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 Telecommunications and Media Informatics Budapest University of Technology and Economics. Traffic models Past and present Complexity notions - PowerPoint PPT PresentationTRANSCRIPT
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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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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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Traffic models
Packet level
Traffic intensity# of packetsBytes
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
Time
Space
Finite variance
Independent increments
Heavy tails(”Noah”)
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.:
1)(
lim22 Hk kc
k
0,),(1)()( xxLxxXPxF
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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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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 < H < 1
• Multifractals: different moments scale differently
,..2,1,...1
1)( kXX
mX mkmmk
mk
)(mX
)(1 mHd
XmX
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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 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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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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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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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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Limit process(Taqqu, Willinger, Sherman, 1997)
tBpMTtCTMK HTM
MT
,),(
1limlim
tcpMTtCTML TM
TM
,),(
1limlim
FractionalBrownian motion
StableLévy motion
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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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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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Thank you for your attention!