Heavy tails, long memory and multifractals in teletraffic modelling

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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

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Outline

• Traffic models
• Past and present
• Complexity notions
• Statistical methods
• Data analysis
• Interdependence
• On-off modelling
• Large queues
• Multifractals

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Traffic models

• Packet level
• Traffic intensity
• of packets
• Bytes
• Fluid

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Past and present applications

• Telephone system
• Human
• Static (averages)
• One timescale

• Data communication
• Machine (fax, web)
• Dynamic (bursts)
• Several timescales

Erlang model
Fractal models
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Notions of complexity
Space
Finite variance
Heavy tails (Noah)
Time
Independent increments
Long-range dependence (Joseph)
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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.

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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!
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Time complexity LRD
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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 lt H lt 1
• Multifractals different moments scale
  differently

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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

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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

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Results (1)
WWW file sizes
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Results (2)
SUNET ATM traffic testing for LRD
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Results (3)
IP packet traffic multifractal test
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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 H0.83, 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 H0.75

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Interdependence of complexity notions
HT

• Large deviation methods in queueing theory

• Gaussian limit theory
• Stationary on-off modelling

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ON-OFF modelling

1. Choose starting state
2. Modify starting period

Stationarity
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ON-OFF aggregation
Cumulative workload
For HT on period
Anick-Mitra-Sondhi
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Limit process
(Taqqu, Willinger, Sherman, 1997)
Fractional Brownian motion
Stable Lévy motion
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Large queues
LDP for fBm
Tail asymptotics for Q
Weibull!
The queue is built up by many bursts of moderate
size.
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Multifractal models

• Multifractal time subordination of monofractal
  processes
• X(t)BY(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

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Thank you for your attention!