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Using Partial Differential Equations Using Partial Differential Equations to Modek TCP Mice and Elephantsin to Modek TCP Mice and Elephantsin
large IP Networkslarge IP Networks
M. Ajmone Marsan, M. Garetto, P. Giaccone,M. Ajmone Marsan, M. Garetto, P. Giaccone,E. Leonardi, E. Leonardi, E.Schiattarella, A. TarelloE.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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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
Consideration
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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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1 2
3
4...
N
URLs/sec
URLs/sec
greedy flows
4N F
23 F
finite flows (mice)
finite flows
greedy flows (elephants)
IP core
Problem statement
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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, ...
Problem statement
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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
Modeling approach
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A simple example:
One bottleneck link
RED buffer
Elephants only (no slow start)
Modeling approach
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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 = Σs Ws(t) (1-P(t)) / Rs(t)
– - C 1{Qk(t)>0}
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
window
w
P(w,t)
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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
33Source: Internet Software Consortium (http://www.isc.org/)
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Source: Internet Traffic Report (http://www.internettrafficreport.com/)
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Source: Internet Traffic Report (http://www.internettrafficreport.com/)
36Source: Sprint ATL (http://ipmon.sprint.com/packstat)April 7th 2003, 2.5 Gbps link
37Source: Sprint ATL (http://ipmon.sprint.com/packstat)April 7th 2003, 2.5 Gbps link
38Source: Sprint ATL (http://ipmon.sprint.com/packstat)April 7th 2003, 2.5 Gbps link
39Source: Sprint ATL (http://ipmon.sprint.com/packstat)April 7th 2003, 2.5 Gbps link
40Source: Sprint ATL (http://ipmon.sprint.com/packstat)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 yearDate: Mon, 23 Jun 2003 09:55:45 -0400 (EDT)From: [email protected]
...Andrew 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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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 FluidApproximation," 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
Literature
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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
Literature
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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.
Literature
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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 MulticlassQueueing Networks”, PfHSN 2002
A.Goel, M.Mitzenmacher, "Exact Sampling of TCP Window States", Infocom 2002
Literature
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Developing accurate analytical models of the behavior of TCP is difficult.
A number of approaches have been proposed, some based on sophisticated modeling tools.
Consideration
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Fluid models – results