Stochastic Differential Equation Modeling and Analysis of TCP Windowsize Behavior

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Stochastic Differential Equation Modeling and
Analysis of TCP - Windowsize Behavior

• Presented by Sri Hari Krishna Narayanan
• (Some slides taken from or based on presentations
  by Vishal Mishra)

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Outline

• Introduction
• TCP window Algorithms
• Poisson counter driven stochastic differential
  equations
• Expressing windowsize changes
• Results
• Statistical tests

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Introduction

• This work is directly related to Ross
  presentation last week. The authors propose a new
  model which is simpler and work with the same
  data as the previous paper to obtain similar
  results.
• TCP is the protocol of choice for communication
  for many applications.
• Modeling TCP is hence important.
• Other applications may use other protocols
• TCP friendliness
• TCP shares the bandwidth fairly amongst hosts
  competing for network bandwidth

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TCP Congestion Control window algorithm

• Window can send W packets at a time
• increase window by one per RTT if no loss, W lt-
  W1 each RTT
• decrease window by half on detection of loss W lt-
  W/2

slide taken from presentation by Vishal Mishra
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TCP Congestion Control window algorithm

• Window can send W packets
• increase window by one per RTT if no loss, W lt-
  W1 each RTT
• decrease window by half on detection of loss W lt-
  W/2

slide taken from presentation by Vishal Mishra
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TCP Congestion Control window algorithm

• Window can send W packets
• increase window by one per RTT if no loss, W lt-
  W1 each RTT
• decrease window by half on detection of loss W lt-
  W/2

slide taken from presentation by Vishal Mishra
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TCP loss indications at the source

• There are two kinds
• Time Outs(TO)
• Triple Acknowledgements (TD)
• Effects on the TCP windowsize
• TO causes windowsize to become 1
• TD causes windowsize to halve
• When there is no packet loss, the windowsize
  increases.

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Other models

• Model TCP from the point of view of the source
• Packets that the source injects into the network
  .
• Each packet has an associated loss probability. p
  
• Identical for each packet
• Can be dependent on factors such as the current
  windowsize

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This model

• Models losses in a network centric way
• The network is the source of the congestion
• Not the packets?
• Losses are events that arrive at the source
• Arrivals are then modeled using statistical
  analysis
• In this case arrivals are modeled as a Poisson
  process.

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SDE based model
slide taken from presentation by Vishal Mishra
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Refinement of SDE model
Window Size is a function of loss rate (l) and
round trip time (R)
W(t) f(l,R)
slide taken from presentation by Vishal Mishra
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Poisson Process

• What is it?
• Process with exponential arrival times
• Arrivals are independent of each other
• Can be used to model natural occurrences
• Spotting fish in the ocean
• Occurrence of soft errors

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Traffic model

• The increase in windowsize
• Rises by 1 for every round trip time (RTT)
• Instead of step increase, the increase is
  considered to be continuous and represented as
  dt/RTT
• Falls by half for TD
• Falls to 1 for a TO

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Poisson counter

• Poisson process N with arrival rate ?
• dN 1 at Poisson arrival
• 0 elsewhere
• EdN ?dt
• This basically means that for ? poisson loss
  events in time dt, there will be ? spikes.

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Poisson Counter Driven Stochastic differential
equations (SDE)

• Dx f(x(t))dt?gi(x(t))dNi
• dW (dt /RTT) (-W/2)dNTD (1-W)dNTO
• First term indicates the additive increase of the
  TCP window
• Second and Third represent the multiplicative
  decrease.

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SDE Graphical Representation
Changing Window size
Time
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What to do with the SDE

• There is a lot of mathematics possible
• This mathematics evaluates the expected value of
  the windowsize and the throughput of the network
  at steady state.
• EW (1/RTT ?TO) /(?TD /2 ?TO )
• R (1/RTT)EW
• (1/RTT)(1/RTT ?TO) /(?TD /2 ?TO )

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Windowsize at steady state
Changing Window size
Time
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Maximum windowsize considerations

• Restricts the maximum value of the windowsize to
  M.
• EW ((1- PWM) /RTT ?TO) /(?TD /2 ?TO )
• What does this mean
• The continuous function rises as long as its
  value is not M.
• In that case it remains constant.
• After some mathematics,
• PWM (2?TO2 ?TO ?TO ?TD ?TO /RTT 2/
  RTT2 2 /RTT )
• (1/RTT1)(2M ?TO M?TD 2 /RTT )

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Windowsize at steady state with maximum window
size
Changing Window size
Time
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Other TCP features

• Slowstart
• Considered unimportant by authors
• Timeout backoff
• Modeled similarly to the maximum window

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Comparison with other models

• This model can be transformed into one involving
  packet loss
• Loss/sec ?TO ?TD
• Packets/sec R
• Loss/packet (Loss/sec) / (Packets/sec)
• (?TO ?TD ) /R

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Comparison with other models

• This model can be transformed into one involving
  no timeouts
• ?TO 0, no arrival of timeouts
• Earlier computation of EW changes
• PWM (2?TO2 ?TO ?TO ?TD ?TO /RTT 2/
  RTT2 2 /RTT )
• (1/RTT1)(2M ?TO M?TD 2 /RTT )
• PWM (2/ RTT2 2 /RTT )
• (1/RTT1)(M?TD 2 /RTT )
• (2/ RTT)
• (M?TD 2 /RTT )
• Similar changes can be made to account for no
  maximum window size

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Results 1
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Results 2
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Results 3
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Results -Analysis

• Closely mirrors earlier work
• Except at low thoughput
• This represente very high loss zone (60-80)
• Does not really matter
• Does not consider 1 hour traces at all
• So why use this model at all?
• Simpler mathematics and analysis
• So how do we get this simple analytical model?

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Trace analysis

• Loss inter arrival events tested for
• Independence
• Lewis and Robinson test for renewal hypothesis
• A sequence of recurrences T1,T2,... is a renewal
  process if the time between recurrences tj Tj
  -j-, j 1, 2,... (T0 0) are independent and
  identically distributed.
• Exponentiality
• Anderson-Darling test
• The Anderson-Darling test is used to test if a
  sample of data came from a population with a
  specific distribution.. The Anderson-Darling
  test is an alternative to the chi-square and
  Kolmogorov-Smirnov goodness-of-fit tests.

www.public.iastate.edu/wqmeeker/
stat533stuff/psnups/chapter16_psnup.pdf
slide based on presentation by Vishal Mishra
http//www.itl.nist.gov/div898/handbook/eda/sect
ion3/eda35e.htm
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Scatter plot of statistic
slide based on presentation by Vishal Mishra
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Experiment 1
slide taken from presentation by Vishal Mishra
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Experiment 2
slide taken from presentation by Vishal Mishra
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Experiment 3
slide taken from presentation by Vishal Mishra
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Experiment 4
slide taken from presentation by Vishal Mishra
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So are there any more magic fits and tests?

• Definitely there are more traces that can fit
  Poisson distribution.
• Motivating Example
• Soft errors
• Cosmic particles hit the chip to cause bit flips
• The existence of these particles can be modeled
  using a Poisson process.
• What about other distributions?
• Definitely, there may be other distributions and
  related mathematics.

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Thank you