# Course Part3

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Hamilton Institute

TCP congestion control

Roughly speaking, TCP operates as follows:

Data packets reaching a destination are

acknowledged by sending an appropriate message to the

sender.

Upon receipt of the acknowledgement, data sources increase

their send rate, thereby probing the network for available

bandwidth, until congestion is encountered.

Network congestion is deduced through the loss of data packets

(receipt of duplicate ACKs or non receipt of ACKs), and

results in sources reducing their send rate drastically (by half).

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Hamilton Institute

TCP congestion control

Congestion control is necessary for a number of reasons,

so that:

catastrophic collapse of the network is avoided under heavy

loads;

each data source receives a fair share of the available

bandwidth;

the available bandwidthB is utilised in an optimal fashion.

interactions of the network sources should not cause

destabilising network side effects such as oscillations or

instability

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Hamilton Institute

TCP congestion control

Hespanhas hybrid model of

TCP traffic.

Loss of packets caused by

queues filling at the

bottleneck link.

TCP sources have two

modes of operation

Additive increase

Multiplicative decrease

Packet-loss detected at

sources one RTT after loss

of packet.

Data s rc 1

Data s rc

Data s rc 2

B ttl ck li k l

R t r R t r

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Hamilton Institute

TCP congestion controlData source 1

Data source n

Data source 2

Bottleneck link l

Router Router

Packet not

being

dropped

Packets

dropped

Packet drop

detected

Half source rate

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Hamilton Institute

TCP congestion controlData source 1

Data source n

Data source 2

Bottleneck link l

Router Router

Queue

not

full

Queue

full

Packet drop

detected

Half source rate

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Hamilton Institute

Modelling the queue not full state

The rate at which the queue grows is easy to determine.

While the queue is not full:

B

QTRTT

BRTT

w

dt

dQ

p

i

!

!

RTTdt

dwi

1!

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Hamilton Institute

Modelling the queue full state

When the queue is full

One RTT later the sources are informed of congestion

RTTdtdw

dt

dQ

1

0

!

!

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Hamilton Institute

TCP congestion control

Queue fills

ONE RTT LATER

QUEUE FULL

RTTdt

dw

dt

dQ

1

0

!

!

BRTT

w

dt

dQ i!

RTTdt

dwi

1!

iiw.w 50!

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Hamilton Institute

TCP congestion control: Example (Hespanha)

Seconds40T

packets250Qcpackets/se1250

p

max

.

B

!

!

!

0 100 200 300 400 500 600 0

50

10 0

15 0

20 0

25 0

30 0

35 0

40 0

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Hamilton Institute

TCP congestion control: Example (Fairness)

Seconds40T

packets250Q

cpackets/se1250

p

max

.

B

!

!

!

0 200 400 600 800 1000 1200 0

10 0

20 0

30 0

40 0

50 0

60 0

70 0

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Ro ert . Shorten Hamilton Institute

Modelling of dynamic systems: Part 3

System Identification

Robert N. Shorten & Douglas Leith

The Hamilton Institute

NUI Maynooth

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Hamilton Institute

Building our first model

Example:Mal

thussla

w of popula

tion growth

Government agencies use population models to plan.

What do you think be a good simple model for population

growth?

Malthuss law states that rate of an unperturbed population

(Y) growth is proportional to the population present.

Introduction

kYdt

dY!

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Hamilton Institute

1800 1820 1840 1860 1880 1900 1920 1940 1960 1980 0

50

100

150

200

250

Y E A R

P op

US Population Growth (m il lions) v. Y ear

1800 1820 1840 1860 1880 1900 1920 1940 1960 1980 1.5

2

2.5

3

3.5

4

4.5

5

5.5

Slope = k

Intercept = ey0

Y E A R

ln(Pop)

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Hamilton Institute

1800 1820 1840 1860 1880 1900 1920 1940 1960 1980 0

50

100

150

200

250

300

350

Y E A R

P op

US Population Growth (m il lions) v. Y ear

M O D E L

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Hamilton Institute

Modelling

Modelling is usually necessary for two reasons: to predictand to control. However to build models we need to do a lot

of work.

Postulate the model structure (most physical systems can be

classified as belonging to the system classes that you have already

seen)

Identify the model parameters;

Validate the parameters (later);

Solve the equations to use the model forprediction and analysis

(now);

Introduction

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Hamilton Institute

Modelling

Modelling is usually necessary for two reasons: to predictand to control. However to build models we need to do a lot

of work.

Postulate the model structure (most physical systems can be

classified as belonging to the system classes that you have already

seen)

Identify the model parameters;

Experiment design

Parameter estimation

Validate the parameters (later);

Solve the equations to use the model forprediction and analysis

(now);

Introduction

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Hamilton Institute

What is parameter estimation?

Parameter identification is the identification of theunknown parameters of a given model.

Usually this involves two steps. The first step is

concerned with obtaining data to allow us to identify the

model parameters.

The second step usually involved using some

mathematical technique to infer the parameters from the

observed data.

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Hamilton Institute

Linear in parameter model structures

The parameter estimation task is simple when the modelis a linear in parameters model form.

For example, in the equation

the unknown parameters appear as coefficients of the

variables (and offset).

The parameters of such equations are estimated using the

principle of least squares.

.baxy !

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Hamilton Institute

The principle of least squares

Karl FriedrickGauss (the greatest mathematician afterHamilton) invented the principle of least squares to

determine the orbits of planets and asteroids.

Gauss stated that the parameters of the models should be

chosen such that the sum of the squares of the

differences between the actually computed values is a

minimum.

For linear in parameter models this principle can be

applied easily.

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Hamilton Institute

The principle of least squares

Karl FriedrickGauss (the greatest mathematician afterHamilton) invented the principle of least squares to

determine the orbits of planets and asteroids.

Gauss stated that the parameters of the models should be

chosen such that the sum of the squares of the

differences between the actually computed values is a

minimum.

For linear in parameter models this principle can be

applied easily.

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Hamilton Institute

The principle of least squares

)y,x(11

)y,x(22

)y,x(kk

x

y

! !

k

iii )yy()b,a(V 1

2

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Hamilton Institute

The principle of least squares: The algebra

For our example: we want to minimize

Hence, we need to solve:

!

!

!

!

m

i ii

m

iii

)baxy(

)yy()b,a(V

1

2

1

2

00 !x

x!

x

x

b

)b,a(V

a

)b,a(V

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Hamilton Institute

The principle of least squares: The algebra

For our example: we want to minimize

Hence, we need to solve the following equations for the

parameters a,b.

012

02

1

1

!!x

x

!!x

x

!

!

))(baxy(b

)b,a(V

)x)(baxy(a

)b,a(V

m

iii

i

m

iii

!!

!!!

!

!

m

ii

m

ii

m

iii

m

ii

m

ii

ymbxa

yxxbxa

11

111

2