Interacting multi-class transmissions in large stochastic networks

Annals of Applied Probability

Volumn 19, Issue 6, 2009, Pages 2334-2361

  Graham, Carl ^a   Robert, Philippe ^b  

^a CENTRE DE MATHÉMATIQUES APPLIQUÉES   (France)

^b INRIA   (France)

Author keywords

Chaoticity; Coupling; Equilibrium distributions; Mean field interaction; Multi class systems; Nonlinear stochastic differential equations

Indexed keywords

EID: 70549083896     PISSN: 10505164     EISSN: 10505164     Source Type: Journal    
DOI: 10.1214/09-AAP614     Document Type: Article

References (33)

1. ADJIH, C., JACQUET, P. and VVEDENSKAYA, N. (2001). Performance evaluation of a single queue under multi-user TCP/IP connections. Technical Report RR-4141, INRIA. Available at http://hal.archives- ouvertes.fr/docs/00/07/24/84/PDF/RR-4141.pdf.
2. ALDOUS, D. J. (1985). Exchangeability and related topics. In École D'été de Probabilités de Saint-Flour, XIII - 1983. Lecture Notes in Mathematics 1117 1-198. Springer, Berlin. MR883646
3. BACCELLI, F., MCDONALD, D. R. and REYNIER, J. (2002). A mean-field model for multiple TCP connections through a buffer implementing RED. Performance Evaluation 49 77-97.
4. BELLOMO, N. and STÖCKER, S. (2000). Development of Boltzmann models in mathematical biology. In Modeling in Applied Sciences. 225-262. Birkhäuser, Boston, MA. MR1763156
5. BEN FRED, S., BONALD, T., PROUTIERE, A., RÉGNIÉ, G. and ROBERTS, J. W. (2001). Statistical bandwidth sharing: A study of congestion at flow level. In Proc. ACM SIGCOMM Comput. Commun. Rev. 111-122. ACM Press, New York.
6. BENACHOUR, S., ROYNETTE, B., TALAY, D. and VALLOIS, P. (1998). Nonlinear self-stabilizing processes. I. Existence, invariant probability, propagation of chaos. Stochastic Process. Appl. 75 173-201. MR1632193
7. BILLINGSLEY, P. (1968). Convergence of Probability Measures. Wiley, New York. MR0233396
8. CERCIGNANI, C., ILLNER, R. and PULVIRENTI, M. (1994). The Mathematical Theory of Dilute Gases. Applied Mathematical Sciences 106. Springer, New York. MR1307620
9. CHAFAI, D., MALRIEU, F. and PAROUX, K. (2009). On the long time behavior of the TCP window size process. Stochastic Process. Appl. To appear.
10. DAVIS, M. H. A. (1993). Markov Models and Optimization. Monographs on Statistics and Applied Probability 49. Chapman and Hall, London. MR1283589
11. DUMAS, V., GUILLEMIN, F. and ROBERT, P. (2002). A Markovian analysis of additive-increase multiplicative-decrease algorithms. Adv. in Appl. Probab. 34 85-111. MR1895332
12. GRAHAM, C. (1992). McKean-Vlasov Itô-Skorohod equations, and nonlinear diffusions with discrete jump sets. Stochastic Process. Appl. 40 69-82. MR1145460
13. GRAHAM, C. (2000). Kinetic limits for large communication networks. In Modeling in Applied Sciences (N. Bellomo and M. Pulvirenti, eds.) 317-370. Birkhäuser, Boston, MA. MR1763158
14. GRAHAM, C. (2008). Chaoticity for multiclass systems and exchangeability within classes. J. Appl. Probab. 45 1196-1203. MR2484171
15. GRAHAM, C. and MÉLÉARD, S. (1994). Chaos hypothesis for a system interacting through shared resources. Probab. Theory Related Fields 100 157-173. MR1296426
16. GRÜNFELD, C. P. (2000). Nonlinear kinetic models with chemical reactions. In Modeling in Applied Sciences (N. Bellomo and M. Pulvirenti, eds.) 173-224. Birkhäuser, Boston, MA. MR1763155
17. GUILLEMIN, F., ROBERT, P. and ZWART, B. (2004). AIMD algorithms and exponential func-tionals. Ann. Appl. Probab. 14 90-117. MR2023017
18. JOURDAIN, B. and MALRIEU, F. (2008). Propagation of chaos and Poincaré inequalities for a system of particles interacting through their CDF. Ann. Appl. Probab. 18 1706-1736. MR2462546
19. JOURDAIN, B., MÉLÉARD, S. and WOYCZYNSKI, W. A. (2008). Nonlinear SDEs driven by Lévy processes and related PDEs. ALEA Lat. Am. J. Probab. Math. Stat. 4 1-29. MR2383731
20. KARPELEVICH, F. I., PECHERSKY, E. A. and SUHOV, Y. M. (1996). Dobrushin's approach to queueing network theory. J. Appl. Math. Stoch. Anal. 9 373-397. MR1429262
21. KELLY, F. P., MAULLOO, A. K. and TAN, D. K. H. (1998). Rate control in communication networks: Shadow prices, proportional fairness and stability. Opsearch 49 237-252.
22. KELLY, F. P. and WILLIAMS, R. J. (2004). Fluid model for a network operating under a fair bandwidth-sharing policy. Ann. Appl. Probab. 14 1055-1083. MR2071416
23. MASSOULIÉ, L. (2007). Structural properties of proportional fairness: Stability and insensitiv-ity. Ann. Appl. Probab. 17 809-839. MR2326233
24. MASSOULIÉ, L. and ROBERTS, J. (1999). Bandwidth sharing: Objectives and algorithms, INFOCOM '99. In Eighteenth Annual Joint Conference of the IEEE Computer and Communications Societies 1395-1403. New York.
25. MCDONALD, D. R. and REYNIER, J. (2006). Mean field convergence of a model of multiple TCP connections through a buffer implementing RED. Ann. Appl. Probab. 16 244-294. MR2209342
26. MÉLÉARD, S. (1996). Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models. In Probabilistic Models for Nonlinear Partial Differential Equations (Montecatini Terme, 1995). Lecture Notes in Mathematics 1627 42-95. Springer, Berlin. MR1431299
27. OTT, T. J., KEMPERMAN, J. H. B. and MATHIS, M. (1996). The stationary behavior of ideal TCP congestion avoidance. Unpublished manuscript.
28. RAGHUNATHAN, V. and KUMAR, P. R. (2007). A counterexample in congestion control of wireless networks. Performance Evaluation 64 399-418.
29. ROBERT, P. (2003). Stochastic Networks and Queues. Applications of Mathematics 52. Springer, Berlin. MR1996883
30. ROGERS, L. C. G. and WILLIAMS, D. (1994). Diffusions, Markov Processes and Martingales, 2nd ed. Wiley Series in Probability and Mathematical Statistics 1. Wiley, Chich-ester. MR1331599
31. SRIKANT, R. (2004). The Mathematics of Internet Congestion Control. Birkhäuser, Boston, MA. MR2018967
32. STRUCKMEIER, J. (2000). Numerical simulation of the Boltzmann equation by particle methods. In Modeling in Applied Sciences (N. Bellomo and M. Pulvirenti, eds.) 371-419. Birkhäuser, Boston, MA. MR1763159
33. SZNITMAN, A.-S. (1991). Topics in propagation of chaos. In École D'Été de Probabilités de Saint-Flour XIX - 1989. Lecture Notes in Mathematics 1464 165-251. Springer, Berlin. MR1108185