Heavy-tailed distributions

Saint-Petersburg State University of Saint-Petersburg State University of TelecommunicationsTelecommunications

Analysis of IP-oriented Multiservice Networks Analysis of IP-oriented Multiservice Networks Characteristics with Consideration of Traffic’s Characteristics with Consideration of Traffic’s

Self-Similarity PropertiesSelf-Similarity Properties

Anatoly M. GalkinAnatoly M. [email protected]@inbox.ru

Adviser: Dr., Professor Gennady G. YanovskyAdviser: Dr., Professor Gennady G. Yanovsky

OUTLINEOUTLINE•Why IP and why self-similarity?Why IP and why self-similarity?•Self-similarity, what is it?Self-similarity, what is it?•Heavy-tailed DistributionsHeavy-tailed Distributions•Self-similarity and NetworksSelf-similarity and Networks•ConclusionsConclusions

Analysis of IP-oriented Multiservice Networks Characteristics Analysis of IP-oriented Multiservice Networks Characteristics with Consideration of Traffic’s Self-Similarity Propertieswith Consideration of Traffic’s Self-Similarity Properties

OUTLINEOUTLINE•Why IP and why self-similarity?Why IP and why self-similarity?•Self-similarity, what is it?Self-similarity, what is it?•Heavy-tailed distributionsHeavy-tailed distributions•Self-similarity and NetworksSelf-similarity and Networks•ConclusionsConclusions


NGNIP Traffic types

NGN – next generation networkNGN – next generation networkNGN is united networkNGN is united network Supports different types of trafficSupports different types of traffic Built on the base of the universal technologyBuilt on the base of the universal technology Divides switching, signaling and managementDivides switching, signaling and management Provides mentioned QoS (quality of service)Provides mentioned QoS (quality of service)

Growth of data services Active introduction of IP networks

Channel switching

Why IP ?Why IP ?

Packet switching

voice

FR

ATM

IP

NGN

...

-Data networks evolution to NGN: the problem of compatibility of technologies and standards (providing traffic transmission of different applications in united transport network)

- Voice networks evolution to NGN: the problem of conversion from Channel Switching to Packet Switching

Why IP ?Why IP ?

Packet network

UTRAN

Mobile network PSTN

LECSDSL

Broadband network

WLLAccess

Separate networks

Core

Control

Applications

Management

Media Gateway

Management system

Home subscribersRemote office/SOHOBusiness subscribersMobile subscribers

Softswitches

Application servers

NGN architectureNGN architecture

2001 year - Conceptual regulations about multiservice 2001 year - Conceptual regulations about multiservice networks structure in Russian communication networksnetworks structure in Russian communication networks

Why IP ?Why IP ?

IP oriented networksIP oriented networks

Type of traffic

Applications RequirementsTransport

layer protocols

Real time

IP telephony, videoconference

Delay sensitivityDelay jitter sensitivityLow losses sensitivity

RSVP, RTP,

RTCP,UDP

Control processes,on-line games

Delay sensitivityDelay jitter sensitivityLosses sensitivity

UDP, TCP

StreamAudio on demandVideo on demand

Internet broadcasting

Low delay sensitivityDelay jitter sensitivityLosses sensitivity

RSVP, SCTP,

UDP,TCP

Elastic

Conference of documentation

Low delay sensitivityLow delay jitter sensitivityHigh losses sensitivity

TCPAnimation, file transfer,

E-mail

Very low delay sensitivityLow delay jitter sensitivityHigh losses sensitivity

Multiservice IP network applications classification of traffic types

Why IP ?Why IP ?

Why self-similarity ?Why self-similarity ?

Problem of NGN is to provide QoS for all types of Problem of NGN is to provide QoS for all types of traffictraffic

QoS depends on service modelQoS depends on service model

Old Markovian models (memory-less), Poisson laws Old Markovian models (memory-less), Poisson laws and Erlang formulas don’t work in new networks.and Erlang formulas don’t work in new networks.

1993 year W. Lenard, M. Taqqu, W. Willinger, D. 1993 year W. Lenard, M. Taqqu, W. Willinger, D. Wilson. “On the Wilson. “On the Self-SimilarSelf-Similar Nature of Ethernet Nature of Ethernet Traffic”Traffic”



FractalsSome mathematicsHurst parameter

Self-similarity, what is it?Self-similarity, what is it?

FractalsFractals 1975 Benua Mandelbrot1975 Benua Mandelbrotfractus (lat.)– consisting of fragmentsfractus (lat.)– consisting of fragments

Fern leaf

0D 1D

2D 3D 1.5D

Fractals property – self-similarityself-similarityFractals are determined by the equations of chaoschaos Chaos deterministic chaosStochastic fractal processes are described by self-Stochastic fractal processes are described by self-similarity of statistical characteristics of the second similarity of statistical characteristics of the second orderorder

Self-similarity, what is it?Self-similarity, what is it?Notations

,...),( 21 XXX

Semi-infinite segment of second-order-stationary stochastic process

,...}2,1{NtIts discrete argument

Its parameters

functionationautocorrelXX

krkr

dispersionXD

averageXM

tkt ))((

)()(

][

][

2

2

,...),( )()(1

)(2mmm XXX AggregatedAggregated process

Nt,m,X...Xm

X tmmtm)m(

t

11

Let r(k) k-L1(k), k 10 L1 – is function slowly varying at infinity

functionationautocorrelkrm )(

Three definitionsThree definitions

1.Exactly second-order self-similar1.Exactly second-order self-similar (es-s) with the parameter H (es-s) with the parameter H=1 =1 ( ( / 2), 0< / 2), 0< <1 <1

If If rrmm((kk) = ) = rr((kk), ), kk ZZ++, , mm {2,3,…} {2,3,…} 2.Second-order asymptotical self-similar2.Second-order asymptotical self-similar (as-s) with the parameter H (as-s) with the parameter H=1 =1 ( ( / 2), 0< / 2), 0< <1 <1 If If 3.Strictly self-similar3.Strictly self-similar (ss-s) with the parameter H (ss-s) with the parameter H=1 =1 ( ( / 2), 0< / 2), 0< <1 <1

If If mm11--HH X X((mm)) = = XX, , mmNN


Nk,kgkrlimm

Process is

In other wordsIn other wordsX is es-s, if the aggregated process XX is es-s, if the aggregated process X(m)(m) is indistinguishable from the initial process X at is indistinguishable from the initial process X at

least in term of statistical characteristics in second order.least in term of statistical characteristics in second order.

X is as-s, if it meets es-s process after it is averaged on blocks of length X is as-s, if it meets es-s process after it is averaged on blocks of length m m and and mm

The relation between ss-s and es-s processes is analogous to relation between second-The relation between ss-s and es-s processes is analogous to relation between second-order stationary process and strictly stationary process order stationary process and strictly stationary process


Hurst parameterHurst parameter

0<H<1 – Hurst parameter (exponent)

Harold Edwin Hurst detected that foodless and fertile years are not random

H=0.5 – Brownian Motion

0<H<0.5 – antipersistence of the process

0.5<H<1 – persistent behaviour of the process or the process has long memory



Parameters of distributionsHeavy tailsParetoWeibullLog-normal

Probability distributionsProbability distributionsX – random valueX – random valueF(x)=P(X<x) – distribution functionF(x)=P(X<x) – distribution functionIt determines probability of random value X<x, where x is certain value It determines probability of random value X<x, where x is certain value

0≤F(x)≤10≤F(x)≤1 f(x)=dF(x)/dx – probability destiny f(x)=dF(x)/dx – probability destiny f(x)≥0f(x)≥0

M[x] – mathematical expectationM[x] – mathematical expectation

D[x] – dispersionD[x] – dispersion,, σ – – root-mean-square deviationroot-mean-square deviation

- quadratic coefficient of variation - quadratic coefficient of variation

dxxfxxM

)(][

Heavy-tailed distributionsHeavy-tailed distributions

)()( ,)()][(][ 2 xDxdxxfxMxxD

22

][][xMxDC



Self-similar processes could be described by so-called Self-similar processes could be described by so-called Heavy-tailed distributionsHeavy-tailed distributionsDefinition Definition The random variable is considered to have heavy-tailed The random variable is considered to have heavy-tailed distribution if with distribution if with 0<a<20<a<2 a – shape parameter , – shape parameter , c – a positive constant – a positive constantLight-tailed distributions (Exponential, Gaussian) have Light-tailed distributions (Exponential, Gaussian) have exponential decrease tailsexponential decrease tailsHeavy-tailed distributions have power law decrease tails Heavy-tailed distributions have power law decrease tails 0<a<2 infinite dispersion0<a<2 infinite dispersion0<a≤1 also infinite average0<a≤1 also infinite averageNetwork interest is the case 1<a<2Network interest is the case 1<a<2Then H=(3-a)/2Then H=(3-a)/2

xxcxZP a ,~][

Pareto distributionPareto distributionHeavy-tailed distributionsHeavy-tailed distributions

xbxb

baxf

xbxbxZPxF

a

a

,)(

,1][)(

1

a is the shape parameter, b is minimum value of x

Pareto Distribution

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0 2 4 6 8 10 12 14

Prob

. den

sity

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

Dis

trib

utio

n fu

nctio

n

Prob. density Distribution function

Pareto distribution is most frequently used (VoIP, FTP, HTTP)

Weibull distributionWeibull distributionHeavy-tailed distributionsHeavy-tailed distributions

0

0

10

,1)(

,

0

0

xxexF

xxexxaxf

a

a

xx

xxa

Weibull Distribution

0

0,1

0,2

0,3

0,4

0,5

0,6

0 1 2 3 4 5 6

Prob

. den

sity

0

0,2

0,4

0,6

0,8

1

1,2

Dist

ribut

ion

func

tion


a is the shape parameter, β is the averaged weight speedx0 is the minimum value of x

Weibull distribution is used for FTP

Log-normal distributionLog-normal distributionLog-normal Distribution

0

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

0,09

0,1

0 5 10 15 20 25

Prob

. den

sity

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

Dist

ribut

ion

func

tion



0,0

0,2

lnexp2

1)( 2

2

x

xmxxxf

It has a finite dispersion but has a subexponential decrease of a tailIt used for call-centers, LANs, etc.



Kendall classificationResearches of networksLimitations for real networksQoS parameters calculationNetwork modeling

KendallKendall classificationclassificationAA//BB//VV//KK//NN

1 S

1

N v

1......

B(x)A(x) K=S+V

A – law of incoming trafficB – law of servicing trafficS – queue sizeV – number of seversK – number of places in systemN – number of sources

If N=∞ then A/B/V/KOften S=∞ → K=∞ then A/B/V

Self-similarity and networksSelf-similarity and networks

Classic teletraffic modelsM/M/1, M/M/V/K , M/D/V etc.M – Poisson law

Model of servicing

xexF 1)(

D – determinate F(x)=const

1993 year W. Lenard, M. Taqqu, 1993 year W. Lenard, M. Taqqu, W. Willinger, D. Wilson. “W. Willinger, D. Wilson. “On the On the Self-SimilarSelf-Similar Nature of Ethernet Nature of Ethernet TrafficTraffic””

The period is 4 yearsThe period is 4 years From 3 pieces of Bellcore networkFrom 3 pieces of Bellcore network

It has been shown that It has been shown that 0.7<H<0.980.7<H<0.98

Poisson Measured


Further researchesFurther researchesNow – about 10000 works about self-similarityNow – about 10000 works about self-similarity

M. Taqqu, W. Willinger, K. Park, M. CrowellM. Taqqu, W. Willinger, K. Park, M. Crowell - research on the - research on the network layer.network layer.W. Willinger, M. Taqqu, R. Sherman, D. Wilson, A. Erramili, O. W. Willinger, M. Taqqu, R. Sherman, D. Wilson, A. Erramili, O. NarayanNarayan - research of the Ethernet traffic on data link layer - research of the Ethernet traffic on data link layer

In S. In S. Molnar’sMolnar’s paper VoIP traffic paper VoIP traffic is observedis observed

K. Park, G. Kim, M. Crovella,V. Almeida, A. de Oliveira, A. B. K. Park, G. Kim, M. Crovella,V. Almeida, A. de Oliveira, A. B. DowneyDowney - research of TCP applications - research of TCP applications

N. Sadek, A. Khotanzad, T. ChenN. Sadek, A. Khotanzad, T. Chen - the - the АТМАТМ traffic traffic

Researches in RussiaResearches in Russia

The interest to self-similarity in Russia was initiated by The interest to self-similarity in Russia was initiated by V.I. NeimanV.I. Neiman

Rigorous mathematics description of self-similar processes is Rigorous mathematics description of self-similar processes is given by given by B. TsibakovB. Tsibakov

Applications of self-similar processes in telecommunications Applications of self-similar processes in telecommunications are presented in the book written by are presented in the book written by O. SheluhinO. Sheluhin

Another works by Another works by A.J. Zaborovski, V.S. Gorodetski, V.V. PetrovA.J. Zaborovski, V.S. Gorodetski, V.V. Petrov

DISTRIBUTION LAWS FOR DIFFERENT TYPES OF TRAFFIC IN IP NETWORKS

Traffic Traffic typetype

Distribution Distribution lawlaw

AuthorsAuthorsАА ВВ

VoIPVoIP PP РР MolnarMolnar

FTP/TCPFTP/TCP PP W and W and LNLN

Van Van MieghemMieghemDowneyDowney

SMTP/TCPSMTP/TCP ММ ММ MolnarMolnar

HTTP/TCPHTTP/TCP PP LN and PLN and PCrovellaCrovella

Van Van Mieghem Mieghem

IPIP PP PP PaxsonPaxson

EthernetEthernet PP PP TaqquTaqqu

ATMATM DD F-ARIMAF-ARIMA SadekSadek

A is law of incoming traffic B is law of of size of protocol data blocks

M is Poisson law

P is Pareto law

LN is lognormal law

F-ARIMA is Fractal Auto-regressive Integrated moving Average

D is determinate

Further researches


Even if one source generate self-similar traffic then aggregated traffic has self-similar properties.

At the network layer aggregated traffic is described with P/P/m most adequately


Insertion of limitation for real values of random quantities

If random value is the size of protocol data block then turn-down of value is [k; L]. k is minimum size L is maximum.

Pareto Distribution

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0 2 4 6 8 10 12 14

Prob

. den

sity

Prob. density

Restricted distribution


L

Insertion of limitation for real values of random quantitiesInsertion of limitation for real values of random quantities

Restricted distribution has a finite parametersRestricted distribution has a finite parametersMx and DxMx and Dx

Then Then 2

22

MxC

- finite value

For Pareto law

kLkLLkxM

1

kLkLLkkLkL

kL 2

2222

12

kLkLLkkLkL

kLLk

kLC 2

222

2

22

121


Now we could calculate QoS parameters – delays and losses

- system load

Delays Losses

21

,22sa CCmP

sm ttt

21,

22sas CC

mt

mPt

st - average time of the packet’s service

t - average time of the packet’s staying in the buffer.

2aC and 2

sC are quadratic coefficients of variation of incoming flow and service time distributions, correspondingly

nbCC

nbCC

loss sa

sa

P22

22

2

12

1

1

nb – buffer size

- average value of the packets’ number in the queue

tm - average value of delay m

m

mm,P

11parameter


The average delay in P/G/1 system for different distribution laws of service time

Loss probability in P/G/1 system for different distributions of service time


Self-similarity boils down to packet losses, delays and congestions

Graphics

Multiservice traffic modelingMultiservice traffic modeling

GPSS General Purpose Simulating SystemGPSS General Purpose Simulating SystemAllows to research discrete models of different typesAllows to research discrete models of different types

NS2 network simulator 2NS2 network simulator 2Object-oriented discrete event simulator. Useful for simulating Object-oriented discrete event simulator. Useful for simulating local and wide area networkslocal and wide area networks


The main advantage – it is free !!!

Excel, MathCAD, MathLAB – non specialized

OPNET, COMNET ect.

ns2ns2Network simulator 2 (ns2) 1996 year Project VINT (Virtual InterNetwork Testbed), organized by DARPA (Defense research project agency)

•Specialized for existing modern technologies •Open source code software•Core modification availability•Ns2 is free product •Result visualization availability

2Mb

2Mb

2Mb2Mb

2Mb

TCP

UDP

TCP

Pareto

Pareto

FIFORED

Pareto

FIFORED

TCP1sinkTCP2sink

UDPnull

Results of modelingResults of modeling

0,2 0,4 0,6 0,80

0,05

0,1

0,15

0,2

0,25

0,3

0

modeling

analysis results

Ploss

Ploss for P/P/m

• AnimationAnimation• Trace fileTrace file

ConclusionsConclusions•NGN is based on multiservice IP-oriented network

•Providing QoS is one of the main problem

•Multiservice IP traffic has a self-similarity properties

•Old distributions (Poisson) don’t work

•IP-traffic has Heavy-tailed distributions (the main is Pareto)

•Self-similarity makes worse QoS parameters

THANK YOU !THANK YOU !

Documents

Heavy-tailed distributions