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In this study by István Maricza at the High-Speed Networks Laboratory, Budapest University of Technology and Economics, the concepts of Heavy Tails (HT), Long-Range Dependence (LRD), and Multifractals (MF) are explored in teletraffic modeling. The research delves into traffic models at various levels, statistical methods, data analysis, complexity notions, interdependence, on-off modeling, and the presence of large queues. It investigates past and current applications, such as the Erlang model and fractal models, to understand the complexities in telecommunication systems. Employing statistical methods like QQ-plot, R/S analysis, and multifractal tests, data from various sources is analyzed to uncover patterns of heavy-tailed distributions, long-range dependence, and multifractal properties. The results show the characteristics of teletraffic phenomena, such as heavy-tailed file sizes, long-range dependence in packet arrival processes, and the absence of long-range dependence in certain traffic types like ATM. The study examines the interplay of complexity notions, large deviation methods, and Gaussian limit theory in understanding queueing systems and the dynamic nature of telecommunication traffic. Multifractal modeling techniques, such as multifractal time subordination and models based on multiplicative cascades, are discussed in the context of teletraffic analysis. Thank you for exploring the intricate world of teletraffic modeling with us!
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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
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
Traffic models • Packet level • Traffic intensity • # of packets • Bytes • Fluid HT, LRD and MF in teletraffic
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
Notions of complexity Space Finite variance Heavy tails (”Noah”) Time Independent increments Long-range dependence (”Joseph”) HT, LRD and MF in teletraffic
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
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
Time complexity: LRD HT, LRD and MF in teletraffic
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
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
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
Results (1) WWW file sizes HT, LRD and MF in teletraffic
Results (2) SUNET ATM traffic: testing for LRD HT, LRD and MF in teletraffic
Results (3) IP packet traffic: multifractal test HT, LRD and MF in teletraffic
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
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
On Off On Off ON-OFF modelling • Choose starting state • Modify starting period Stationarity: HT, LRD and MF in teletraffic
On Off ON-OFF aggregation Cumulative workload: For HT on period: Anick-Mitra-Sondhi HT, LRD and MF in teletraffic
Limit process (Taqqu, Willinger, Sherman, 1997) Fractional Brownian motion Stable Lévy motion HT, LRD and MF in teletraffic
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
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
Thank you for your attention! HT, LRD and MF in teletraffic
