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Analysis of selfsimilar Traffic Using Multiplexer

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Title: Analysis of selfsimilar Traffic Using Multiplexer


1
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

2
Overview
  • Background Knowledge
  • Motivation Model description
  • Results Analysis
  • Achievements of the Research
  • Questions from the Research

3
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.

4
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..
5
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.

6
Background knowledge
7
Self-Similar traffic V.S. Poisson Traffic
8
LRD V.S. SRD
  • LRD traffic streams are highly correlated at
    every time scales.
  • SRD traffic streams has negative exponentially
    distributed inter arrival times.

9
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.

10
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.

11
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.

12
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.

13
Traffic Pattern for input sources
14
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.

15
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.

16
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.

17
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.

18
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.

19
Results Analysis (100 million timeslot)
  • This is the final result of the outputs of the
    MUX\DEMUX network.

20
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.

21
Multiplexed Demultiplexed outputs
22
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.

23
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.

24
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.

25
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.

26
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.

27
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.

28
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.

29
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.

30
Thank you !
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