reviewch1-probability models in computer and electrical engineering

8/3/2019 ReviewCh1-Probability Models in Computer and Electrical Engineering

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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

22

(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

20

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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