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

1

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

2

Outline

Dimensioning IP networks

Queuing network models

Fluid approaches

Conclusions

3

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

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, ...

5

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

6

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

7

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

8

A simple example:

One bottleneck link

RED buffer

Elephants only (no slow start)

Modeling approach

9

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

10

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

11

IP network model

Rs(t) = PDs + Qk(t)/C

Where:

• PDs is the propagation delay for source s

12

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

13

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

14

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)

15

Basic source model

Where:

16

Mice Source Equations

17

Fluid models – extensions

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

18

Fluid models – results

19

Fluid models – model results

20

Fluid models – NS results

21

Fluid models – model results

22

Fluid models – NS results

23


24


25


26


27


28


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

29

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

30

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

31

Thank You !

32

Outline

The Internet today

Dimensioning IP networks

Queuing network models

Fluid approaches

Conclusions

33Source: Internet Software Consortium (http://www.isc.org/)

34

Source: Internet Traffic Report (http://www.internettrafficreport.com/)

35

Source: Internet Traffic Report (http://www.internettrafficreport.com/)

36Source: Sprint ATL (http://ipmon.sprint.com/packstat)April 7th 2003, 2.5 Gbps link

http://ipmon.sprint.com/packstat

41

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....

42

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

43

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

44

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

45

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

46

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

47


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

Documents

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