Risk Analysis Workshop April 14, 2004 HT, LRD and MF in teletraffic1 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

Risk Analysis Workshop April 14, 2004 HT, LRD and MF in teletraffic2 Outline Traffic models Past and present Complexity notions Statistical methods Data analysis Interdependence On-off modelling Large queues Multifractals

Risk Analysis Workshop April 14, 2004 HT, LRD and MF in teletraffic3 Traffic models  Packet level  Traffic intensity  # of packets  Bytes  Fluid

Risk Analysis Workshop April 14, 2004 HT, LRD and MF in teletraffic4 Past and present: applications Telephone system Human Static (averages) One timescale Data communication Machine (fax, web) Dynamic (bursts) Several timescales Erlang modelFractal models

Risk Analysis Workshop April 14, 2004 HT, LRD and MF in teletraffic5 Notions of complexity Time Space Finite variance Independent increments Heavy tails (”Noah”) Long-range dependence (”Joseph”)

Risk Analysis Workshop April 14, 2004 HT, LRD and MF in teletraffic6 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 dependent if its autocorrelation function decays hyperbolically, i.e.:

Risk Analysis Workshop April 14, 2004 HT, LRD and MF in teletraffic7 Space complexity Exponential  Phone call lengths  Inter-call times  Classical buffer sizes Heavy tailed  FTP/WWW file sizes  Modem session lengths  CPU time usage Classical theory cannot explain large buffers!

Risk Analysis Workshop April 14, 2004 HT, LRD and MF in teletraffic8 Time complexity: LRD

Risk Analysis Workshop April 14, 2004 HT, LRD and MF in teletraffic9

Risk Analysis Workshop April 14, 2004 HT, LRD and MF in teletraffic10

Risk Analysis Workshop April 14, 2004 HT, LRD and MF in teletraffic11 Definitions (2) Let be the m-aggregated process of a process X: –X is second order self-similar if –H is the Hurst parameter, 0.5 < H < 1 Multifractals: different moments scale differently

Risk Analysis Workshop April 14, 2004 HT, LRD and MF in teletraffic12 Investigated data Synthetic control data (fBm generated by random Midpoint Displacement method) WWW file download sizes –Data measured at Boston University –Own client based measurements IP packet arrival flow –Berkeley Labs ATM packet arrival flow –SUNET ATM network

Risk Analysis Workshop April 14, 2004 HT, LRD and MF in teletraffic13 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

Risk Analysis Workshop April 14, 2004 HT, LRD and MF in teletraffic14 Results (1) WWW file sizes

Risk Analysis Workshop April 14, 2004 HT, LRD and MF in teletraffic15 Results (2) SUNET ATM traffic: testing for LRD

Risk Analysis Workshop April 14, 2004 HT, LRD and MF in teletraffic16 Results (3) IP packet traffic: multifractal test

Risk Analysis Workshop April 14, 2004 HT, LRD and MF in teletraffic17 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

Risk Analysis Workshop April 14, 2004 HT, LRD and MF in teletraffic18 Interdependence of complexity notions HT LRD Large buffers Gaussian limit theory Stationary on-off modelling Large deviation methods in queueing theory

Risk Analysis Workshop April 14, 2004 HT, LRD and MF in teletraffic19 ON-OFF modelling OnOff OnOff 1.Choose starting state 2.Modify starting period Stationarity:

Risk Analysis Workshop April 14, 2004 HT, LRD and MF in teletraffic20 ON-OFF aggregation Anick-Mitra-Sondhi OnOff Cumulative workload: For HT on period:

Risk Analysis Workshop April 14, 2004 HT, LRD and MF in teletraffic21 Limit process (Taqqu, Willinger, Sherman, 1997) Fractional Brownian motion Stable Lévy motion

Risk Analysis Workshop April 14, 2004 HT, LRD and MF in teletraffic22 Large queues The queue is built up by many bursts of moderate size. Server fBm  LDP for fBmTail asymptotics for Q Weibull!

Risk Analysis Workshop April 14, 2004 HT, LRD and MF in teletraffic23 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

Risk Analysis Workshop April 14, 2004 HT, LRD and MF in teletraffic24 Thank you for your attention!