HT, LRD and MF in teletraffic 1

Risk Analysis Workshop April 14, 2004

Heavy tails, long memory and multifractals in

teletraffic modelling

István MariczaHigh Speed Networks Laboratory

Department of Telecommunicationsand Media Informatics

Budapest University of Technology and Economics

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Risk Analysis Workshop April 14, 2004

Outline• Traffic models

• Past and present

• Complexity notions

• Statistical methods

• Data analysis

• Interdependence

• On-off modelling

• Large queues

• Multifractals

t

x

et

2

2

2

1

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Risk Analysis Workshop April 14, 2004

Traffic models

Packet level

Traffic intensity# of packetsBytes

Fluid

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Risk Analysis Workshop April 14, 2004

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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Risk Analysis Workshop April 14, 2004

Notions of complexity

Time

Space

Finite variance

Independent increments

Heavy tails(”Noah”)

Long-range dependence (”Joseph”)

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Risk Analysis Workshop April 14, 2004

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.:

1)(

lim22 Hk kc

k

0,),(1)()( xxLxxXPxF

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Risk Analysis Workshop April 14, 2004

Space complexity

ExponentialPhone call lengths

Inter-call times

Classical buffer sizes

Heavy tailedFTP/WWW file sizes

Modem session lengths

CPU time usage

Classical theory cannot explain large buffers!

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Risk Analysis Workshop April 14, 2004

Time complexity: LRD

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Risk Analysis Workshop April 14, 2004

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Risk Analysis Workshop April 14, 2004

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Risk Analysis Workshop April 14, 2004

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

,..2,1,...1

1)( kXX

mX mkmmk

mk

)(mX

)(1 mHd

XmX

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Risk Analysis Workshop April 14, 2004

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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Risk Analysis Workshop April 14, 2004

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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Risk Analysis Workshop April 14, 2004

Results (1)WWW file sizes

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Risk Analysis Workshop April 14, 2004

Results (2)SUNET ATM traffic: testing for LRD

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Risk Analysis Workshop April 14, 2004

Results (3)

IP packet traffic: multifractal test

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Risk Analysis Workshop April 14, 2004

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

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Risk Analysis Workshop April 14, 2004

Interdependence of complexity notions

HT LRD Large buffers

•Gaussian limit theory•Stationary on-off modelling

•Large deviation methods in queueing theory

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Risk Analysis Workshop April 14, 2004

ON-OFF modelling

On Off

On Off

1. Choose starting state

2. Modify starting periodStationarity: OFFON

ONp

dxxFxFx

0

1 1)(~

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Risk Analysis Workshop April 14, 2004

ON-OFF aggregation

Anick-Mitra-Sondhi

On OffCumulative workload:

duuWtCTt M

iiTM

0 1

,

1,

1

kconstkTM

For HT on period:

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Risk Analysis Workshop April 14, 2004

Limit process(Taqqu, Willinger, Sherman, 1997)

tBpMTtCTMK HTM

MT

,),(

1limlim

tcpMTtCTML TM

TM

,),(

1limlim

FractionalBrownian motion

StableLévy motion

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Risk Analysis Workshop April 14, 2004

Large queues

CttBQt

)(sup0

The queue is built up by many bursts of moderate size.

Server

fBm

LDP for fBm

xIxBt

tvP

tv tt

log

1lim

Tail asymptotics for Q

xv

xCIbQP

bv xb

)(inflog

1lim

Weibull!

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Risk Analysis Workshop April 14, 2004

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 marginalsnegative values

• Models based on multiplicative cascades:

simple to generatephysical explanationseveral parameters

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Risk Analysis Workshop April 14, 2004

Thank you for your attention!