Modelling of Real Network Traffic by Phase-Type distribution
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1 Modelling of Real Network Traffic by Phase-Type distribution Andriy Panchenko Dresden University of Technology Juli Würzburger Workshop "IP Netzmanagement, IP Netzplanung und Optimierung" 1
2 Outline 1. Problem overview 2. Fitting algorithm 3. Experiments Juli Würzburger Workshop "IP Netzmanagement, IP Netzplanung und Optimierung" 2
3 Problem overview Traffic Need of a traffic model simple but adequate applicable to further performance analysis Juli Würzburger Workshop "IP Netzmanagement, IP Netzplanung und Optimierung" 3
4 Problem overview Failure of Poisson process - Interarrivals are not exponentially distributed - Heavy tailed distributions - Presence of huge values that are generated with small probability but have a great impact on the lower moments of pdf Juli Würzburger Workshop "IP Netzmanagement, IP Netzplanung und Optimierung" 4
5 Problem overview Traffic modeling goals Goal: to develop a traffic model simple but adequate applicable to further analysis (possibly by means of existing tools and techniques) Approaches Separate modeling of each transmitting process and their further aggregation (simulation) Statistical description of traffic processes (analytical model) Juli Würzburger Workshop "IP Netzmanagement, IP Netzplanung und Optimierung" 5
6 Problem overview Modeling compromise Real traffic traces? Model parameterisation Model Poisson FGn, Wavelets Markovian processes? New techniques Existing tools and analysis techniques Juli Würzburger Workshop "IP Netzmanagement, IP Netzplanung und Optimierung" 6
7 Problem overview Phase-Type (PH) distribution CTMC with N states described by a generator matrix: D0 includes local transitions rates and negative diagonal elements d1 includes transition rates accompanied by an arrival Absorbing state j k N+1 i D0 i,j (rates) h d1 i π is the stationary distribution (π Q=0) τ the distribution after an arrival complete definition of PH-distribution Juli Würzburger Workshop "IP Netzmanagement, IP Netzplanung und Optimierung" 7
8 Problem overview Properties of PH-distribution distribution function probability density function k-th moment Juli Würzburger Workshop "IP Netzmanagement, IP Netzplanung und Optimierung" 8
9 Problem overview Randomisation technique computing of CTMC transient measures Randomisation where Randomisation rate Poisson probability Juli Würzburger Workshop "IP Netzmanagement, IP Netzplanung und Optimierung" 9
10 Problem overview General fitting task Adjust the free parameters of PH-distribution Maximize the density function over all values Juli Würzburger Workshop "IP Netzmanagement, IP Netzplanung und Optimierung" 10
11 Fitting algorithm EM-algorithm (Expectation-Maximization) t sample; O model parameters EM iterative algorithm for maximum likelihood parameter estimation if data are incomplete or hidden. Problem: find maximum likelihood estimate O ML Principle: sequential mapping such that E-step: evaluation of conditional expectation M-step: maximization Differentials w.r.t. model parameters O Generalized EM Juli Würzburger Workshop "IP Netzmanagement, IP Netzplanung und Optimierung" 11
12 Fitting algorithm Exploited features Aggregated trace (approx.) Fixed initial distribution Reduced Likelihood Exploit the randomization technique Juli Würzburger Workshop "IP Netzmanagement, IP Netzplanung und Optimierung" 12
13 Fitting algorithm Empirical pdf Represent trace t as set of tuples 1: Identical intervals - Not enough intervals - Heavy tails 2: Log-scaled intervals Juli Würzburger Workshop "IP Netzmanagement, IP Netzplanung und Optimierung" 13
14 Steps of our EM-algorithm 1. start with some PH-distribution 2. compute the likelihood of the PH-distribution according to the each element in the trace 3. compute the new transition probabilities in the matrices for the given likelihood 4. check for convergence and stop or repeat steps Juli Würzburger Workshop "IP Netzmanagement, IP Netzplanung und Optimierung" 14
15 Fitting algorithm EM-steps Forward likelihood y Backward likelihood x k Juli Würzburger Workshop "IP Netzmanagement, IP Netzplanung und Optimierung" 15
16 Fitting algorithm EM-steps Collect weighted values of likelihood Normalize Juli Würzburger Workshop "IP Netzmanagement, IP Netzplanung und Optimierung" 16
17 Fitting algorithm Algorithm framework Input: Trace, Initial distribution, Randomization rate Build aggregated trace Encode matrices REPEAT Calculate forward / backward likelihoods FOR all intervals Calculate likelihood matrices END for Collect and normalize likelihood matrices in UNTIL converges Decode matrices Juli Würzburger Workshop "IP Netzmanagement, IP Netzplanung und Optimierung" 17
18 Results Influence of initial distribution uniform (1,0,,0) random Juli Würzburger Workshop "IP Netzmanagement, IP Netzplanung und Optimierung" 18
19 Results Influence of matrix form dense \\ upper triang Juli Würzburger Workshop "IP Netzmanagement, IP Netzplanung und Optimierung" 19
20 Results Fitting of heavy-tailed distributions Juli Würzburger Workshop "IP Netzmanagement, IP Netzplanung und Optimierung" 20
21 Conclusion good capture of main part as well as tail behaviour of empirical distribution a larger number of phases yields a better likelihood value best approximation was achieved with initial vector (1,0,,0) matrix form: upper triangular or with main and first upper diagonals h.t.d. properties are captured with relatively compact models with 5 states Juli Würzburger Workshop "IP Netzmanagement, IP Netzplanung und Optimierung" 21
22 Thank You! Juli Würzburger Workshop "IP Netzmanagement, IP Netzplanung und Optimierung" 22
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