Origins of Long Range Dependence Myths and Legends

1
Origins of Long Range Dependence Myths and
Legends

• Aleksandar Kuzmanovic
• 01/08/2001

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Outline

• Definitions
• Why is LRD important?
• Heavy tails
• Producing self-similar traffic
• Physical interpretation in LAN and WAN networks
• Different hypothesis from around 10 papers

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On the Self-Similar Nature of Ethernet Traffic,
W. Willinger, 1994
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Definitions

• Long range dependent process
• if its autocorrelation function is nonsummable
• Self-similar process
• scaling behavior of finite dimensional
  distributions
• X(m(1-H))X(m) in distribution
• Second order self-similar process
• aggregated processes possess the same
  non-degenerate AC functions as the original
  process
• X and (m(1-H))X(m) have the same AC function
• Self-similar processes have hyperbolically
  decaying autocorrelation functions - LRD can be
  characterized by a single parameter H

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Heavy tails (Noah effect)

• Heavy-tailed distributions
• LLCD
• Pareto a typical example

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Producing Self-Similar Traffic

• 1. Multiplexing ON/OFF sources that have a fixed
  rate in ON periods and ON/OFF period lengths that
  are heavy tailed.
• Aggregate traffic is fBm with
• 2. queue model
• implies that multiplexing constant-rate
  connections with Poisson connection arrivals and
  a heavy-tailed distribution for connection
  lifetimes would result in self-similar traffic
• 3. Inter-arrival packet times are i.i.d. Pareto
  with
• and then consider the corresponding count process
  (the number of arrivals in consecutive
  intervals), we have pseudo self-similar traffic
  (Paxson, Floyd) (or even self-similar (L.
  Lipsky)?)

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Questions we want to answer

• What physical activity causes LRD?
• What is the role of protocols (TCP and MAC layer
  protocols)?
• What is the role of limited resources (i.e.
  bandwidth)?
• What model fits best to each of the assumptions?
• What is the largest time-scale over which the
  correlation is present?
• Self-similarity vs. pseudo self-similarity and
  relevance

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Statistical Analysis of Ethernet LAN Traffic at
the Source Level, W. Willinger, 1997, I
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Statistical Analysis of Ethernet LAN Traffic at
the Source Level, W. Willinger, 1997, II

• Model 1 (heavy tailed ON/OFF activity at the
  source level) is widely accepted
• Result proven theoretically
• Noah effect (heavy-tailed periods)
• ON periods alpha 1.7
• OFF periods alpha 1.2
• TCP traffic measured most of the time...
• Higher load - H increases
• WAN measurements do not fit into this model
• connection typically do not stay long

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Wide Area Traffic The Failure of Poisson
Modeling, V. Paxson, S. Floyd, 1995

• Summary of ways to produce LRD traffic
• WAN (TCP) traffic for TELNET and FTP applications
• TELNET connection arrivals appear to be Poisson,
  but packet arrivals are not
• Single TELNET connection is LRD
• Model 3 Inter-arrival times are i.i.d. Pareto
• Aggregate is also LRD, but there is no analytical
  proof ()
• FTP traffic also LRD, yet non of the models fit
  because of limited resources.
• Aggregated traffic is not fBm (single H is not
  enough)

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Explaining WWW Traffic Self-Similarity, M.
Crovella, 1995

• WWW traffic is self-similar
• but only when load is high (i.e. in busiest
  hours)
• Authors force model 1 (ON/OFF model)
• The distribution of
• transfer times (alpha 1.21)
• user requests for documents (alpha 1.06)
• document sizes available in the Web (alpha
  1.05)
• user think times (alpha 1.5)
• H increases as the load increases (same as in LAN)

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On the Relationships betw. file sizes, tran.
prot. and s-s netw. traffic, M. Crovella, 1996

• Model 1 The success of this simple model is
  surprising given that it ignores non-linarities
  arising in real networks
• Hypothesis
• Heavy tailed file size distributions together
  with TCP is responsible for LRD
• if UDP is used, there is little or no LRD
• Explanation
• In some sense, the effect of the unaccounted for
  nonlinearity is reflected back as a stretching in
  time effect, thus conforming to the models
  original suppositions
• Other interesting stuff mix of Pareto and exp.
  background traffic

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On the Propagation of LRD in the Internet, A.
Veres, 2000, I

• Not about roots, but about propagation of
  self-similarity by TCP
• A(t) C - B(t)
• TCP is a linear system beyond a characteristic
  time scale
• if it adapts well to a background traffic, it
  itself becomes self-similar

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On the Propagation of LRD in the Internet, A.
Veres, 2000, II

• Experimental proof
• NY-Budapest file transfer, source is not LRD -
  traffic is LRD (H0.76)
• Max time scale 8 min
• Also, if there is number of on-off TCP
  connections, they can spread LRD
• W. Willinger obviously does not like this paper
• This is a fraud and has no relevance for LRD
  observed on link level...
• Protocols have no impact on LRD, they just have
  to send the data generated by applications...

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TCP Congestion Control and Heavy-Tails, M.
Crovella, 2000, I

• Switch to Model 3 (Heavy-tailed inter-packet
  arrivals)
• Although heavy-tailed flow lengths are commonly
  associated with heavy-tailed file sizes, there is
  no strong correlation between file sizes and
  transmission times
• It has been shown that TCP can show heavy-tailed
  inter-arrival times under some
• conditions
• Because most of the
• connections are short
• lived (!) only slow start
• and exp. back-off were
• considered

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TCP Congestion Control and Heavy-Tails, M.
Crovella, 2000, II

• Simple Markov chain model for exp. backoff and
  slow start with pr. of loss parameter
• State probability with different loss rates
• For alpha to be
• between 1 and 2,
• p has to be between
• 1/8 and 1/4
• ...but for different model
• p increases gt
• H increases

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TCP Congestion Control and Heavy-Tails, M.
Crovella, 2000, III

• Pathological TCP connections 15 packets
• Analytical model not that good (borders are
  loose)
• For this set-up, correlation up to 1000 sec
• For larger file sizes, up to 200-300 sec
• Under certain conditions, heavy tailed
  transmission times can occur even in the absence
  of any variability in file sizes
• Future work to consider the variability in
  round-trip time estimation

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On the Autocorrelation Structure of TCP Traffic,
Don Towsley, 2000, I

• Answer to previous two papers
• TCP can create self-similarity but over finite
  range of time scales - pseudo self similarity
• but everything in nature is finite (thus
  pseudo)
• Also criticize pathological model of previous
  paper, but they themselves use pathological model
  of different kind (always packets model)
• Separate Markovian models for Congestion
  avoidence (CA) and Time Out (TO) models
• Simulated these two models with different loss
  probability parameters

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On the Autocorrelation Structure of TCP Traffic,
Don Towsley, 2000, II

• Range of time scales observed from the simulation
  (26RTT(2.5 to 10)) gt 29RTT
• Explanation on why aggregate is self-similar
• independent bottlenecks (at the edge)
• aggregate of independent pseudo-self-similar
  flows should be self-similar itself ()

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On the Autocorrelation Structure of TCP Traffic,
Don Towsley, 2000, III

• !About Veres paper
• compute loss probability (0.08 to 0.14)
• TO model predicts H0.69-0.72 (really measured
  0.74)
• Time scale goes up to 26 RTO (also near measured
  value)
• Experiments (file transfers)
• North-South America
• Measurements p 0.13, H 0.77, ts (27 to
  28)RTT
• TO model p 0.12, H 0.72, ts (27 to
  29)RTT
• East - West Coast
• Measurements p 0.018, H 0.86, ts 26RTT
• CA model p 0.018, H 0.75, ts
  24RTT
• One should be careful when attributing the origin
  of traffic characteristics to a specific cause

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Protocols Can Make Traffic Appear Self-Similar,
Jon Peha, 1997. I

• How basic retransmission mechanism can cause
  self-similarity
• No model, only experimental investigation
• Simple single queue (bottleneck) model
• Input traffic - Poisson retransmissions are
  bursty
• As time-scale gets larger, burstiness from
  original Poisson traffic decreases, but
  burstiness from retransmissions stays the same!
• Unlikely that traffic from retransmission
  mechanism cause truly self similar traffic,
  rather pseudo self-similarity

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Protocols Can Make Traffic Appear Self-Similar,
Jon Peha, 1997. II

• Pictorial
• proof

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Protocols Can Make Traffic Appear Self-Similar,
Jon Peha, 1997. III

• Cut-off time scales observed
• 150Mbps link rate, 500 bits packets, RTT 60 msec
• TS 5 minutes
• 10Mbps Ethernet, No. of retransmissions5, To125
• TS in range of minutes
• For larger To, it is possible to reach time
  scales measured at Bellcore
• I have computed cut-off time-scale for Veres
  paper
• 128 Kbps, Tout10RTT2 sec, TS8min
• If this effect is found to be as strong in more
  complex models, this could be a significant cause

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The Second-order Characteristics of TCP,
J.Y.Boudec, 1996, I

• Pseudo self similarity (TS20-30 sec)
• Minimum bottleneck bandwidth 34Mbps (?)
• Two main reasons (both heavy-tailed)
• Burst length arrivals
• Round trip time
• Real network measurements
• Figure - missing

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The Second-order Characteristics of TCP,
J.Y.Boudec, 1996, II

• Even for 34Mbps link and utilization of 25, the
  arrival bursts are eliminated and the inter
  packet times are dependent on the round trip
  times
• The aggregate of TCP connections have the same H
  as a single TCP connection ()
• It seems likely that the heavy tailed
  distributions observed in Willingers work were a
  result of, among other things, the heavy tailed
  distribution of a round trip time

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More on RTTs

• Why are round trip times heavy-tailed?
• Because of TCP congestion control?
• Because of retransmissions?
• Because of variety of destinations?
• It can be heavy-tailed even without any
  congestion protocol or different destinations!
• Measurement and Analysis of LRD Behavior of
  Internet Packet Delay, M. Borella, Infocom 97
• Constant UDP transmissions - LRD response
• Is cross-traffic heavy-tailed?
• Or multiple bottlenecks assumption?
• Simple example (not through bandwidth adaptation,
  but through RTT adaptation)

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Summary

• Heavy-tailed parameters
• File sizes
• Connection life-times
• Inter-arrival packet times
• Document sizes available in the web
• User think times
• TELNET packet arrivals
• Round trip times

• Pseudo self-similarity
• it should be clear that the range of time scales
  covered is far beyond dominant time scales, and
  as long as packet loss is concerned, this is
  relevant

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Conclusions

• One should be careful when attributing the origin
  of traffic characteristics to a specific cause
• There is more than one physical activity causing
  LRD
• Protocols (TCP) influence is more than relevant
• Time scales covered are relevant in both
  generation, time-stretching and propagation
  hypothesis
• Model 3 (inter-arrival times i.i.d. Pareto) plus
  heavy-tailed file sizes (introducing congestion)
  is promising
• Analytical proof for aggregate is missing
  (simulation proof reported in 3 papers)
• Round-trip times hypothesis might be promising -
  supports Veres idea in a slightly different way