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Heavy tails, long memory and multifractals in teletraffic modelling

Heavy tails, long memory and multifractals in teletraffic modelling. István Maricza High Speed Networks Laboratory Department of Telecommunications and Media Informatics Budapest University of Technology and Economics. Traffic models Past and present Complexity notions

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Heavy tails, long memory and multifractals in teletraffic modelling

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  1. Heavy tails, long memory and multifractals in teletraffic modelling István Maricza High Speed Networks Laboratory Department of Telecommunicationsand Media Informatics Budapest University of Technology and Economics HT, LRD and MF in teletraffic

  2. Traffic models Past and present Complexity notions Statistical methods Data analysis Interdependence On-off modelling Large queues Multifractals Outline HT, LRD and MF in teletraffic

  3. Traffic models • Packet level • Traffic intensity • # of packets • Bytes • Fluid HT, LRD and MF in teletraffic

  4. Telephone system Human Static (averages) One timescale Data communication Machine (fax, web) Dynamic (bursts) Several timescales Past and present: applications Erlang model Fractal models HT, LRD and MF in teletraffic

  5. Notions of complexity Space Finite variance Heavy tails (”Noah”) Time Independent increments Long-range dependence (”Joseph”) HT, LRD and MF in teletraffic

  6. Definitions (1) • A distribution is heavy tailed with parameter  if its distribution function satisfies where L(x) is a slowly varying function. • A stationary process is long range dependentif its autocorrelation function decays hyperbolically, i.e.: HT, LRD and MF in teletraffic

  7. Exponential Phone call lengths Inter-call times Classical buffer sizes Heavy tailed FTP/WWW file sizes Modem session lengths CPU time usage Space complexity Classical theory cannot explain large buffers! HT, LRD and MF in teletraffic

  8. Time complexity: LRD HT, LRD and MF in teletraffic

  9. HT, LRD and MF in teletraffic

  10. HT, LRD and MF in teletraffic

  11. Definitions (2) • Let be the m-aggregated process of a processX: • Xis second orderself-similarif • His theHurst parameter, 0.5 < H < 1 • Multifractals: different moments scale differently HT, LRD and MF in teletraffic

  12. Investigated data • Synthetic control data (fBm generated by random Midpoint Displacement method) • WWW file download sizes • Data measured at Boston University • Own clientbased measurements • IP packet arrival flow • Berkeley Labs • ATM packet arrival flow • SUNET ATM network HT, LRD and MF in teletraffic

  13. Employed statistical methods • Heavy tail modelling • QQ-plot, • Hill plot and De Haan moment estimator • Long range dependence • Variance-time plot • R/S analysis • Periodogram plot and Whittle estimator • Multifractal tests • Absolute moment method • Wavelet-based method HT, LRD and MF in teletraffic

  14. Results (1) WWW file sizes HT, LRD and MF in teletraffic

  15. Results (2) SUNET ATM traffic: testing for LRD HT, LRD and MF in teletraffic

  16. Results (3) IP packet traffic: multifractal test HT, LRD and MF in teletraffic

  17. Summary of results • Sizes of downloaded WWW files exhibit the heavy tail property and are well approximated by a Pareto distribution with parameter =0.7 • The IP packet arrival process exhibits long range dependence and second order asymptotic self-similarity with Hurst parameter H=0.83, as well as the multifractal property. • The SUNET ATM traffic does not exhibit the long range dependence property, although it is consistent with the second order asymptotic self-similarity property with H=0.75 HT, LRD and MF in teletraffic

  18. LRD Large buffers Interdependence of complexity notions HT • Large deviation methods in queueing theory • Gaussian limit theory • Stationary on-off modelling HT, LRD and MF in teletraffic

  19. On Off On Off ON-OFF modelling • Choose starting state • Modify starting period Stationarity: HT, LRD and MF in teletraffic

  20. On Off ON-OFF aggregation Cumulative workload: For HT on period: Anick-Mitra-Sondhi HT, LRD and MF in teletraffic

  21. Limit process (Taqqu, Willinger, Sherman, 1997) Fractional Brownian motion Stable Lévy motion HT, LRD and MF in teletraffic

  22. fBm Server Large queues LDP for fBm Tail asymptotics for Q Weibull! The queue is built up by manybursts of moderate size. HT, LRD and MF in teletraffic

  23. Multifractal models • Multifractal time subordination of monofractal processes: X(t)=B[Y(t)], where B(t) is a monofractal process (fBm), Y(t) is a multifractal process. • Gaussian marginals • negative values • Models based on multiplicative cascades: • simple to generate • physical explanation • several parameters HT, LRD and MF in teletraffic

  24. Thank you for your attention! HT, LRD and MF in teletraffic

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