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Transcript of Probability Review Thinh Nguyen. Probability Theory Review Sample space Bayes’ Rule Independence...
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