1 chapters 9 self-similartraffic. chapter 9 – self-similar traffic 2 introduction- motivation...
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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
Chapter 9 – Self-Similar Traffic3
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
Chapter 9 – Self-Similar Traffic4
Sierpinski triangleSierpinski triangle
……can be foundcan be found
……in chaos.in chaos.
OrderOrder
Chapter 9 – Self-Similar Traffic5
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 ??????
Chapter 9 – Self-Similar Traffic8
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?
Chapter 9 – Self-Similar Traffic9
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”
Chapter 9 – Self-Similar Traffic11
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
Chapter 9 – Self-Similar Traffic12
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
Chapter 9 – Self-Similar Traffic13
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
Chapter 9 – Self-Similar Traffic14
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
Chapter 9 – Self-Similar Traffic15
Pareto and Exponential Pareto and Exponential Examples Density Functions Examples Density Functions ComparedCompared
f f ((xx) = ) = kkkk xx
+1+1
Chapter 9 – Self-Similar Traffic16
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
Chapter 9 – Self-Similar Traffic17
Mean Waiting Time (Delay) – Mean Waiting Time (Delay) – Ethernet/ISDN StudyEthernet/ISDN Study
Chapter 9 – Self-Similar Traffic18
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
Chapter 9 – Self-Similar Traffic19
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)
Chapter 9 – Self-Similar Traffic20
Self-Similar Storage Model Self-Similar Storage Model (Norros)(Norros)
Chapter 9 – Self-Similar Traffic21
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?
Chapter 9 – Self-Similar Traffic22
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