.. Point-fixe, 191 11.2.1 Le point fixe exact pour les inactifs, 11.2.2 Point fixe approximé pout les utilisateurs actifs . . . . . . . 192 11.2.3 Formules autour du point fixe et normalisation, 0193.

T. La-modélisation-fine-de, évolution de la taille de la file d'attente se fait comme dans le chapitre 8 avec les mêmes notations. Ainsi nous faisons l'hypothèse simplificatrice que TCP fonctionne dans le mode d

. La-dégradation-est-pire-que-ce and . Qu, on aurait pu attendre, ceci à cause d'un effet de levier sur le taux d'activité des utilisateurs. Dans un second temps nous tentons d'expliquer les turbulences observées dans l'article [9]. Il s'agit de montrer qu'un routeur devant un site WEB peut se trouver dans une situation où à même comportement des utilisateurs, deux situations sont possibles, l'une stable, p.autre oscillatoire

C. Adjih, P. Jacquet, and N. Vvedenskaya, Performance evaluation of a single queue under multi-user tcp/ip connections, 2002.
URL : https://hal.archives-ouvertes.fr/inria-00072110

E. Altman and T. Jimenez, Ns simulator course for beginners www-sop.inria.fr/mistral/personnel/Eitan.Altman/COURS-NS/n3.pdf. [4] , Ns simulator course for beginners, 2002.
DOI : 10.2200/s00397ed1v01y201112cnt010

G. Appenzeller, N. Mckeown, J. Sommers, and P. Barford, Recent results on sizing router buffers, Network Systems Design Conference, 2004.

J. Aweya, M. Ouellette, Y. M. Delfin, and A. Chapman, A load adaptive mechanism for buffer management, Computer Networks, vol.36, issue.5-6, 2000.
DOI : 10.1016/S1389-1286(01)00176-1

F. Baccelli and T. Bonald, Window flow control in FIFO networks with cross traffic, Queuing Syst, Theory Appl, vol.32, issue.1-3, pp.195-231, 1999.

F. Baccelli and P. Bremaud, Elements of queuing theory, 2003.

F. Baccelli, A. Chaintreau, D. D. Vleeschauwer, and D. R. Mcdonald, A mean-field analysis of short lived interacting tcp flows, Proceedings of the joint international conference on Measurement and modeling of computer systems, pp.343-354, 2004.

F. Baccelli and D. Hong, AIMD, fairness and fractal scaling of TCP traffic, Proceedings.Twenty-First Annual Joint Conference of the IEEE Computer and Communications Societies, pp.219-230, 2000.
DOI : 10.1109/INFCOM.2002.1019264

URL : https://hal.archives-ouvertes.fr/inria-00072468

F. Baccelli and K. B. Kim, TCP throughput analysis under transmission error and congestion losses, IEEE INFOCOM 2004, 2004.
DOI : 10.1109/INFCOM.2004.1354700

URL : https://hal.archives-ouvertes.fr/inria-00071626

F. Baccelli, K. B. Kim, and D. Vleeschauwer, Analysis of the competition between wired, DSL and wireless users in an access network, Proceedings IEEE 24th Annual Joint Conference of the IEEE Computer and Communications Societies., 2005.
DOI : 10.1109/INFCOM.2005.1497906

F. Baccelli, K. B. Kim, and D. R. Mcdonald, Equilibria of a class of transport equations arising in congestion control, Queueing Systems, vol.210, issue.1, 2005.
DOI : 10.1007/s11134-006-9001-x

URL : https://hal.archives-ouvertes.fr/inria-00070357

F. Baccelli and D. Mcdonald, A square root formula for the rate of non-persistent TCP flows, Next Generation Internet Networks, 2005, 2004.
DOI : 10.1109/NGI.2005.1431673

URL : https://hal.archives-ouvertes.fr/inria-00070699

F. Baccelli, D. R. Mcdonald, and J. Reynier, A mean-field model for multiple tcp connections through a buffer implementing RED, Perform, Eval, vol.49, issue.1-4, pp.77-97, 2002.

A. Bain, Fluid limits for congestion control in networks UL : Order in Manuscripts Room, Classmark : PhD.27674, 2004.

H. Balakrishnan, S. Padmanabhan, V. N. Seshan, and R. H. Katz, A comparison of mechanisms for improving TCP performance over wireless links, IEEE/ACM Transactions on Networking, issue.5, pp.756-769, 1997.

S. B. Fredj, T. Bonald, A. Proutière, G. Régnié, and J. W. Roberts, Statistical bandwidth sharing : a study of congestion at flow level, SIGCOMM '01 : Proceedings of the 2001 conference on Applications, technologies, architectures, and protocols for computer communications, pp.111-122, 2001.
URL : https://hal.archives-ouvertes.fr/hal-01266470

P. Billingsley, Convergence of probability measures, 1968.
DOI : 10.1002/9780470316962

T. Bonald and L. Massoulié, Impact of fairness on internet performance, SIGME- TRICS '01 : Proceedings of the, ACM SIGMETRICS international conference on Measurement and modeling of computer systems, pp.82-91, 2001.

R. Braden, RFC 1122 -requirements for internet hosts ? communication layers, 1989.

L. S. Brakmo and L. L. Peterson, TCP Vegas: end to end congestion avoidance on a global Internet, IEEE Journal on Selected Areas in Communications, vol.13, issue.8, 1995.
DOI : 10.1109/49.464716

P. Brémaud, Point processes and queues : Martingale dynamics, 1981.
DOI : 10.1007/978-1-4684-9477-8

M. Bursenberg, Delay differential equations and dynamical systems, Sprin- ger

A. Chaintreau and D. Vleeschauwer, A closed form formula for long-lived tcp connections throughput, Perform, Eval, vol.49, issue.14, pp.57-76, 2002.

C. Chang and Z. Liu, A Bandwidth Sharing Theory for a Large Number of HTTP-Like Connections, IEEE/ACM Transactions on Networking, vol.12, issue.5, pp.952-962, 2004.
DOI : 10.1109/TNET.2004.836118

M. Christiansen, K. Jeffay, D. Ott, and F. D. Smith, Tuning RED for web traffic, Proc. of ACM/SIGCOMM, 2000.

D. A. Dawson, Critical dynamics and fluctuations for a mean-field model of cooperative behavior, Journal of Statistical Physics, vol.11, issue.1, pp.29-85, 1983.
DOI : 10.1007/BF01010922

S. Deb and R. Srikant, Rate-based versus queue-based models of congestion control, 2004.

X. Deng, S. Yi, G. Kesidis, and C. Das, A control theoretic approach in designing adaptive aqm schemes, 2003.

A. Desolneux, L. Moisan, and J. M. , Morel, Seeing, thinking and knowing, chapter gestalt theory and computer vision, 2004.

P. Donnelly and T. Kurtz, Particle Representations for Measure-Valued Population Models, The Annals of Probability, vol.27, issue.1, pp.166-205, 1999.
DOI : 10.1214/aop/1022677258

R. M. Dudley, Real analysis and probability, 2002.
DOI : 10.1017/CBO9780511755347

V. Dumas, F. Guillemin, and P. Robert, A Markovian analysis of additive-increase multiplicative-decrease algorithms, Advances in Applied Probability, vol.III, issue.01, pp.85-111, 2002.
DOI : 10.1145/122431.122434

S. N. Ethier and T. G. Kurtz, Markov processes : Characterization and convergence, 1986.
DOI : 10.1002/9780470316658

W. Feller, An introduction to probability theory and its applications

W. Feng, D. Kandlur, D. Saha, and K. Shin, A self-configuring RED gateway, IEEE INFOCOM '99. Conference on Computer Communications. Proceedings. Eighteenth Annual Joint Conference of the IEEE Computer and Communications Societies. The Future is Now (Cat. No.99CH36320), 1999.
DOI : 10.1109/INFCOM.1999.752150

S. Floyd, Connections with multiple congested gateways in packet-switched networks part 1, ACM SIGCOMM Computer Communication Review, vol.21, issue.5, pp.30-47, 1991.
DOI : 10.1145/122431.122434

S. Floyd and T. Henderson, RFC 2582 -the newreno modification to tcp's fast recovery algorithm, 1999.

S. Floyd, T. Henderson, and A. Gurtov, RFC 3782 -the newreno modification to tcp's fast recovery algorithm, 2004.

S. Floyd and V. Jacobson, Random early detection gateways for congestion avoidance, IEEE/ACM Transactions on Networking, vol.1, issue.4, pp.397-413, 1993.
DOI : 10.1109/90.251892

S. Floyd, J. Mahdavi, M. Mathis, and M. Podolsky, RFC 2883 -an extension to the selective acknowledgement (sack) option for tcp, 2000.

S. Floyd, S. Ratnasamy, and S. Shenker, Modifying tcp's congestion control for high speeds, 2002.

R. Gibbens and F. Kelly, Resource pricing and the evolution of congestion control, Automatica, vol.35, issue.12, pp.1969-1985, 1999.
DOI : 10.1016/S0005-1098(99)00135-1

C. Graham, Chaoticity on path space for a queuing network with selection of the shortest queue among several, Journal of Applied Probabilities, issue.37, pp.198-211, 2000.

C. Graham and S. Méléard, Chaos hypothesis for a system interacting through shared ressources, Probability Theory and Related fields, pp.157-173, 1994.

H. C. Gromoll, A. L. Puha, and R. J. Williams, The fluid limit of a heavily loaded processor sharing queue, The Annals of Applied Probability, vol.12, issue.3, pp.797-859, 2001.
DOI : 10.1214/aoap/1031863171

F. Guillemin, P. Robert, and B. Zwart, Aimd algorithms and exponential functionals, Annals of Applied Probability, vol.14, issue.1, pp.90-117, 2004.
URL : https://hal.archives-ouvertes.fr/inria-00072141

C. V. Hollot, V. Misra, D. Towsley, and W. Gong, A control theoretic analysis of RED, Proceedings IEEE INFOCOM 2001. Conference on Computer Communications. Twentieth Annual Joint Conference of the IEEE Computer and Communications Society (Cat. No.01CH37213), p.10, 2001.
DOI : 10.1109/INFCOM.2001.916647

D. Hong and D. Lebedev, Many tcp user asymptotic analysis of the aimd model
URL : https://hal.archives-ouvertes.fr/inria-00072390

]. P. Hurley, L. Boudec, and P. J. , Thiran, A note on the fairness of additive increase and multiplicative decrease, p.16, 1999.

. Ikeda, . Watanabe, and K. Fukushima, Itô's stochastic calculus and probability theory
DOI : 10.1007/978-4-431-68532-6

N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, North Holland and Kodansha, 1981.

V. Jacobson, Congestion avoidance and control, SIGCOMM '88 : Symposium proceedings on Communications architectures and protocols, pp.314-329, 1988.

J. Jacod and A. Shiryaev, Limit theorems for stochastic processes, 1987.
DOI : 10.1007/978-3-662-02514-7

C. Jin, D. X. Wei, and S. H. Low, Fast tcp : motivation, architecture, algorithms, performance, Proc. of IEEE Infocom, 2004.

I. Karatzas and S. E. Shreve, Brownian motion and stochastic calculus, 1991.
DOI : 10.1007/978-1-4612-0949-2

F. Kelly, Charging and rate control for elastic traffic, European Transactions on Telecommunications, vol.13, issue.1, pp.33-37, 1997.
DOI : 10.1002/ett.4460080106

F. Kelly, A. Maulloo, and D. Tan, Rate control for communication networks: shadow prices, proportional fairness and stability, Journal of the Operational Research Society, vol.49, issue.3, pp.237-252, 1998.
DOI : 10.1057/palgrave.jors.2600523

F. Kelly and R. Williams, Fluid model for a network operating under a fair bandwidth-sharing policy, Annals of Applied Probabilities, vol.14, pp.1055-1083, 2004.

T. Kelly, Scalable TCP, ACM SIGCOMM Computer Communication Review, vol.33, issue.2, 2002.
DOI : 10.1145/956981.956989

A. Kherani and A. Kumar, Stochastic models for throughput analysis of randomly arriving elastic flows in the internet Closed loop analysis of the bottleneck buffer under adaptive window controlled transfer of HTTP-like traffic, Proceedings of IEEE INFOCOM, 2002.

K. B. Kim, Design of feedback controls supporting TCP based on modern control theories, IEEE Transactions on Automatic Control, 2004.

K. B. Kim and S. H. Low, Design of receding horizon AQM in stabilizing TCP with multiple links and heterogeneous delays, Proc. of 4th Asian Control Conference, 2002.

L. Kleinrock, Queueing systems, volume i et ii, 1975.

D. E. Knuth, The art of computer programming) sorting and searching, 1998.

H. Kunita, Stochastic flows and stochastic differential equations

S. Kunniyur and R. Srikant, End?to?end congestion control schemes : utility functions , random losses and ECN marks, Proc. of IEEE Infocom, 2000.

P. Kuusela, P. Lassila, J. Virtamo, and P. Key, Modeling red with idealized tcp sources., IFIP Conference on performance modelling and evaluation of ATM and IP networks, 2001.

T. V. 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, issue.3, pp.336-350, 1997.
DOI : 10.1109/90.611099

J. Boudec, Rate adaptation, congestion control and fairness : A tutorial

A. Legout, Contrôle de congestion multipoint pour les réseaux best effort, 2000.

J. Liu, I. Matta, and M. Crovella, End-to-end interface of loss nature in a hybrid wired, Proc. of the workshop (WiOpt'03), 2003.

Y. Liu, F. Lo, P. Misra, and D. Towsley, Fluid models and solutions for large-scale ip networks, 2003.

S. H. Low and D. E. Lapsley, Optimization flow control. I. Basic algorithm and convergence, IEEE/ACM Transactions on Networking, vol.7, issue.6, pp.861-874, 1999.
DOI : 10.1109/90.811451

L. Massoulié and J. Roberts, Bandwidth sharing: objectives and algorithms, IEEE/ACM Transactions on Networking, vol.10, issue.3, pp.320-328, 2002.
DOI : 10.1109/TNET.2002.1012364

F. Mathieu, Graphes du web, mesures d'importance à la pagerank, 2004.
URL : https://hal.archives-ouvertes.fr/tel-00667563

M. Mathis, J. Semke, J. Mahdavi, and T. Ott, The macroscopic behavior of the TCP congestion avoidance algorithm, ACM SIGCOMM Computer Communication Review, vol.27, issue.3, pp.67-82, 1997.
DOI : 10.1145/263932.264023

M. May, T. Bonald, and J. Bolot, Analytic evaluation of RED performance, Proc. of IEEE Infocom, 2000.
URL : https://hal.archives-ouvertes.fr/hal-01248747

D. R. Mcdonald and J. Reynier, Mean field convergence of a model of multiple TCP connections through a buffer implementing RED, www. eleves.ens.fr/home/jreynier/Recherche/PresentationPerf2002. [108], pp.244-294, 2002.
DOI : 10.1214/105051605000000700

URL : https://hal.archives-ouvertes.fr/inria-00071733

B. Miller, K. Avrachenkov, K. Stepanyan, and G. Miller, Flow control as stochastic optimal control problem with incomplete information, Proc. of IEEE Infocom, 2004.
URL : https://hal.archives-ouvertes.fr/inria-00070759

A. Misra and T. J. Ott, Performance sensitivity and fairness of ecn-aware "modified tcp, International IFIP-TC6 Networking Conference Proceedings, 2002.

V. Misra, W. B. Gong, and D. Towsley, Fluid-based analysis of a network of AQM routers supporting TCP flows with an application to RED, Proc. of ACM/SIG- COMM, 2000.

J. Mo, R. La, V. Anantharam, and J. Walrand, Analysis and comparison of TCP Reno and Vegas, IEEE INFOCOM '99. Conference on Computer Communications. Proceedings. Eighteenth Annual Joint Conference of the IEEE Computer and Communications Societies. The Future is Now (Cat. No.99CH36320), 1999.
DOI : 10.1109/INFCOM.1999.752178

. O. Diekmann, The cell size distribution and semigroups of linear operators., Lecture notes in biomathematics : The dynamics of physiologically structured populations

T. Ott, M. J. Kemperman, and . Mathis, The stationary behavior of ideal tcp congestion avoidance

T. J. Ott, T. V. Lakshman, and L. Wong, SRED: stabilized RED, IEEE INFOCOM '99. Conference on Computer Communications. Proceedings. Eighteenth Annual Joint Conference of the IEEE Computer and Communications Societies. The Future is Now (Cat. No.99CH36320), 1999.
DOI : 10.1109/INFCOM.1999.752153

P. Tinnakornsrisuphap and A. Makowski, Many flow asymptotics for tcp with ecn

J. Padhye, V. Firoiu, D. Towsley, and J. F. Kurose, Modeling TCP Reno performance: a simple model and its empirical validation, IEEE/ACM Transactions on Networking, vol.8, issue.2, pp.133-145, 2000.
DOI : 10.1109/90.842137

J. Padhye, . Firoiu, V. Towsley, and J. D. Kurose, Modeling tcp throughput : a simple model and its empirical validation, SIGCOMM '98 : Proceedings of the ACM SIG- COMM '98 conference on Applications, technologies, architectures, and protocols for computer communication, pp.303-314, 1998.

V. Paxson and S. Floyd, Wide-area traffic : The failure of poisson modelling, IEEE/ACM Trans. Networking, pp.226-255, 1995.

S. Pilosof, R. Ramjee, D. Raz, Y. Shavitt, and P. Sinha, Understanding TCP fairness over wireless LAN, IEEE INFOCOM 2003. Twenty-second Annual Joint Conference of the IEEE Computer and Communications Societies (IEEE Cat. No.03CH37428), 2003.
DOI : 10.1109/INFCOM.2003.1208924

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=

M. Porter, COMPETITIVE STRATEGY, Measuring Business Excellence, vol.1, issue.2, 1980.
DOI : 10.1108/eb025476

A. Proutière, Insensibilité et bornes stochastiques dans les réseaux de files d'attente . application à la modélisation des réseaux de télécommunications au niveau flot, 2004.

B. Radunovic, J. Y. Le, and . Boudec, A unified framework for max-min and minmax fairness with applications, Proceedings of 40th Annual Allerton Conference on Communication, Control, and Computing, 2002.

G. Raina and D. Wischik, How good are deterministic fluid models of internet congestion control ?, INFOCOM, 2002.

K. Ramakrishnan, S. Floyd, and D. Black, RFC 3168 -the addition of explicit congestion notification (ecn) to ip, 2001.

K. K. Ramakrishnan and R. Jain, A binary feedback scheme for congestion avoidance in computer networks with a connectionless network layer, SIGCOMM '88 : Symposium proceedings on Communications architectures and protocols, pp.303-313, 1988.

J. Reynier, Simulation des équations du champ moyen pour http

J. Roberts, U. Mocci, and J. T. , Editors) Virtamo, Broadbend network traffic : Performance evaluation and design of broadband multiservice networks, final report of action cost 242 (lecture notes in computer science

V. Rosolen, O. Bonaventure, and G. Leduc, A RED discard strategy for ATM networks and its performance evaluation with TCP/IP traffic, ACM SIGCOMM Computer Communication Review, vol.29, issue.3, 1999.
DOI : 10.1145/505724.505728

S. Schenker, L. Zhang, and D. D. Clark, Some observations on the dynamics of a congestion control algorithm, ACM SIGCOMM Computer Communication Review, vol.20, issue.5, pp.30-39, 1990.
DOI : 10.1145/381906.381931

W. Stevens, Tcp/ip illustrated : the protocols, 1999.

D. Stoyan, W. S. Kendall, and J. Mecke, Stochastic Geometry and Its Applications., Biometrics, vol.45, issue.2, 1995.
DOI : 10.2307/2531521

A. S. Sznitman, Topics in propagation of chaos, École d'Été de Probabilités de Saint-Flour, pp.165-251, 1989.

J. W. Tang and S. H. Low, Is fair allocation always inefficient, IEEE INFOCOM 2004, 2004.
DOI : 10.1109/INFCOM.2004.1354479

E. C. Titchmarsh, The theory of functions, 1932.

F. Verhulst, Nonlinear differential equations and dynamical systems

C. Villamizar and C. Song, High performance TCP in ANSNET, ACM SIGCOMM Computer Communication Review, vol.24, issue.5, pp.45-60, 1994.
DOI : 10.1145/205511.205520

G. Vinnicombe, ON THE STABILITY OF NETWORKS OPERATING TCP-LIKE CONGESTION CONTROL, Proc. of 15st IFAC World Congress on Automatic Control, 2002.
DOI : 10.3182/20020721-6-ES-1901.00370

M. Vojnovic, J. Y. Le-boudec, and C. Boutremans, Global fairness of additiveincrease and multiplicative-decrease with heterogeneous round-trip times, Proceedings of the Conference of the IEEE Computer and Communications Societies, pp.1303-1312, 2000.

. Wiki, Comment ça marche ? encyclopédie informatique, www. commentcamarche.net consulté en juin 2005. [149] , Technical historis of the internet and other network protocols, 2005.

D. Wischik and N. Mckeown, Part I, ACM SIGCOMM Computer Communication Review, vol.35, issue.3
DOI : 10.1145/1070873.1070884

L. Xu, K. Harfoush, and I. Rhee, Binary increase congestion control for fast longdistance networks, Proc. of IEEE Infocom, 2004.

L. Zhang, S. Shenker, and D. D. , Clark, Observations on the dynamics of a congestion control algorithm : The effects of two-way traffic, pp.133-147, 1991.