Prof' K'J'Blow, Dr' Marc Eberhard and Dr' Scott Fowler

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Significance of Joint Density Plots in Markov
Internet Traffic Modelling
AHMED D. SHAIKH
Prof. K.J.Blow, Dr. Marc Eberhard and Dr. Scott
Fowler Adaptive Communications Networks Research
Group Electronic Engineering Aston University
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Outline
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Outline
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Traffic Modelling Approaches

• Two types of Approaches
• Black Box models
• Internal structure is unknown. Opaque to user.
• Examples HMM, MMPP, BMAP
• White Box models
• Transparent structure. Has a physical meaning.
• Examples Classic Markov Models, On-Off models

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Markov Models An Introduction

• Probabilistic models defining a stochastic
  process with finite number of states observing
  the Markov Property.
• Transitions occur with a fixed
• transition rate Rij.
• States can model activities of traffic
• sources on a network.
• Inter-Arrival times are exponentially
• distributed.
• Packet level statistics obtained from Monte Carlo
  simulations are expressed in IPT (Inter-Packet
  times)

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Outline
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Simple Two State Markov Traffic Model

• The sequence of packets will be ABABABABABABAB..

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Two state Model (Analytical analysis contd..)

• Two state models will have equal number of
  visits to each state.
• So, V1 V2 0.5
• Probability densities of time spent in each
  state
• The Probability Density function of IPT for a two
  state model is

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Two state Markov Model (Numerical vs. Analytical
results )
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Two state Markov Model (Numerical vs. Analytical
results Symmetric rates)
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Higher order Statistics for Markov Models

• Higher Order Distributions
• Markov Models can also produce higher order
  statistics.
• Possible to study the sequence of IPTs and a
  variety of other unique features associated with
  the network traffic statistics.
• The Joint Density function for the two state
  Markov Model is given by

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Second Order Statistics Joint Density (Results
for Symmetric 2-state model)
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Higher Order Statistics Joint Density(Results
for Asymmetric 2-state Model)
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Outline
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N-state models with Poisson statistics
The general form equation for the IPT PDF of
N-state Markov Models where every state is
emitting packets is PDF (N-state) V1 P1(t)
V2 P2(t) V3 P3(t)... VN PN(t)
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Outline
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Two state Model with non-Poisson statistics

• The sequence of packets is AAAAAAAAAAAA
• The PDF equation for the IPT is

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PDF for the two state model with only one state
emitting packets
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Joint Density 2 state model with one packet
emitting state / source
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PDF for IPT for N-state Markov Models with only
one state emitting packets
The general form analytical equation of the PDF
of IPT for Markov loop Models with only one state
emitting packets is
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Use of Gamma Markov Models
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Taking it further - A Gaussian Markov Model

• Now in the general equation of the Gamma
  distribution, we know that as N approaches
  infinity, the gamma distribution can be
  approximated by a normal or Gaussian
  distribution.
• Gives a normal distribution with mean
• Variance
• Gaussian Distribution PDF.

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Gaussian Markov Models
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Outline
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Modelling Real World Example IP Traffic
Measurement at UDP Port 15010 - VoIP
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Fitting a Gaussian Markov Model
Gaussian Model(PDF) V1 Gaussian(µ1,s1) V2
Gaussian(µ2,s2) V6 Gaussian(µ6,s6)
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Comparing the Joint Densities
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Outline
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Understanding Packet Sequences from Joint Density
Results
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The significance of the Joint Density Plots

• Let us consider a 3 1 states Model where V1
  V2 V3 1/3. (Markov Model A)
• Packet sequence can be ABBACACABCABBACAACA.

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PDF and Joint Density Markov Model A
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Markov Model B

• Let us now consider a 3 state Loop Model where V1
  V2 V3 1/3. (Markov Model B)
• Packet sequence must be ABCABCABCABCABCABC..

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PDF and Joint Density Markov Model B
Observation Two different models have the same
PDFs yet different Joint Densities. The Joint
density Plots give more statistical details on
Packet Sequences.
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Outline
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Understanding the curve of periodicity
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Modelling Periodic Events with Markov Models
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Small ? for Markov Models C and D - S? model
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Large ? for Markov Models C and D - L? model
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Multiple Periodicity
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Use of S? and L? model sets to model measured
results
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Use of S? and L? model sets to model measured
results
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Outline
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Summary and Conclusions

• Summary
• Observed first and second order statistics for
  N-state Markov Models with Poisson and
  Non-Poisson statistics and confirmed our
  anlaytical understanding of the models with
  simulated results.
• Established the significance of the Joint Density
  Plots and explored the use of simple Markov
  models to model unique features of Joint Density
  Traffic Statistics Results.
• Conclusions
• The Joint Density Plot contains much more
  statistical information on the activities and
  nature of the traffic sources than the PDF.
• Modelling PDFs alone will result in reproducing
  first order statistics. Use of Joint Density
  Plots is Recommended to model source behaviour.
  Simple Markov Models can be used to model the
  unique features of Joint Densities.

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Thank you!

• Questions or comments?
• The man himself
• Andrey Markov
• (1856 - 1922)