Network Traffic Self-Similarity
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Transcript of Network Traffic Self-Similarity
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CMPT 855 Module 7.01
Network Traffic Self-SimilarityNetwork Traffic Self-Similarity
Carey WilliamsonCarey Williamson
Department of Computer ScienceUniversity of Saskatchewan
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CMPT 855 Module 7.02
IntroductionIntroduction
A recent measurement study has shown that A recent measurement study has shown that aggregate Ethernet LAN traffic is aggregate Ethernet LAN traffic is self-similarself-similar [Leland et al 1993] [Leland et al 1993]
A statistical property that is very different A statistical property that is very different from the traditional Poisson-based modelsfrom the traditional Poisson-based models
This presentation: definition of network This presentation: definition of network traffic self-similarity, Bellcore Ethernet traffic self-similarity, Bellcore Ethernet LAN data, implications of self-similarityLAN data, implications of self-similarity
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CMPT 855 Module 7.03
Measurement MethodologyMeasurement Methodology
Collected lengthy traces of Ethernet LAN Collected lengthy traces of Ethernet LAN traffic on Ethernet LAN(s) at Bellcoretraffic on Ethernet LAN(s) at Bellcore
High resolution time stampsHigh resolution time stamps Analyzed statistical properties of the resulting Analyzed statistical properties of the resulting
time series datatime series data Each observation represents the number of Each observation represents the number of
packets (or bytes) observed per time interval packets (or bytes) observed per time interval (e.g., 10 4 8 12 7 2 0 5 17 9 8 8 2...)(e.g., 10 4 8 12 7 2 0 5 17 9 8 8 2...)
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CMPT 855 Module 7.04
Self-Similarity: The IntuitionSelf-Similarity: The Intuition
If you plot the number of packets observed If you plot the number of packets observed per time interval as a function of time, then per time interval as a function of time, then the plot looks ‘‘the same’’ regardless of the plot looks ‘‘the same’’ regardless of what interval size you choose what interval size you choose
E.g., 10 msec, 100 msec, 1 sec, 10 sec,...E.g., 10 msec, 100 msec, 1 sec, 10 sec,... Same applies if you plot number of bytes Same applies if you plot number of bytes
observed per interval of timeobserved per interval of time
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CMPT 855 Module 7.05
Self-Similarity: The IntuitionSelf-Similarity: The Intuition In other words, self-similarity implies a In other words, self-similarity implies a
‘‘fractal-like’’ behaviour: no matter what ‘‘fractal-like’’ behaviour: no matter what time scale you use to examine the data, you time scale you use to examine the data, you see similar patternssee similar patterns
Implications:Implications: Burstiness exists across many time scalesBurstiness exists across many time scales No natural length of a burstNo natural length of a burst Traffic does not necessarilty get ‘‘smoother” Traffic does not necessarilty get ‘‘smoother”
when you aggregate it (unlike Poisson traffic)when you aggregate it (unlike Poisson traffic)
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CMPT 855 Module 7.06
Self-Similarity: The MathematicsSelf-Similarity: The Mathematics Self-similarity is a rigourous statistical Self-similarity is a rigourous statistical
property (i.e., a lot more to it than just the property (i.e., a lot more to it than just the pretty ‘‘fractal-like’’ pictures)pretty ‘‘fractal-like’’ pictures)
Assumes you have time series data with Assumes you have time series data with finite mean and variance (i.e., covariance finite mean and variance (i.e., covariance stationary stochastic process)stationary stochastic process)
Must be a Must be a very long very long time series time series (infinite is best!)(infinite is best!)
Can test for presence of self-similarityCan test for presence of self-similarity
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CMPT 855 Module 7.07
Self-Similarity: The MathematicsSelf-Similarity: The Mathematics Self-similarity manifests itself in several Self-similarity manifests itself in several
equivalent fashions:equivalent fashions: Slowly decaying varianceSlowly decaying variance Long range dependenceLong range dependence Non-degenerate autocorrelationsNon-degenerate autocorrelations Hurst effectHurst effect
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CMPT 855 Module 7.08
Slowly Decaying VarianceSlowly Decaying Variance
The variance of the sample decreases more The variance of the sample decreases more slowly than the reciprocal of the sample sizeslowly than the reciprocal of the sample size
For most processes, the variance of a For most processes, the variance of a sample diminishes quite rapidly as the sample diminishes quite rapidly as the sample size is increased, and stabilizes soonsample size is increased, and stabilizes soon
For self-similar processes, the variance For self-similar processes, the variance decreases decreases very slowlyvery slowly, even when the , even when the sample size grows quite largesample size grows quite large
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CMPT 855 Module 7.09
Variance-Time PlotVariance-Time Plot
The ‘‘variance-time plot” is one means The ‘‘variance-time plot” is one means to test for the slowly decaying to test for the slowly decaying variance propertyvariance property
Plots the variance of the sample versus the Plots the variance of the sample versus the sample size, on a log-log plotsample size, on a log-log plot
For most processes, the result is a straight For most processes, the result is a straight line with slope -1line with slope -1
For self-similar, the line is much flatterFor self-similar, the line is much flatter
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Variance-Time PlotVariance-Time PlotV
aria
nce
m
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Variance-Time PlotVariance-Time PlotV
aria
nce
m
Variance of sampleon a logarithmic scale
0.0001
0.001
10.0
0.01
100.0
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Variance-Time PlotVariance-Time PlotV
aria
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m
Sample size mon a logarithmic scale
1 10 100 10 10 10 104 5 6 7
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Variance-Time PlotVariance-Time PlotV
aria
nce
m
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Variance-Time PlotVariance-Time PlotV
aria
nce
m
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Variance-Time PlotVariance-Time PlotV
aria
nce
m
Slope = -1for most processes
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Variance-Time PlotVariance-Time PlotV
aria
nce
m
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Variance-Time PlotVariance-Time PlotV
aria
nce
m
Slope flatter than -1for self-similar process
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CMPT 855 Module 7.018
Long Range DependenceLong Range Dependence
Correlation is a statistical measure of the Correlation is a statistical measure of the relationship, if any, between two random relationship, if any, between two random variablesvariables
Positive correlation: both behave similarlyPositive correlation: both behave similarly Negative correlation: behave as oppositesNegative correlation: behave as opposites No correlation: behaviour of one is No correlation: behaviour of one is
unrelated to behaviour of otherunrelated to behaviour of other
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CMPT 855 Module 7.019
Long Range Dependence (Cont’d)Long Range Dependence (Cont’d)
Autocorrelation is a statistical measure of the Autocorrelation is a statistical measure of the relationship, if any, between a random relationship, if any, between a random variable and itself, at different time lagsvariable and itself, at different time lags
Positive correlation: big observation usually Positive correlation: big observation usually followed by another big, or small by smallfollowed by another big, or small by small
Negative correlation: big observation usually Negative correlation: big observation usually followed by small, or small by bigfollowed by small, or small by big
No correlation: observations unrelated No correlation: observations unrelated
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CMPT 855 Module 7.020
Long Range Dependence (Cont’d)Long Range Dependence (Cont’d)
Autocorrelation coefficient can range Autocorrelation coefficient can range between +1 (very high positive correlation) between +1 (very high positive correlation) and -1 (very high negative correlation)and -1 (very high negative correlation)
Zero means no correlationZero means no correlation Autocorrelation function shows the value of Autocorrelation function shows the value of
the autocorrelation coefficient for different the autocorrelation coefficient for different time lags ktime lags k
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Autocorrelation FunctionAutocorrelation Function
+1
-1
0
lag k0 100Aut
ocor
rela
tion
Coe
ffic
ient
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Autocorrelation FunctionAutocorrelation Function
+1
-1
0
lag k0 100Aut
ocor
rela
tion
Coe
ffic
ient
Maximum possible positive correlation
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Autocorrelation FunctionAutocorrelation Function
+1
-1
0
lag k0 100Aut
ocor
rela
tion
Coe
ffic
ient
Maximum possiblenegative correlation
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Autocorrelation FunctionAutocorrelation Function
+1
-1
0
lag k0 100Aut
ocor
rela
tion
Coe
ffic
ient
No observedcorrelation at all
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Autocorrelation FunctionAutocorrelation Function
+1
-1
0
lag k0 100Aut
ocor
rela
tion
Coe
ffic
ient
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Autocorrelation FunctionAutocorrelation Function
+1
-1
0
lag k0 100Aut
ocor
rela
tion
Coe
ffic
ient
Significant positivecorrelation at short lags
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Autocorrelation FunctionAutocorrelation Function
+1
-1
0
lag k0 100Aut
ocor
rela
tion
Coe
ffic
ient
No statistically significantcorrelation beyond this lag
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CMPT 855 Module 7.028
Long Range Dependence (Cont’d)Long Range Dependence (Cont’d) For most processes (e.g., Poisson, or For most processes (e.g., Poisson, or
compound Poisson), the autocorrelation compound Poisson), the autocorrelation function drops to zero very quickly function drops to zero very quickly (usually immediately, or exponentially fast)(usually immediately, or exponentially fast)
For self-similar processes, the For self-similar processes, the autocorrelation function drops very slowly autocorrelation function drops very slowly (i.e., hyperbolically) toward zero, but may (i.e., hyperbolically) toward zero, but may never reach zeronever reach zero
Non-summable autocorrelation functionNon-summable autocorrelation function
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Autocorrelation FunctionAutocorrelation Function
+1
-1
0
lag k0 100Aut
ocor
rela
tion
Coe
ffic
ient
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Autocorrelation FunctionAutocorrelation Function
+1
-1
0
lag k0 100Aut
ocor
rela
tion
Coe
ffic
ient
Typical short-rangedependent process
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Autocorrelation FunctionAutocorrelation Function
+1
-1
0
lag k0 100Aut
ocor
rela
tion
Coe
ffic
ient
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Autocorrelation FunctionAutocorrelation Function
+1
-1
0
lag k0 100Aut
ocor
rela
tion
Coe
ffic
ient
Typical long-range dependent process
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Autocorrelation FunctionAutocorrelation Function
+1
-1
0
lag k0 100Aut
ocor
rela
tion
Coe
ffic
ient
Typical long-range dependent process
Typical short-rangedependent process
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CMPT 855 Module 7.034
Non-Degenerate AutocorrelationsNon-Degenerate Autocorrelations For self-similar processes, the For self-similar processes, the
autocorrelation function for the aggregated autocorrelation function for the aggregated process is indistinguishable from that of the process is indistinguishable from that of the original processoriginal process
If autocorrelation coefficients match for all If autocorrelation coefficients match for all lags k, then called lags k, then called exactlyexactly self-similar self-similar
If autocorrelation coefficients match only If autocorrelation coefficients match only for large lags k, then called for large lags k, then called asymptoticallasymptotically y self-similarself-similar
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Autocorrelation FunctionAutocorrelation Function
+1
-1
0
lag k0 100Aut
ocor
rela
tion
Coe
ffic
ient
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Autocorrelation FunctionAutocorrelation Function
+1
-1
0
lag k0 100Aut
ocor
rela
tion
Coe
ffic
ient
Original self-similarprocess
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Autocorrelation FunctionAutocorrelation Function
+1
-1
0
lag k0 100Aut
ocor
rela
tion
Coe
ffic
ient
Original self-similarprocess
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Autocorrelation FunctionAutocorrelation Function
+1
-1
0
lag k0 100Aut
ocor
rela
tion
Coe
ffic
ient
Original self-similarprocess
Aggregated self-similarprocess
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CMPT 855 Module 7.039
AggregationAggregation
Aggregation of a time series X(t) means Aggregation of a time series X(t) means smoothing the time series by averaging the smoothing the time series by averaging the observations over non-overlapping blocks observations over non-overlapping blocks of size m to get a new time series X (t)of size m to get a new time series X (t)m
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CMPT 855 Module 7.040
Aggregation: An ExampleAggregation: An Example
Suppose the original time series X(t) Suppose the original time series X(t) contains the following (made up) values:contains the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1...2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1... Then the aggregated series for Then the aggregated series for m = 2 m = 2 is:is:
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CMPT 855 Module 7.041
Aggregation: An ExampleAggregation: An Example
Suppose the original time series X(t) Suppose the original time series X(t) contains the following (made up) values:contains the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1...2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1... Then the aggregated series for m = 2 is:Then the aggregated series for m = 2 is:
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CMPT 855 Module 7.042
Aggregation: An ExampleAggregation: An Example
Suppose the original time series X(t) Suppose the original time series X(t) contains the following (made up) values:contains the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1...2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1... Then the aggregated series for m = 2 is:Then the aggregated series for m = 2 is:
4.54.5
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CMPT 855 Module 7.043
Aggregation: An ExampleAggregation: An Example
Suppose the original time series X(t) Suppose the original time series X(t) contains the following (made up) values:contains the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1...2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1... Then the aggregated series for m = 2 is:Then the aggregated series for m = 2 is:
4.5 8.04.5 8.0
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CMPT 855 Module 7.044
Aggregation: An ExampleAggregation: An Example
Suppose the original time series X(t) Suppose the original time series X(t) contains the following (made up) values:contains the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1...2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1... Then the aggregated series for m = 2 is:Then the aggregated series for m = 2 is:
4.5 8.0 2.54.5 8.0 2.5
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CMPT 855 Module 7.045
Aggregation: An ExampleAggregation: An Example
Suppose the original time series X(t) Suppose the original time series X(t) contains the following (made up) values:contains the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1...2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1... Then the aggregated series for m = 2 is:Then the aggregated series for m = 2 is:
4.5 8.0 2.5 5.04.5 8.0 2.5 5.0
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CMPT 855 Module 7.046
Aggregation: An ExampleAggregation: An Example
Suppose the original time series X(t) Suppose the original time series X(t) contains the following (made up) values:contains the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1...2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1... Then the aggregated series for m = 2 is:Then the aggregated series for m = 2 is:
4.5 8.0 2.5 5.0 6.0 7.5 7.0 4.0 4.5 5.0...4.5 8.0 2.5 5.0 6.0 7.5 7.0 4.0 4.5 5.0...
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CMPT 855 Module 7.047
Aggregation: An ExampleAggregation: An Example
Suppose the original time series X(t) Suppose the original time series X(t) contains the following (made up) values:contains the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1...2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1...
Then the aggregated time series for Then the aggregated time series for m = 5 m = 5 is:is:
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CMPT 855 Module 7.048
Aggregation: An ExampleAggregation: An Example
Suppose the original time series X(t) Suppose the original time series X(t) contains the following (made up) values:contains the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1...2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1...
Then the aggregated time series for m = 5 is:Then the aggregated time series for m = 5 is:
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CMPT 855 Module 7.049
Aggregation: An ExampleAggregation: An Example
Suppose the original time series X(t) Suppose the original time series X(t) contains the following (made up) values:contains the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1...2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1...
Then the aggregated time series for m = 5 is:Then the aggregated time series for m = 5 is:
6.06.0
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CMPT 855 Module 7.050
Aggregation: An ExampleAggregation: An Example
Suppose the original time series X(t) Suppose the original time series X(t) contains the following (made up) values:contains the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1...2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1...
Then the aggregated time series for m = 5 is:Then the aggregated time series for m = 5 is:
6.0 4.46.0 4.4
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CMPT 855 Module 7.051
Aggregation: An ExampleAggregation: An Example
Suppose the original time series X(t) Suppose the original time series X(t) contains the following (made up) values:contains the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1...2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1...
Then the aggregated time series for m = 5 is:Then the aggregated time series for m = 5 is:
6.0 4.4 6.4 4.8 ...6.0 4.4 6.4 4.8 ...
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CMPT 855 Module 7.052
Aggregation: An ExampleAggregation: An Example
Suppose the original time series X(t) Suppose the original time series X(t) contains the following (made up) values:contains the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1...2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1...
Then the aggregated time series for Then the aggregated time series for m = 10 m = 10 is:is:
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CMPT 855 Module 7.053
Aggregation: An ExampleAggregation: An Example
Suppose the original time series X(t) Suppose the original time series X(t) contains the following (made up) values:contains the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1...2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1...
Then the aggregated time series for m = 10 is:Then the aggregated time series for m = 10 is:
5.25.2
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CMPT 855 Module 7.054
Aggregation: An ExampleAggregation: An Example
Suppose the original time series X(t) Suppose the original time series X(t) contains the following (made up) values:contains the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1...2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1...
Then the aggregated time series for m = 10 is:Then the aggregated time series for m = 10 is:
5.2 5.65.2 5.6
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CMPT 855 Module 7.055
Aggregation: An ExampleAggregation: An Example
Suppose the original time series X(t) Suppose the original time series X(t) contains the following (made up) values:contains the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1...2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1...
Then the aggregated time series for m = 10 is:Then the aggregated time series for m = 10 is:
5.2 5.6 ...5.2 5.6 ...
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Autocorrelation FunctionAutocorrelation Function
+1
-1
0
lag k0 100Aut
ocor
rela
tion
Coe
ffic
ient
Original self-similarprocess
Aggregated self-similarprocess
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CMPT 855 Module 7.057
The Hurst EffectThe Hurst Effect For almost all naturally occurring time For almost all naturally occurring time
series, the rescaled adjusted range statistic series, the rescaled adjusted range statistic (also called the (also called the R/S statisticR/S statistic) for sample size ) for sample size n obeys the relationshipn obeys the relationship
E[R(n)/S(n)] = c n E[R(n)/S(n)] = c n
where:where:
R(n) = max(0, W , ... W ) - min(0, W , ... W )R(n) = max(0, W , ... W ) - min(0, W , ... W )
S (n) is the sample variance, andS (n) is the sample variance, and
W = W = X - k X for k = 1, 2, ... nX - k X for k = 1, 2, ... n
H
11 nn
2
k i ni =1
k
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CMPT 855 Module 7.058
The Hurst Effect (Cont’d)The Hurst Effect (Cont’d)
For models with only short range For models with only short range dependence, H is almost always 0.5dependence, H is almost always 0.5
For self-similar processes, 0.5 < H < 1.0For self-similar processes, 0.5 < H < 1.0 This discrepancy is called the This discrepancy is called the Hurst EffectHurst Effect, ,
and H is called the and H is called the Hurst parameterHurst parameter Single parameter to characterize Single parameter to characterize
self-similar processesself-similar processes
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CMPT 855 Module 7.059
R/S Statistic: An ExampleR/S Statistic: An Example
Suppose the original time series X(t) Suppose the original time series X(t) contains the following (made up) values:contains the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 12 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1 There are 20 data points in this exampleThere are 20 data points in this example
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CMPT 855 Module 7.060
R/S Statistic: An ExampleR/S Statistic: An Example
Suppose the original time series X(t) Suppose the original time series X(t) contains the following (made up) values:contains the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 12 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1 There are 20 data points in this exampleThere are 20 data points in this example For R/S analysis with For R/S analysis with n = 1n = 1, you get 20 , you get 20
samples, each of size 1:samples, each of size 1:
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CMPT 855 Module 7.061
R/S Statistic: An ExampleR/S Statistic: An Example
Suppose the original time series X(t) Suppose the original time series X(t) contains the following (made up) values:contains the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 12 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1 There are 20 data points in this exampleThere are 20 data points in this example For R/S analysis with n = 1, you get 20 For R/S analysis with n = 1, you get 20
samples, each of size 1:samples, each of size 1:
Block 1: X = 2, W = 0, R(n) = 0, S(n) = 0Block 1: X = 2, W = 0, R(n) = 0, S(n) = 0n 1
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CMPT 855 Module 7.062
R/S Statistic: An ExampleR/S Statistic: An Example
Suppose the original time series X(t) Suppose the original time series X(t) contains the following (made up) values:contains the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 12 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1 There are 20 data points in this exampleThere are 20 data points in this example For R/S analysis with n = 1, you get 20 For R/S analysis with n = 1, you get 20
samples, each of size 1:samples, each of size 1:
Block 2: X = 7, W = 0, R(n) = 0, S(n) = 0Block 2: X = 7, W = 0, R(n) = 0, S(n) = 0n 1
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CMPT 855 Module 7.063
R/S Statistic: An ExampleR/S Statistic: An Example
Suppose the original time series X(t) Suppose the original time series X(t) contains the following (made up) values:contains the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 12 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1 For R/S analysis with For R/S analysis with n = 2n = 2, you get 10 , you get 10
samples, each of size 2:samples, each of size 2:
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CMPT 855 Module 7.064
R/S Statistic: An ExampleR/S Statistic: An Example
Suppose the original time series X(t) contains Suppose the original time series X(t) contains the following (made up) values:the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 12 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1 For R/S analysis with n = 2, you get 10 For R/S analysis with n = 2, you get 10
samples, each of size 2:samples, each of size 2:
Block 1: X = 4.5, W = -2.5, W = 0, Block 1: X = 4.5, W = -2.5, W = 0,
R(n) = 0 - (-2.5) = 2.5, S(n) = 2.5,R(n) = 0 - (-2.5) = 2.5, S(n) = 2.5,
R(n)/S(n) = 1.0R(n)/S(n) = 1.0
n 1 2
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CMPT 855 Module 7.065
R/S Statistic: An ExampleR/S Statistic: An Example
Suppose the original time series X(t) contains Suppose the original time series X(t) contains the following (made up) values:the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 12 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1 For R/S analysis with n = 2, you get 10 For R/S analysis with n = 2, you get 10
samples, each of size 2:samples, each of size 2:
Block 2: X = 8.0, W = -4.0, W = 0, Block 2: X = 8.0, W = -4.0, W = 0,
R(n) = 0 - (-4.0) = 4.0, S(n) = 4.0,R(n) = 0 - (-4.0) = 4.0, S(n) = 4.0,
R(n)/S(n) = 1.0R(n)/S(n) = 1.0
n 1 2
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CMPT 855 Module 7.066
R/S Statistic: An ExampleR/S Statistic: An Example
Suppose the original time series X(t) Suppose the original time series X(t) contains the following (made up) values:contains the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 12 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1 For R/S analysis with For R/S analysis with n = 3n = 3, you get 6 , you get 6
samples, each of size 3:samples, each of size 3:
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CMPT 855 Module 7.067
R/S Statistic: An ExampleR/S Statistic: An Example
Suppose the original time series X(t) Suppose the original time series X(t) contains the following (made up) values:contains the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 12 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1 For R/S analysis with n = 3, you get 6 For R/S analysis with n = 3, you get 6
samples, each of size 3:samples, each of size 3:Block 1: X = 4.3, W = -2.3, W = 0.4, W = 0 Block 1: X = 4.3, W = -2.3, W = 0.4, W = 0 R(n) = 0.3 - (-2.3) = 2.6, S(n) = 2.05,R(n) = 0.3 - (-2.3) = 2.6, S(n) = 2.05,R(n)/S(n) = 1.30R(n)/S(n) = 1.30
n 1 2 3
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CMPT 855 Module 7.068
R/S Statistic: An ExampleR/S Statistic: An Example
Suppose the original time series X(t) Suppose the original time series X(t) contains the following (made up) values:contains the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 12 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1 For R/S analysis with n = 3, you get 6 For R/S analysis with n = 3, you get 6
samples, each of size 3:samples, each of size 3:Block 2: X = 5.7, W = 6.3, W = 5.6, W = 0 Block 2: X = 5.7, W = 6.3, W = 5.6, W = 0 R(n) = 6.3 - (0) = 6.3, S(n) = 4.92,R(n) = 6.3 - (0) = 6.3, S(n) = 4.92,R(n)/S(n) = 1.28R(n)/S(n) = 1.28
n 1 2 3
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CMPT 855 Module 7.069
R/S Statistic: An ExampleR/S Statistic: An Example
Suppose the original time series X(t) Suppose the original time series X(t) contains the following (made up) values:contains the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 12 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1 For R/S analysis with For R/S analysis with n = 5n = 5, you get 4 , you get 4
samples, each of size 5:samples, each of size 5:
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CMPT 855 Module 7.070
R/S Statistic: An ExampleR/S Statistic: An Example
Suppose the original time series X(t) contains Suppose the original time series X(t) contains the following (made up) values:the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 12 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1 For R/S analysis with n = 5, you get 4 samples, For R/S analysis with n = 5, you get 4 samples,
each of size 4:each of size 4:
Block 1: X = 6.0, W = -4.0, W = -3.0,Block 1: X = 6.0, W = -4.0, W = -3.0,
W = -5.0 , W = 1.0 , W = 0, S(n) = 3.41,W = -5.0 , W = 1.0 , W = 0, S(n) = 3.41,
R(n) = 1.0 - (-5.0) = 6.0, R(n)/S(n) = 1.76R(n) = 1.0 - (-5.0) = 6.0, R(n)/S(n) = 1.76
n 1 2
3 4 5
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CMPT 855 Module 7.071
R/S Statistic: An ExampleR/S Statistic: An Example
Suppose the original time series X(t) contains Suppose the original time series X(t) contains the following (made up) values:the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 12 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1 For R/S analysis with n = 5, you get 4 samples, For R/S analysis with n = 5, you get 4 samples,
each of size 4:each of size 4:
Block 2: X = 4.4, W = -4.4, W = -0.8,Block 2: X = 4.4, W = -4.4, W = -0.8,
W = -3.2 , W = 0.4 , W = 0, S(n) = 3.2,W = -3.2 , W = 0.4 , W = 0, S(n) = 3.2,
R(n) = 0.4 - (-4.4) = 4.8, R(n)/S(n) = 1.5R(n) = 0.4 - (-4.4) = 4.8, R(n)/S(n) = 1.5
n 1 2
3 4 5
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CMPT 855 Module 7.072
R/S Statistic: An ExampleR/S Statistic: An Example
Suppose the original time series X(t) Suppose the original time series X(t) contains the following (made up) values:contains the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 12 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1 For R/S analysis with For R/S analysis with n = 10n = 10, you get 2 , you get 2
samples, each of size 10:samples, each of size 10:
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CMPT 855 Module 7.073
R/S Statistic: An ExampleR/S Statistic: An Example
Suppose the original time series X(t) Suppose the original time series X(t) contains the following (made up) values:contains the following (made up) values:
2 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 12 7 4 12 5 0 8 2 8 4 6 9 11 3 3 5 7 2 9 1 For R/S analysis with For R/S analysis with n = 20n = 20, you get 1 , you get 1
sample of size 20:sample of size 20:
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CMPT 855 Module 7.074
R/S PlotR/S Plot
Another way of testing for self-similarity, Another way of testing for self-similarity, and estimating the Hurst parameterand estimating the Hurst parameter
Plot the R/S statistic for different values of n, Plot the R/S statistic for different values of n, with a log scale on each axiswith a log scale on each axis
If time series is self-similar, the resulting plot If time series is self-similar, the resulting plot will have a straight line shape with a slope H will have a straight line shape with a slope H that is greater than 0.5that is greater than 0.5
Called an R/S plot, or R/S pox diagramCalled an R/S plot, or R/S pox diagram
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R/S Pox DiagramR/S Pox DiagramR
/S S
tati
stic
Block Size n
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R/S Pox DiagramR/S Pox DiagramR
/S S
tati
stic
Block Size n
R/S statistic R(n)/S(n)on a logarithmic scale
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R/S Pox DiagramR/S Pox DiagramR
/S S
tati
stic
Block Size n
Sample size non a logarithmic scale
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R/S Pox DiagramR/S Pox DiagramR
/S S
tati
stic
Block Size n
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R/S Pox DiagramR/S Pox DiagramR
/S S
tati
stic
Block Size n
Slope 0.5
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R/S Pox DiagramR/S Pox DiagramR
/S S
tati
stic
Block Size n
Slope 0.5
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R/S Pox DiagramR/S Pox DiagramR
/S S
tati
stic
Block Size n
Slope 0.5
Slope 1.0
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R/S Pox DiagramR/S Pox DiagramR
/S S
tati
stic
Block Size n
Slope 0.5
Slope 1.0
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R/S Pox DiagramR/S Pox DiagramR
/S S
tati
stic
Block Size n
Slope 0.5
Slope 1.0Self-similarprocess
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R/S Pox DiagramR/S Pox DiagramR
/S S
tati
stic
Block Size n
Slope 0.5
Slope 1.0Slope H (0.5 < H < 1.0)(Hurst parameter)
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CMPT 855 Module 7.085
SummarySummary Self-similarity is an important mathematical Self-similarity is an important mathematical
property that has recently been identified as property that has recently been identified as present in network traffic measurementspresent in network traffic measurements
Important property: burstiness across many Important property: burstiness across many time scales, traffic does not aggregate welltime scales, traffic does not aggregate well
There exist several mathematical methods There exist several mathematical methods to test for the presence of self-similarity, to test for the presence of self-similarity, and to estimate the Hurst parameter Hand to estimate the Hurst parameter H
There exist models for self-similar trafficThere exist models for self-similar traffic