Analysis of selfsimilar Traffic Using Multiplexer

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Analysis of self-similar Traffic Using
Multiplexer Demultiplexer Loaded with
Heterogeneous ON/OFF Sources

• Huai Huang
• Dept. of Electronic Engineering
• Queen Mary, University of London

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Overview

• Background Knowledge
• Motivation Model description
• Results Analysis
• Achievements of the Research
• Questions from the Research

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Background Knowledge

• Traditional Poisson-based models for Voice and
  Early Data Networks (before early 1990s)
• Packet arrivals Call arrivals (Poisson)
• Exponential holding times
• Traditional network traffic models, most of
  which assume Markovian characteristics, have been
  used extensively as an attractive means to the
  simulation and control of the networks before the
  early 1990s in many cases they prove adequate
  for evaluating network performance and show their
  practicality.

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Background knowledge

• Big Bang from 1993
• On the Self-Similar Nature of Ethernet Traffic
  Will E. Leland, Walter Willinger, Daniel V.
  Wilson, Murad S. Taqqu
• Extract from abstract
• We demonstrate that Ethernet local area network
  (LAN) traffic is statistically self-similar, that
  none of the commonly used traffic models is able
  to capture this fractal behavior, that such
  behavior has serious implications for the design,
  control, and analysis of high-speed
• Evidence of Self-similarity and Long-Range
  Dependence in network traffics
• Burstiness on multiple time scales
• Highly variable traffic
• Heavy-tailed distributions of file sizes and
  corresponding transmission times

That Changed Everything..
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Background Knowledge

• Self-Similarity
• Let X (Xk kgt0) be stationary process
  representing the amount of data transmitted in
  consecutive short time periods.
• Let Xk (m) 1/m ? km i(k-1)m1 Xi where m 1
  denote the m aggregated process.
• X is self-similar if X and m1-H X(m) have the
  same variance and autocorrelation ( with Hurst
  parameter H ).
• Long-range Dependency ( LRD )
• Autocorrelation r(k) ? k -ß, as k ? ?, which
  means the process follows a power law, rather
  than exponential decaying.( 0ltßlt1 )
• H1-ß/2, so self-similar process shows long-range
  dependency if 0.5ltHlt1
• Heavy-tailed Distribution
• A distribution of a random variable P is said to
  be heavy-tailed if P X gt x x -a , as x ? ?
  0 lt alt 2
• If a 1, the distribution has an infinite mean.
  If a 2, the distribution has an infinite
  variance.

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Background knowledge
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Self-Similar traffic V.S. Poisson Traffic
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LRD V.S. SRD

• LRD traffic streams are highly correlated at
  every time scales.
• SRD traffic streams has negative exponentially
  distributed inter arrival times.

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Heavy-tailed V.S. Exponential

• The PDF of the Pareto (Heavy-tailed)distribution
  decays slowly as the batch size increases. In
  log-log plot, it decays linearly and have very
  big batch size.
• While the PDF of the Exponential distribution
  decays very fast as the batch size increases.

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Background knowledge

• Multiplexer is a key element of the modern
  high-speed flow networks in that statistical
  multiplexing allows increasing network
  utilization considerably. It allows statistical
  multiplexing of different sources to make
  efficient use of the network resources.
• Modelling the multiplexer loaded with
  heterogeneous sources has been done to get the
  performance evaluation of the aggregate traffic.
  These studies get many useful results.

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Motivation Model description

• However, most of them just considered
  multiplexing the traffic, and didnt investigate
  the statistic features of the individual traffic
  flows after they divided by the demultiplexer.
  Actually, it is very interesting and valuable to
  study on the related issues.

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ON/OFF Model for traffic generation

• We choose ON/OFF model is because it is practical
  and popular for network traffic modeling, and
  matches very well with the real network
  activities active and silent.
• We use two methods to generate the ON/OFF input
  traffic using the Pareto and Exponential
  functions, and using the chaotic maps.

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Traffic Pattern for input sources
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Results Analysis ( 1 million run time)

• Take case 0110 as an example, from the
  simulation, we can obtain the statistic features
  of the ON and OFF periods for both the inputs and
  outputs.
• From the figures we can see the outputs share the
  same attribute as the inputs. The input is
  0110, and the output is 0110 too.
• We can also get the statistics of the buffer
  state and delay time from the simulation.

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Simulation results in brief ( 1 million )

• A tick or cross in the column Unclear about the
  Output indicates whether or not there is the
  need for further investigating about the output.
• A tick or cross in the column Big Delays for the
  packets through the server indicates whether or
  not there are big delays for the packets, and
  that means whether or not we need apply some
  control algorithms on the server to reduce the
  big delays.
• In the table, 0 means the sojourn time of the
  traffic is exponentially distributed 1 means
  the sojourn time of the traffic is Pareto
  distributed We use 2 denotes the statistic
  feature is not 0 or 1, or we are just unsure
  about what it is.

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The traffic Pattern 2 in the results

• We highlight the 2, and we use the log-log plot
  and the lineal-log plot with different scales to
  show whats the difference between 0,1 and
  2.
• In the log-log plots, We can clearly see from the
  graphs that the highlighted 2 looks like
  exponential distribution, and it doesnt have any
  sojourn time larger than 100 timeslots.
• In the linear-log plot, the Exponential-distribute
  d traffic looks like a straight line, but the
  highlighted 2 turns outside just like the
  Pareto distribution.

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Results Analysis (100 million timeslot)

• Though the simulations on the magnitude of 100
  millions, we clear out the ambiguous situations.
• Although we have 2 in the final results, but
  in here, the 2 is not unclear. It is just
  another kind of the traffic which is not behaving
  like Pareto or Exp.

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Results Analysis (100 million timeslot)

• As we can see in the graphs, the 2 is almost as
  the Exp distributed before its probability
  reaches 0.0001, after that, it looks like a
  straight line as Pareto distributed input source
  but with much smaller tails.

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Results Analysis (100 million timeslot)

• This is the final result of the outputs of the
  MUX\DEMUX network.

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Results Analysis

• Validation of the simulation result
• The Multiplexed outputs, does it agree with the
  results done by other people?
• The Queue Analysis, same question.
• Using different Parameters for the Simulation
• Highlight on some interesting cases to
  investigate further.
• Find out more subtle interaction between the
  traffic sources, especially about the
  Heavy-tailed sojourn time of the traffic.

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Multiplexed Demultiplexed outputs
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Queue Analysis of the simulation

• We find that if the ON period distribution is
  Pareto distributed in any of the input sources,
  the Probability Density Function (PDF) of the
  queue decays like a straight line, otherwise, it
  decays exponentially fast.

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Sim using Different Parameters

• We choose 0101, 0110, 0111, 1011, 1111 to study
  on, and we reach a rough conclusion below
• The MUX/DEMUX network doesn't change the
  attribute of the heavy-tailed distribution of the
  OFF period very much.
• The MUX/DEMUX network tends to change the
  attribute of the heavy-tailed distribution of the
  ON period a lot.
• If a heavy-tailed ON sojourn-time traffic
  multiplexed with a exponential ON sojourn-time
  traffic, usually, the heavy-tailed ON will be
  less burst than the original traffic.
• If a heavy-tailed ON sojourn-time traffic
  multiplexed with a another heavy-tailed ON
  sojourn-time traffic, usually, the lighter one
  will remain almost the same. Meanwhile, the
  heavier one will be less burst than the original
  traffic, in some cases, can change from the
  heavy-tailed distribution to the exponential
  distribution.
• As an exception in 4, for case 1010, both of the
  heavy-tailed ON sojourn-time are changed from
  heavy-tailed distribution to the exponential
  distribution.

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ACF Analysis of the Simulation

• We are not only interested in the tail
  distribution of the traffic, but we are also very
  interested in the LRD and SRD attributes of the
  traffic.
• We use autocorrelation function (ACF) to measure
  the LRD or SRD attributes of the traffic. And we
  divide the ten cases into two groups
• Group 1 The outputs share the same pattern with
  the inputs.
• Group 2 The outputs are different with the
  inputs.

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ACF Analysis of the Sim ( Group 1)

• For the cases in the first group, we find that
  the correlation structure of the outputs remain
  the same as the inputs, just as they do in the
  distribution of the ON/OFF sojourn-time. The
  example figure is the case 0001.

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ACF Analysis of the Sim ( Group 2)

• For the case 0010(Output 0000), we can easily
  find the output two was changed into a correlated
  traffic by the queue, while the output one shared
  the same pattern with the input one.
• For the case 0011(Output 2001), the result is
  very similar to the case 0010.

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ACF Analysis of the Sim ( Group 2)

• For the case 0111(Output 2111 or 1111), we can
  easily find both of the outputs share the same
  pattern with the inputs.

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ACF Analysis of the Sim ( Group 2)

• For the case 1010(Output 0000), we can clearly
  see there exist strong correlation within the
  sources. And another interesting phenomenon about
  this case is the AutoCorrelation Function of the
  outputs go up and down from the beginning, appear
  as two separate line in the log-log scale, and
  finally converge to one line.
• For the case 1011(Output 1001), the result is
  very similar to the case 1010.

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Achievements of the Research

• We have successfully obtained the detailed and
  accurate results for the whole situation of the
  10 cases for two kinds of traffic source models
  one, traffic sources generated by Pareto and
  Exponential functions and two, traffic sources
  generated by chaotic maps.
• We analyzed the subtle interaction of the traffic
  sources by using different parameters and reach a
  conclusion.
• We find some new traffic sources dont have
  heavy-tailed distribution, but at the same time,
  possess the LRD correlation structure. These
  sources can not be modeled with the chaotic maps
  or random processes as far as we know.

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Thank you !