customer:

Expected cost paid by each customer:2

][][ 2SEQEC

2

][][][

2SEQECQE

)1(2

][][

2

SEQE

At a given time t, the customers pay at a rate equal to the sum of the remaining service times of all the customer in the queue. The queue begin first come-first served, this sum is equal to the queueing time of a customer who would enter the queue at time t.

1

][ QET

The customers pat at rate since each customer pays on the average and customers go through the queue per unit time.

CC