# 1 Using Partial Differential Equations to Modek TCP Mice and Elephantsin large IP Networks M. Ajmone Marsan, M. Garetto, P. Giaccone, E. Leonardi, E.Schiattarella,

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1 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

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 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 greedy flows 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 dW s (t)/dt = 1/R s (t) – W s (t) s (t) / 2 Where: W s (t) is the average window R s (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 dQ k (t)/dt = Σ s W s (t) (1-P(t)) / R s (t) – - C 1 {Qk(t)>0} Where: Q k (t) is the length of queue k at time t

11 IP network model R s (t) = PD s + Q k (t)/C Where: PD s 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 Fluid models – results

24 Fluid models – results

25 Fluid models – results

26 Fluid models – results

27 Fluid models – results

28 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.

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

33 Source: 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/)

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

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

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

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

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

41 And still growing... Subject: [news] Internet still growing 70 to 150 per cent per year Date: Mon, 23 Jun 2003 09:55:45 -0400 (EDT) From: CAnet-NEWS@canarie.ca... 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 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 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 Multiclass Queueing 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 Fluid models – results

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