Using%20Partial%20Differential%20Equations%20to%20Modek%20TCP%20Mice%20and%20Elephantsin%20large%20IP%20Networks

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Using Partial Differential Equations to Modek TCP
Mice and Elephantsin large IP Networks

• M. Ajmone Marsan, M. Garetto, P. Giaccone,
• E. Leonardi, E.Schiattarella, A. Tarello
• Politecnico di Torino - Italy

Hong-Kong March 7-11 , 2004 TANGO
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Outline

• Dimensioning IP networks
• Queuing network models
• Fluid approaches
• Conclusions

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Consideration

• Over 90 of all Internet traffic is due to TCP
  connections
• TCP drives both the network behavior and the
  performance perceived by end-users
• Analytical models of TCP are a must for IP
  network design and planning

4
Consideration

• Accurate TCP models must consider
• closed loop behavior
• short-lived flows
• multi-bottleneck topologies
• AQM schemes (or droptail)
• QoS approaches, two-way traffic, ...

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Problem statement
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finite flows (mice)
1
greedy flows
? URLs/sec
IP core
? URLs/sec
finite flows
3
N
greedy flows (elephants)
4
...
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Problem statement
Input variables only primitive network
parameters

• IP network channel data rates, node distances,
  buffer sizes, AQM algorithms (or droptail), ...
• TCP number of elephants, mice establishment
  rates and file length distribution, segment size,
  max window size, ...

Output variables

• IP network link utilizations, queuing delays,
  packet loss probabilities, ...
• TCP average elephant window size and
  throughput, average mice completion times, ...

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Modeling approach

• Abandon a microscopic view of the IP network
  behavior, and model packet flows and other system
  parameters as fluids
• The system is described with differential
  equations
• Solutions are obtained numerically

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Modeling approach

• A simple example
• One bottleneck link
• RED buffer
• Elephants only (no slow start)

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TCP model
dWs(t)/dt 1/Rs(t) Ws(t) ?s(t) / 2

• Where
• Ws(t) is the average window
• Rs(t) is the average round trip time
• ?s(t) is the congestion indication rate
• of TCP sources of class s at time t

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IP network model
dQk(t)/dt Ss Ws(t) (1-P(t)) / Rs(t) - C
1Qk(t)gt0

• Where
• Qk(t) is the length of queue k at time t

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IP network model
Rs(t) PDs Qk(t)/C

• Where
• PDs is the propagation delay for source s

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Problems
Difficult to deal with mice since the start time
of each mouse detemines the window dynamics over
time. One equation shoud be written for each
mouse Difficult to consider droptail buffers due
to the intrinsic burstiness of the loss process
experienced by sources
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Problems
Difficult to deal with mice since the start time
of each mouse detemines the window dynamics over
time. One equation shoud be written for each
mouse
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Our Approach

• Consider a population of TCP sources P(w,t) is
  the number of TCP flows that at time t have
  window not greater than w.
  
  .

Partial differential equations are obtained
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Basic source model
Where
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Mice Source Equations
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Fluid models extensions

• Slow start (mice)
• Finite window
• Threshold
• Fast recovery
• Droptail buffers
• Core network topologies

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Fluid models results
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Fluid models model results
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Fluid models NS results
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Fluid models model results
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Fluid models NS results
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Fluid models results
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Fluid models results
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Fluid models results
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Fluid models results
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Fluid models results
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Fluid models results
We obtained results for the GARR network with
over one million TCP flows, and link capacities
up to 2.5 Gb/s.
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Conclusions

• Fluid models today seem the most promising
  approach to study large IP networks
• Tools for the model development and solution are
  sought
• Efficient numerical techniques are a challenge

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Conclusions

• The modeling paradigms to study the Internet
  behaviour are changing
• This is surely due to scaling needs, but
  probably also corresponds to a new phase of
  maturity in Internet modeling
• Reliable predictions of the behavior of
  significant portions of the Internet are within
  our reach

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Thank You !
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Outline

• The Internet today
• Dimensioning IP networks
• Queuing network models
• Fluid approaches
• Conclusions

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Source Internet Software Consortium
(http//www.isc.org/)
34
Source Internet Traffic Report
(http//www.internettrafficreport.com/)
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Source Internet Traffic Report
(http//www.internettrafficreport.com/)
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Source Sprint ATL (http//ipmon.sprint.com/packst
at) April 7th 2003, 2.5 Gbps link
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Source Sprint ATL (http//ipmon.sprint.com/packst
at) April 7th 2003, 2.5 Gbps link
38
Source Sprint ATL (http//ipmon.sprint.com/packst
at) April 7th 2003, 2.5 Gbps link
39
Source Sprint ATL (http//ipmon.sprint.com/packst
at) April 7th 2003, 2.5 Gbps link
40
Source Sprint ATL (http//ipmon.sprint.com/packst
at) April 7th 2003, 2.5 Gbps link
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And still growing ...
Subject news Internet still growing 70 to 150
per cent per year Date Mon, 23 Jun 2003 095545
-0400 (EDT) From CAnet-NEWS_at_canarie.ca ... Andre
w Odlyzko, director of the Digital Technology
Center at the University of Minnesota, ... says
Internet traffic is steadily growing about 70
percent to 150 percent per year. On a conference
call yesterday to discuss the results, he said
traffic growth slowed moderately over the last
couple of years, but it had mostly remained
constant for the past five years. ...
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Literature
V. Misra, W. Gong, D. Towsley, "Stochastic
Differential Equation Modeling and Analysis of
TCP Windowsize Behavior, Performance'99 T.
Bonald, "Comparison of TCP Reno and TCP Vegas via
Fluid Approximation," INRIA report no. 3563,
November 1998 V. Misra, W. Gong, D. Towsley, "A
Fluid-based Analysis of a Network of AQM Routers
Supporting TCP Flows with an Application to RED,
SIGCOMM 2000
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Literature
Y.Liu, F.Lo Presti, V.Misra, D.Towsley, Y.Gu,
"Fluid Models and Solutions for Large-Scale IP
Networks", SIGMETRICS 2003 F. Baccelli, D.Hong,
"Interaction of TCP flows as Billiards, Infocom
2003 F.Baccelli, D.Hong, "Flow Level Simulation
of Large IP Networks, Infocom 2003
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Literature
T. Lakshman and U. Madhow, "The performance of
TCP/IP for networks with high bandwidth-delay
products and random loss," IEEE/ACM Transactions
on Networking, vol. 5, no. 3, 1997. M.Ajmone
Marsan, E.de Souza e Silva, R.Lo Cigno, M.Meo,
An Approximate Markovian Model for TCP over
ATM, UKPEW '97 J. Padhye, V. Firoiu, D.
Towsley, J. Kurose, "A Stochastic Model of TCP
Reno Congestion Avoidance and Control, UMASS
CMPSCI Technical Report, Feb 1999.
45
Literature
C.Casetti, M.Meo, A New Approach to Model the
Stationary Behavior of TCP Connections, Infocom
2000 M.Garetto, R.Lo Cigno, M.Meo, E.Alessio,
M.Ajmone Marsan, Modeling Short-Lived TCP
Connections with Open Multiclass Queueing
Networks, PfHSN 2002 A.Goel, M.Mitzenmacher,
"Exact Sampling of TCP Window States", Infocom
2002
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Consideration
Developing accurate analytical models of the
behavior of TCP is difficult. A number of
approaches have been proposed, some based on
sophisticated modeling tools.
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Fluid models results