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242-164 Introduction to QueueingNetworks : Engineering Approach
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ChapterChapter 11 Probability Models inProbability Models inComputer and ElectricalComputer and Electrical
EngineeringEngineering
Assoc. Prof. Thossaporn KamolphiwongCentre for Network Research (CNR)
Department of Computer Engineering, Faculty of EngineeringPrince of Songkla University, Thailand
Email : [email protected]
Outline
Mathematical Models
Deterministic Model
Probability Models
Example of Probability Models
2
Mathematical Models
System work in a chaotic environment
Probability Models:
Make sense out of the chaos
Build system
efficient, reliable, cost-effective
Introduction to
Theory underlying probability models
Basic techniques used in the development ofmodels
3
Mathematical Models(cont.)
Model is an approximate representation of aphysical situation
Mathematical models are used when theobservational phenomenon hasmeasurableproperties.
Deterministic Models
Probability Models
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Modeling Process
Formulatehypothesis
Define experiment totest hypothesis
Physicalprocess/system
Model
Sufficientagreement?
All aspects of interestinvestigated?
Stop
PredictionObservations
Yes No5
Deterministic Models
The conditions under which an experiment iscarried out determine the exact outcome ofexperiment.
In deterministic mathematical models, theso ution speci ies t e exact outcome o t eexperiment eg. Circuit theory
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Probability Models
Many systems of interest involve phenomenathat exhibit unpredictable variation andrandomness
Random experiment : an experiment in which
e ou come var es n an unpre c a e as on.
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Example of Random Experiment
Example1
A ball is selected from an urn containing identicalballs, labeled 0, 1 and 2.
from the set S = {0, 1, 2}
The set S of the possible outcomes is calledSample space
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Graph of outcome
e
4
3
2
Trial number
Outco
1009080-2
10 706050403020
1
0
-1
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Statistical regularity
Statistical regularity
Manyprobability models are based on the factthat averages obtained in long sequences ofrepetitions (trials) of random experiments
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Relative frequency
Example Experiment from example1 is repeated n
times under identical conditions.
Let N0(n), N1(n), and N2(n) be the number of times
in which the outcomes are balls 0, 1 and 2respective y
Relative frequency of outcome kbe define by
n
nNnf kk
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Probability
By statistical regularity, fk(n) varies less and lessabout a constant value as n is made large, that
is,
n lim
The constant pk is called the probability of theoutcome k
kkn
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Graph of Relative Frequency
0.5
0.6
0.7
0.8
0.9
1
Frequency Ball no. 0
Ball no. 2
Ball no. 1
0
0.1
0.2
0.3
0.4
0 10 20 30 40
Relative
Number of Trial
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Properties of Relative Frequency
Suppose that a random experiment has K
possible outcomes S = {1, 2, , K}
k , , ,
0 < fk(n) < 1 for k = 1, 2, , K
nnN
K
k
k 1
11
K
k
k nf14
Event
Event : Any outcome of experiment satisfyingcertain condition
Event E : an-even numbered of balls is selected
nfnfn
nNnN
n
nNnf EE
20
20
nNnNnNE 20
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Axiom of Probability
Axiom 1 : 0
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Detailed Example: A Packet VoiceTransmission System
A communication system is required to transmit 48simultaneous conversations from city A to city Busing packets of voice information. The speechof each speaker is converted into voltage
bundled into packets of information thatcorrespond to 10-millisecond (ms) segments ofspeech. A source and destination address isappended to each voice packet before it is
transmitted17
(Continue)
Simple Design
Transmit 48 packets every 10 ms in eachdirection
Inefficient design
On average 2/3 packets contain silence (no speechinformation)
48 speakers produce 48/3 = 16 active packets
Need another system transmits M < 48 packetsever 10 ms
18
(Continue)
Active
1
Multiplexer
SilenceN
To Site B
M packets/10 ms
N packets/10 ms19
(Continue)
Let
A : number of active packets in 10 ms
IfA < M active packet are transmitted
IfA > M unable to transmit all activepackets
A Mof active packet are selected at random anddiscard
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(Continue)
Experiment is repeated n times
A(j) : outcome injth trial
N(n) : number of trials in which the number of
active packets is k
Relative frequency :
n
nNnf kk
480lim
kpnf kkn
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(Continue)
Active packet are produced n
Sample mean : 1 n
njAA
48
0
1
1
k
k
j
nkNn
n
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(Continue)
Probabilities for number of active speakers ina group of 48
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Other Example
Communication over Unreliable channels
Every Tseconds, the transmitter accepts a binary
input, namely, a 0 or a 1, and transmits acorresponding signal. At the end of the Tsecon s, t e receiver ma es a ecision as towhat the input was, based on the signal it hasreceived. Most communications systems areunreliable in the sense that the decision of thereceiver is not always the same as thetransmitter input.
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(Continue)
1 -
0 0001 111
1
0Input
1
0
Output
-
CoderBinary
channelDecoder
Binaryinformation
Deliveredinformation
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Example : Processing of RandomSignals
Processing of Random Signals
Signal S(t) corrupted with noiseN(t)
Y(t) = S(t) +N(t)
The measure of quality : signal-to-noise ratio(SNR)
tNtS
tSSNR
ofpoweraverageofpoweraverage
ofpoweraverage
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Example : Reliability Systems
Reliability Systems
C1 C3C2
C1
C3
C2
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Example : Resource SharingSystems
Resource Sharing Systems
Multi-user computer system : Queueing System
Terminals
Queue System
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Average Resp. Time
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15ponseTime25
Number of Users0 10 20 30 40 50
10
5
AverageRe
0
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Internet Scale System
Internet Scale System
Internet
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Throughput performance
hput
0.8
1.2
Throughput performance of multi-usercomputer system
Number of users
Throu
0 10 20 30 40 50
0.4
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Reference
1. Alberto Leon-Garcia, Probability and RandomProcesses for Electrical Engineering, 3rd
edition, Addision-Wesley Publishing, 2008
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