Self-Similarity

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Self-Similarity

• Chapter 8

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Queuing Theory Implications

• In section 7.8 for traffic merging we assumed
  that if two Poisson streams were merged, the
  resulting stream would have a mean equal to the
  sum of the two streams
• If the traffic is not Poisson, what happens when
  we merge streams
• If it is self similar, the variance tends to
  increase, causing buffer overflow

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Self Similarity

• No natural length of a burst, at every time
  scale, similar-looking traffic bursts are
  evident.
• Aggregating streams of such traffic intensifies
  the self-similarity instead of smoothing it
• Aggregation causes more burstiness and requires
  larger buffers

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Cantor Set
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Self Similar Intuition

• Structure repeats at all scales
• Queue size builds up more than would be expected
• Aggregation of self-similar sources is not
  smooth
• Just as Stochastic processes are invariant to
  time, self-similar processes are invariant to
  scale.

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Self-Similarity
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Formal Definition

• A stochastic process is self similar with
  parameter H if

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m-aggregated time series
Self-similar if autocorrelation remains constant
for all m
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Bellcore 1992 ethernet study
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Bellcore Study

• 4 years 1989-1992, IP Traffic NFS, email etc

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Noticing Self-similarity

• the variance of the sample mean decreases more
  slowly than the reciprocal of the sample size
  (slowly decaying variances), i.e., var(x (m) )
  decreases more slowly than 1/m (1/mB), as m ,
  with 0 lt B lt 1
• the autocorrelations decay hyperbolically rather
  than exponentially fast, implying a non-summable
  autocorrelation function Sk r (k)
  (long-rangedependence),
• Normally r (m) (k) 0, as m ( k 1, 2, . .
  . ). t 2N/K 1, . . .

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Statistics
H1
rautocorrelation svariance
H.5
Degree of aggregation Estimate H0.79
Variance decreases linearly on log-log plot
indicating that the variance is decreasing more
slowly than the reciprocal of m
Line with slope-1 estimate slope-.4 estimate
H.8
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Statistics
Periodgram plot, spectral density centered around
0
Hurst parameter using three previously mentioned
methods
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Statistics
Estimate from Pox diagram Estimate from variance
Estimates of H for low, medium, high use periods
of the day for 1989 August Busy15 (120 Hosts)
1989 October Busy30 after upgrade to RISC
machines (140 Hosts)
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Statistics
1990 (1200 hosts) Link between 2 Labs at Bellcore
and Internet
1992 More Routers, Firewall (600 hosts outside of
router)
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Hurst Parameter
High traffic Medium traffic Low traffic
Number of packets
95 Confidence Interval
Number of bytes
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Other Self-similar traffic

• WWW Traffic
• SS7 Traffic
• TCP, FTP, Telnet
• VBR Video

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Mean Waiting Time
Poor Correlation
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Self-similar Storage Model
Self-similar requires larger queues
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Self-Similar Model
For H1/2, reduces to normal buffer requirements
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Joseph Effect
Joseph in Egypt, 7 years of plenty followed by 7
years of drought, self similar on Nile, difficult
to predict the size of storage tank necessary
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Joseph Effect

• Although there were periodic fluxuations in
  rainfall, in this case a minimum was reflected at
  different time scales up to 7 years.
• What about the next scale up (2000 years)
• Can you apply the Hurst parameter to the
  millenium? (1000 years of peace are coming)
• Could you model the amount of food storage to
  prepare for?