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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    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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    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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    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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    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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    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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    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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    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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    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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    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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    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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    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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    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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    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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    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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    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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    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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    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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    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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    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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    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