Chapters 9

1
Chapters 9

• Self-Similar
• Traffic

2
Introduction- Motivation

• Validity of the queuing models we have studied
  depends on the Poisson nature of data traffic
• Recent studies show that Internet traffic
  patterns are often bursty and self-similar, not
  Poisson (exponential)
• Patterns appear through time
• Understanding of the characteristics of
  self-similar traffic is essential to enable
  analysis of modern networks

3
Introduction- Motivation
Exponential Distribution
Exponential Density
EX ?X 1/?
F(x) PrX?x 1 e-?x
4
Order
Sierpinski triangle
can be found
in chaos.
5
Self-Similar Time Series Example
0 8 24 32 72 80 96
104 216 224 240 248 288 296 312
320 648 656 672 680 720 728 744 752
864 872 888 896 936 944 960 968
6
Self-Similar Time Series Example

• Characteristics
• Bursty
• Repeating
• Pattern independent of scale

A phenomenon that is self-similar looks the same
or behaves the same when viewed at different
degrees of aggregation.
7
Cantor Set 5 levels of recursion

• Construction of Cantor Sets
• Begin with closed interval 0,1
• Remove the open middle third of the interval
• Repeat for each succeeding interval.

• Properties
• Has structure at arbitrarily small scales
• The structure repeats

These properties do not hold indefinitely, but
over a large range of scales, many real phenomena
exhibit self-similarity.
8
Relevance in Networking

• Clustering (a.k.a. burstiness) is common in
  network traffic patterns
• patterns are typically persistent through time
• the clusters may themselves be clustered
• Poisson traffic demonstrates clustering in short
  term, but smoothes over long term (known as the
  memoryless property)
• During bursts due to clustering, queue sizes in
  switches/routers may build up more than the
  classical M/M/1 model predicts
• impact on buffer sizes?

9
Relevance in Networking

• Bottom line -
• The M/M/1 queuing model assumes that arrivals and
  service times are exponential and, therefore,
  memoryless
• real network traffic patterns might not be
  memoryless, therefore the M/M/1 model might be
  optimistic

10
Self-Similar Stochastic Processes
g(t) is similar to g(t aT), a 0, ?1, ?2,
g(t) is not similar to g(t aT), a 0, ?1, ?2,
11
Continuous-Time Self-Similar Stochastic Processes

• Continuous time 0 ?t ? ?
• A stochastic process x(t) is statistically
  self-similar with parameter H (0.5 ? H ? 1), if
  for a ? 0
• a -H x(at) has the same statistical properties as
  x(t)
• In other words
• Ex(t) mean
• Varx(t) variance
• Rx(t, s)
  autocorrelation

12
Hurst Parameter (H)

• Measure of the persistence of a statistical
  phenomenon.. the measure of the long-range
  dependence of a stochastic process
• H 0.5 indicates absence of long-term dependence
• As H approaches 1, the greater the degree of
  long-term dependence
• Note Brownian motion process B(t) is
  self-similar with H 0.5

SEE Appendix 9A
13
Heavy-Tailed Distributions
14
Pareto Heavy-Tailed Distribution

• Characteristics
• f(x) F(x) 0 (x ? k)
• F(x) 1 - (x ? k, ? ? 0)
• f(x) (x ? k, ? ? 0)
• EX k (? ? 1)
• Note that for k 1,
• ? ? 2, infinite variance
• ? ? 1, infinite mean and variance

15
Pareto and Exponential Examples Density Functions
Compared
16
Examples of Self-Similar Data Traffic

• LAN (Ethernet)
• self-similar, H 0.9
• Pareto fit for ? 1.2
• World-Wide Web
• browser traffic nicely fits Pareto distribution
  for 1.16 ? ? ? 1.5
• file sizes on WWW seem to fit this distribution
  as well
• TCP - FTP, TELNET
• session arrivals approximate Poisson
• traffic patterns are bursty heavy-tailed

17
Mean Waiting Time (Delay) Ethernet/ISDN Study
18
Findings/Implications?

• Higher loads lead to higher degrees of
  self-similarity
• performance issues most relevant at high loads
• Traditional Poisson modeling of traffic proven
  inadequate
• leads to inaccurate queuing analysis results
• increased delays, buffer size requirements
• applicable to ATM, frame relay, 100BaseT
  switches, WAN routers, etc.
• note excessive cell loss in first generation ATM
  switches
• Pareto modeling yields better (more conservative)
  results

19
Self-Similar Modeling (Norros)

• Attempts to develop reliable analytical model of
  self-similar behavior
• uses Fractional Brownian Motion (FBM) process
  (sect. 9.2) as basis
• Buffer size can often be estimated using
• q ?1/2(1-H) / (1-?)H/(1-H)
• (note that for H 0.5, this simplifies to
  ?/(1-?), the M/M/1 model)

20
Self-Similar Storage Model (Norros)
21
Findings/Implications?

• Buffer requirements much higher at lower levels
  of utilization for higher degrees of
  self-similarity (higher H)
• THEREFORE
• If higher levels of utilization are required,
  much larger buffers are needed for self-similar
  traffic than would be predicted using classical
  modeling

So, what does all this really mean?
22
Modeling and Estimating Self-Similarity

• Task determine if time series of data is
  actually is self-similar, and estimate the value
  of H
• Variance Time Plot Varx(m) vs. m
• R/S Plot logR/S vs. N
• Periodogram spectral density estimate
• Whittles Estimator estimate H, assuming
  self-similarity