Probability Review

Thinh Nguyen

Probability Theory Review

Sample spaceBayes’ RuleIndependenceExpectationDistributions

Sample Space - Events

Sample PointThe outcome of a random experiment

Sample Space SThe set of all possible outcomesDiscrete and Continuous

EventsA set of outcomes, thus a subset of SCertain, Impossible and Elementary

Set OperationsUnionIntersectionComplement

PropertiesCommutation

Associativity

Distribution

De Morgan’s Rule

A B∪A 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∞ ∞

==

⎡ ⎤=⎢ ⎥

⎣ ⎦∑U

[ ]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 partitionof 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

|

|

|

jj

j jn

k kk

P A BP B A

P A

P A B P B

P A B P B=

⎡ ⎤∩⎣ ⎦⎡ ⎤ =⎣ ⎦

⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦=

∑

likelihood priorposteriorevidence

×=

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 numberAssign a numerical value to the outcome of the experiment

DefinitionA 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 1XxF x

→∞=

( )lim 0XxF 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

ContinuousProbability Density Function

DiscreteProbability 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 xδ= −∑

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 Xσ ⎡ ⎤= = −⎣ ⎦

[ ] [ ]Std X Var Xσ= =

[ ] 0Var c =

[ ] [ ]2Var cX c Var X=

[ ] [ ]Var X c Var X+ =

Queuing Theory

Example

nnSend a file over the internetSend a file over the internet

packet link

buffer

Modemcard

(fixed rate)

Delay Models

time

place

A

B

Cpr

opag

atio

n

tran

smis

sion

Com

puta

tion

(Que

uing

)

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 serviceA (possibly infinite) pool of customersSome 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 time distribution

B = the service time distribution

c = the number of servers

d = the queue size limit

A/B/c/d

M for Markovian (exponential) distribution

D for Deterministic distribution

G for General (arbitrary) distribution

:Where A and B can be

nnomitted 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]

λ = mean arrival ratet = kδ ; k = 0,1,…. ; δ→0

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

Pr(a(t) = n) = e-λ t (λ t)n/n!

nnReadily amenable for analysisReadily amenable for analysisnnReasonable 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 : τn = tn - t n-1

• τn in Poisson process with mean arrival rate λ, are exponentially distributed,

Pr(τn ≤ t) = 1 - e-λ t

Service times are exponentially distributed, with mean

service rate µ:

Pr(Sn ≤ s) = 1 - e-µs

System Features

Service times are independentservice 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 x P x x tP x

P x t P xP x

e eP x

e ee

e ee

x t x

x t x

x

x t

x

(( ) ) ( ( ))( )

( ( )) ( )( )

( ) ( )( )

( )( )

( )

( )

τ τ ττ

τ ττ τ

λ λ

λ λ

λ

λ λ

λ

− < > =< < +

>

=< + − <

>=

− − −− <

=− +− −

=−

− + −

− + −

−

− −

−

1 11

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 tP t t tP t t t

t

P t t P t dP tdt

t

n n

n

n

n nn

( ) ( )[( )( ) ]( )[( )( )]( )[( )( )]

,

( ) ( ) ( )

+ = − − ++ −+ −

→

+ = +

+

−

∆ ∆ ∆ ∆ ∆∆ ∆∆ ∆

∆

∆ ∆

1 111

0

1

1

µ λ µ λµ λλ µ

as Taylor series

Steady State Analysis

Substituting for

Steady stateP0

P t tP P P

P

n

n n n

( )( )

++ = +

=

+ −

∆λ µ µ λ

λ µ

1 1

1

Markov Chains

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

λ λ λ

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

PP P

P P PP P

n

n n

n n n

( )

( )

λ µλ µ

λ µ λ µλ µ

++

+ = +=

− +

− +

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

= =

= + −

= + − = + −

λµ

ρ

µ λ µ λλµ

λµ

ρ ρ

( )

( )

Substituting P1

P P PP P P P

P Pnn

2 0 02

0 0 02

0

0

1= + −

= + − =

=

ρ ρ ρ

ρ ρ ρ ρ

ρ

( )

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

of higher states

What about P0

P P P P

P P

P

nn

n

n

n

n

nn

=

∞

=

∞

=

∞

∑ = = ∑ = ∑ =−

=−

→ = −

= −

0 00 0

0

0

00

11

11

1

1

ρ ρρ

ρρ

ρ ρ( )

Queue determined by ρλµ

=

E(n), Average Queue Size

ρρ

ρρρρ

-1=

)1()1()(000∑∑∑∞

=

∞

=

∞

=

−=−===n

n

n

n

nn nnnPnEq

Selecting Buffers

E(N) ρ1/3 .251 .53 .759 .9

For large utilization, buffers grow exponentially

Throughput

Throughput=utilization/service time = ρ/Ts

For ρ=.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 λ=

Applying Little’s Law

)1()1(/

)1()( so 1

1)1(

1)1(

)()(

or or )()(Delay Average M/M/1

ρρλµλ

ρλρ

µ

λµρµρλρ

λ

λλλ

−=

−=

−==

−=

−=

−==

===

ss

qw

TTET

nETE

TqTwTEnE

Probability of Overflow

P n N pnn N

n

n N

N( ) ( )> = ∑ = − ∑ == +

∞

= +

∞+

1 1

11 ρ ρ ρ

Buffer with N Packets

1 with )1(1

)1(1

)1( and 1

1

111

1+N1

110

1

00

00

<<−=−−

=

−−

=−−

=

⎥⎦

⎤⎢⎣

⎡−

−===

+

++

+

==∑∑

ρρρρρρ

ρρρ

ρρ

ρρρ

NN

N

N

N

n

nN

NN

n

nN

nn

p

pp

ppp

ExampleGiven

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

Find Utilization

Probability of having 4 packets in the queue

ρ λµ

= = =10001100

0 91.

PP P P P

44

1 2 3 5

1 062082 075 068 056

= − == = = =

( ) .. , . , . , .

ρ ρ

Example

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

0411

yprobabilit loss cell buffers fixed 12With 28.)12(

buffers infiniteWith

99.91

)(

112

12

12

112

=

=−−

=

==>

=−

=

+

+

nP

.)(P

nP

nE

ρρρ

ρ

ρρ

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

R/N

R/NR/N

R

λµ −=

1T

NT

NNT =

−=

λµ1'

Network of M/M/1 Queues

1µ1γ 2µ

3γ3µ4γ

2γ

211 γγλ += 3212 γγγλ ++= 313 γγλ +=

ii

iiL

λµλ−

= 321 LLLL ++= 321 γγγγ ++=

∑= −

=J

i ii

iT1

1λµ

λγ

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 :

QS

Total cost paid by a customer:

Expected cost paid by each customer:2

][][ 2SEQEC +=µ

⎟⎟⎠

⎞⎜⎜⎝

⎛+==

2][][][

2SEQECQEµ

λλ

)1(2][][

2

ρλ

−=

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.

µ1][ += QET

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

CλC λ