Chapters 9

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Chapters 9Chapters 9

Self-SimilarSelf-SimilarTrafficTraffic

Chapter 9 – Self-Similar Traffic2

Introduction- MotivationIntroduction- Motivation Validity of the queuing models we Validity of the queuing models we

have studied depends on the have studied depends on the Poisson naturePoisson nature of data traffic of data traffic

Recent studies show that Internet Recent studies show that Internet traffic patterns are often traffic patterns are often bursty and bursty and self-similarself-similar, not Poisson , not Poisson (exponential)(exponential)

– Patterns appear through timePatterns appear through time Understanding of the characteristics Understanding of the characteristics

of self-similar traffic is essential to of self-similar traffic is essential to enable enable analysis of modern networksanalysis of modern networks


Introduction- MotivationIntroduction- Motivation

F(x) = Pr[XF(x) = Pr[Xx] = 1 – ex] = 1 – e--xx

Exponential DistributionExponential Distribution Exponential DensityExponential Density

E[X] = X = 1/

f(x) = F(x) = f(x) = F(x) = e e --

xxdd

dxdx


Sierpinski triangleSierpinski triangle

……can be foundcan be found

……in chaos.in chaos.

OrderOrder


Self-Similar Time Series Self-Similar Time Series ExampleExample 0 8 24 32 72 80 96 104 216 224 240 248 288 296 312 3200 8 24 32 72 80 96 104 216 224 240 248 288 296 312 320648 656 672 680 720 728 744 752 864 872 888 896 936 944 960 968648 656 672 680 720 728 744 752 864 872 888 896 936 944 960 968

0 72 216 288 648 720 864 9360 72 216 288 648 720 864 936

0 216 648 8640 216 648 864

3283287272

144144

144144 7272

72727272 ??????

216216 432432 216216 ??????


Relevance in NetworkingRelevance in Networking Clustering (a.k.a. burstiness) is Clustering (a.k.a. burstiness) is

common in network traffic patternscommon in network traffic patterns– patterns are typically persistent through timepatterns are typically persistent through time– the clusters may themselves be clusteredthe clusters may themselves be clustered

Poisson traffic demonstrates clustering Poisson traffic demonstrates clustering in short term, but smoothes over long in short term, but smoothes over long term (known as the “memoryless” term (known as the “memoryless” property)property)

During bursts due to clustering, queue During bursts due to clustering, queue sizes in switches/routers may build up sizes in switches/routers may build up more than the classical M/M/1 model more than the classical M/M/1 model predictspredicts

– impact on buffer sizes?impact on buffer sizes?


Relevance in NetworkingRelevance in NetworkingBottom line -Bottom line -

The The M/M/1M/M/1 queuing model queuing model assumes that arrivals and assumes that arrivals and service times are exponential service times are exponential and, therefore, “memoryless”and, therefore, “memoryless”

… … real network traffic patterns real network traffic patterns might not bemight not be memoryless, memoryless, therefore the M/M/1 model therefore the M/M/1 model might be “optimistic”might be “optimistic”


Continuous-Time Self-Similar Continuous-Time Self-Similar Stochastic ProcessesStochastic Processes Continuous time: 0 Continuous time: 0 tt A stochastic process x(A stochastic process x(tt) is statistically ) is statistically

self-similar with parameter self-similar with parameter H H (0.5 (0.5 HH 1), if for a 1), if for a 0: 0:

a a --H H x(x(atat) has the same statistical ) has the same statistical properties as x(properties as x(tt))

In other words:In other words: E[x(E[x(tt)] = )] = meanmean

Var[x(Var[x(tt)] = )] = variancevariance

RRxx((t, st, s) = ) = autocorrelationautocorrelation

E[x(E[x(atat)])]aaHH

Var[x(Var[x(atat)])]aa22HH

RRxx((at, asat, as))aa22HH


Hurst Parameter (H) Hurst Parameter (H) Measure of the persistence of a Measure of the persistence of a

statistical phenomenon….. the statistical phenomenon….. the measure of the long-range dependence measure of the long-range dependence of a stochastic processof a stochastic process

H = 0.5 indicates absence of long-term H = 0.5 indicates absence of long-term dependencedependence

As H approaches 1, the greater the As H approaches 1, the greater the degree of long-term dependencedegree of long-term dependence

Note: Note: Brownian motion processBrownian motion process B( B(tt) is ) is self-similar with H = 0.5self-similar with H = 0.5

SEE Appendix 9ASEE Appendix 9A


Heavy-Tailed Distributions Heavy-Tailed Distributions It is possible to define It is possible to define self-similar self-similar

stochastic processesstochastic processes with heavy-tailed with heavy-tailed distributions distributions

Leads to Leads to manageablemanageable distribution distribution models models

Can be used to characterize probability Can be used to characterize probability densities for traffic processes, e.g., densities for traffic processes, e.g., packet interarrival timespacket interarrival times and and burst burst lengthslengths

Distribution of random variable X is Distribution of random variable X is heavy-tailed if:heavy-tailed if:1 – F(x) = Pr[X 1 – F(x) = Pr[X xx] ~ , as ] ~ , as xx , 0 , 0

11x x


Pareto Heavy-Tailed Pareto Heavy-Tailed DistributionDistribution Characteristics:Characteristics:

f(f(xx) = F() = F(xx) = 0 () = 0 (xx k k))

F(F(xx) = 1 - () = 1 - (xx kk, , 0) 0)

f(x) = (f(x) = (xx kk, , 0) 0)

E[X] = E[X] = kk ( ( 1) 1) Note that for Note that for kk = 1, = 1,

2, infinite variance2, infinite variance 1, infinite mean and variance1, infinite mean and variance

k k

xx

k k +1+1

xx kk

-1-1


Pareto and Exponential Pareto and Exponential Examples Density Functions Examples Density Functions ComparedCompared

f f ((xx) = ) = kkkk xx

+1+1


Examples of Self-Similar Examples of Self-Similar Data TrafficData Traffic LAN (Ethernet):LAN (Ethernet):

– self-similar, H = 0.9self-similar, H = 0.9– ParetoPareto fit for fit for = 1.2 = 1.2

World-Wide Web: World-Wide Web: – browser traffic nicely browser traffic nicely fits Paretofits Pareto distribution distribution

for 1.16 for 1.16 1.5 1.5– file sizes on WWW seem to fit this file sizes on WWW seem to fit this

distribution as welldistribution as well TCP - FTP, TELNET:TCP - FTP, TELNET:

– session arrivals approximate session arrivals approximate PoissonPoisson– traffic patterns are bursty… traffic patterns are bursty… heavy-tailedheavy-tailed


Mean Waiting Time (Delay) – Mean Waiting Time (Delay) – Ethernet/ISDN StudyEthernet/ISDN Study


Findings/Implications?Findings/Implications? Higher loads lead to higher degrees of self-Higher loads lead to higher degrees of self-

similaritysimilarity – performance issues most relevant at high performance issues most relevant at high

loadsloads Traditional Poisson modeling of traffic Traditional Poisson modeling of traffic

proven inadequate proven inadequate – leads to inaccurate queuing analysis resultsleads to inaccurate queuing analysis results– increased delays, buffer size requirementsincreased delays, buffer size requirements– applicable to ATM, frame relay, 100BaseT applicable to ATM, frame relay, 100BaseT

switches, WAN routers, etc.switches, WAN routers, etc.– note excessive cell loss in first generation note excessive cell loss in first generation

ATM switchesATM switches Pareto modeling yields better (more Pareto modeling yields better (more

conservative) resultsconservative) results


Self-Similar Modeling (Norros)Self-Similar Modeling (Norros) Attempts to develop reliable analytical Attempts to develop reliable analytical

model of self-similar behaviormodel of self-similar behavior –uses Fractional Brownian Motion (FBM) uses Fractional Brownian Motion (FBM) process (sect. 9.2) as basisprocess (sect. 9.2) as basis

Buffer size can often be estimated using:Buffer size can often be estimated using:

q = q = 1/[2(1-H)]1/[2(1-H)] / (1- / (1-))H/(1-H)H/(1-H)

(note that for H = 0.5, this simplifies to (note that for H = 0.5, this simplifies to /(1-/(1-)), the M/M/1 model), the M/M/1 model)


Self-Similar Storage Model Self-Similar Storage Model (Norros)(Norros)


Findings/Implications?Findings/Implications? Buffer requirements much higher at lower Buffer requirements much higher at lower

levels of utilization for higher degrees of levels of utilization for higher degrees of self-similarity (higher H)self-similarity (higher H)

THEREFORE…THEREFORE… If higher levels of utilization are required, If higher levels of utilization are required,

much larger buffers are needed for self-much larger buffers are needed for self-similar traffic than would be predicted using similar traffic than would be predicted using classical modelingclassical modeling

So, what does all this really mean?So, what does all this really mean?


Modeling and Estimating Modeling and Estimating Self-SimilaritySelf-Similarity

Task: determine if time series of Task: determine if time series of data is actually is self-similar, and data is actually is self-similar, and estimate the value of Hestimate the value of H

– Variance Time Plot:Variance Time Plot: Var[x Var[x((mm))] vs. ] vs. mm– R/S Plot:R/S Plot: log[ log[R/SR/S] vs. ] vs. NN– Periodogram: Periodogram: spectral density spectral density

estimateestimate– Whittle’s Estimator: Whittle’s Estimator: estimate H, estimate H,

assuming self-similarityassuming self-similarity

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