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

Adaptive Communications Networks Research Group

Electronic Engineering

Aston University

Significance of Joint Density Plots in Markov Internet Traffic Modelling

AHMED D. SHAIKH

Outline

Outline

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

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)

Outline

Simple Two State Markov Traffic Model

The sequence of packets will be ABABABABABABAB…..

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:

Two state Markov Model (Numerical vs. Analytical results )

Two state Markov Model (Numerical vs. Analytical results –

Symmetric rates)

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:

Second Order Statistics – Joint Density (Results for Symmetric 2-

state model)

Higher Order Statistics – Joint Density

(Results for Asymmetric 2-state Model)

Outline

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)

Outline

Two state Model with non-Poisson statistics

The sequence of packets is AAAAAAAAAAAA……

The PDF equation for the IPT is:

PDF for the two state model with only one state emitting packets

Joint Density – 2 state model with one packet emitting state / source

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:

Use of Gamma Markov Models

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.

Gaussian Markov Models

Outline

Modelling Real World Example – IP Traffic Measurement at UDP Port

15010 - VoIP

Fitting a Gaussian Markov Model

Gaussian Model(PDF) = V1* Gaussian(μ1,σ1) + V2 * Gaussian(μ2,σ2) + …+V6 * Gaussian(μ6,σ6)

Comparing the Joint Densities

Outline

Understanding Packet Sequences from Joint Density Results

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……….

PDF and Joint Density – Markov Model A

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…..

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.

Outline

Understanding the curve of periodicity

Modelling Periodic Events with Markov Models

Small ∆ for Markov Models C and D - S∆ model

Large ∆ for Markov Models C and D - L∆ model

Multiple Periodicity

Use of S∆ and L∆ model sets to model measured results

Use of S∆ and L∆ model sets to model measured results

Outline

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.

Thank you!

Questions or comments?

The man himself:

Andrey Markov

(1856 - 1922)