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A Markovian analysis of additive-increase multiplicative-decrease algorithms

Part of: Special processes Computer system organization Operations research and management science

Published online by Cambridge University Press:  19 February 2016

Vincent Dumas ,
Fabrice Guillemin  and
Philippe Robert
Vincent Dumas*
Affiliation:
Université Bordeaux
Fabrice Guillemin*
Affiliation:
France Télécom
Philippe Robert*
Affiliation:
INRIA
*
∗ Postal address: INRIA, Domaine de Voluceau, BP 105, 78153 Le Chesnay Cedex, France.
∗∗ Postal address: France Télécom R&D, DAC/CPN, 2, Avenue Pierre Marzin, 22 300 Lannion, France.
∗ Postal address: INRIA, Domaine de Voluceau, BP 105, 78153 Le Chesnay Cedex, France.
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Abstract

The additive-increase multiplicative-decrease (AIMD) schemes designed to control congestion in communication networks are investigated from a probabilistic point of view. Functional limit theorems for a general class of Markov processes that describe these algorithms are obtained. The asymptotic behaviour of the corresponding invariant measures is described in terms of the limiting Markov processes. For some special important cases, including TCP congestion avoidance, an important autoregressive property is proved. As a consequence, the explicit expression of the related invariant probabilities is derived. The transient behaviour of these algorithms is also analysed.

Keywords

TCPconvergence of invariant probabilitiesautoregressive processeshitting timesnetwork protocols

MSC classification

Primary: 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Secondary: 90B18: Communication networks 68M12: Network protocols
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 34 , Issue 1 , March 2002 , pp. 85 - 111
Copyright
Copyright © Applied Probability Trust 2002 

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Footnotes

Partly supportedby contract 00-1B320 with France Télécom andthe Future andEmer ging Technologies programme of the EU under contract number IST-1999-14186 (ALCOM-FT).

References

Adjih, C., Jacquet, P. and Vvedenskaya, N. (2001). Performance evaluation of a single queue under multi-user TCP/IP connections. Tech. Rep. RR-4141, INRIA.Google Scholar
Allman, M., Paxson, V. and Stevens, W. (1999). TCP congestion control. Request for Comment No. 2581. Available at http://www.rfc-editor.org/.Google Scholar
Altman, E., Avrachenko, K. and Barakat, C. (2000). A stochastic model of TCP/IP with stationary random losses. In Proc. ACM-SIGCOMM'00 (Stockholm, 2000) Vol. 4, pp. 231–242.Google Scholar
Altman, E., Avrachenkov, K., Barakat, C. and Nunez-Queija, R. (2001). State-dependent M/G/1 type queueing analysis for congestion control in data networks. In Proc. IEEE INFOCOM (Anchorage, Alaska, April 2001).CrossRefGoogle Scholar
Asmussen, S. (1987). Applied Probability and Queues. John Wiley, Chichester.Google Scholar
Baccelli, F. and Hong, D. (2000). TCP is max-plus linear. In Proc. ACM-SIGCOMM'00 (Stockholm, 2000) Vol. 4, pp. 219–230.Google Scholar
Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Brown, P. (2000). Resource sharing of TCP connections with different round trip times. In Proc. IEEE Infocom (Tel-Aviv, Israel, March 2000).Google Scholar
Dumas, V., Guillemin, F. and Robert, P. (2001). Limit results for Markovian models of TCP. Preprint, INRIA. Available at http://algo.inria.fr/robert/. In Proc Globecom '01 (IEEE Global Telecommunications Conference, November 2001, San Antonio).Google Scholar
Durrett, R. (1996). Probability: Theory and Examples, 2nd edn. Duxbury Press, Belmont, CA.Google Scholar
Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G. (1955). Higher Transcendental Functions, Vol. III. McGraw-Hill, New York.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. John Wiley, New York.Google Scholar
Floyd, S. (1991). Connections with multiple congested gateways in packet-switched networks, Part 1: One way traffic. Comput. Commun. Rev. 21, 30–47.CrossRefGoogle Scholar
Floyd, S., Handley, M. and Padhye, J. (2000). A comparison of equation-based AIMD congestion control. Preprint, ICSI Centre for Internet Research, Berkeley, CA.Google Scholar
Jacobson, V. (1988). Congestion avoidance and control. In Proc. SIGCOMM '88 Symposium: Communications Architectures and Protocols (August 1988), Association for Computing Machinery, New York.Google Scholar
Kelly, F. P. (2001). Mathematical modeling of the internet. In Mathematics Unlimited—2001 and Beyond (eds Engquist, B. and Schmid), W., Springer, Berlin, pp. 685–702.CrossRefGoogle Scholar
Madhavi, J. and Floyd, S. (1997). TCP-friendly unicast rate-based flow control. End2end-interest mailing list, January 1997. Available at http://www.postel.org/mailman/listinfo/end2end-interest/.Google Scholar
Meyn, S. and Tweedie, R. (1993). Markov Chains and Stochastic Stability. Springer, Berlin.CrossRefGoogle Scholar
Nummelin, E. (1984). General Irreducible Markov Chains and Nonnegative Operators. Cambridge University Press.Google Scholar
Ott, T. J., Kemperman, J. and Mathis, M. (1996). The stationary behavior of ideal TCP congestion avoidance. Unpublished manuscript.Google Scholar
Padhye, J., Firoiu, V., Towsley, D. and Kurose, J. (2000). Modeling TCP throughput: a simple model and its empirical validation. IEEE/ACM Trans. Networking 8, 133–145.Google Scholar
Roberts, J. and Massoulié, L. (1999). Bandwidth sharing and admission control for elastic traffic. In Proc. Infocom'99.Google Scholar
Rogers, L. C. G. and Williams, D. (1987). Diffusions, Markov Processes, and Martingales, Vol. 2: Itô Calculus. John Wiley, New York.Google Scholar
Stevens, W. R. (1994). TCP-IP Illustrated, Vol. 1: The Protocols. Addison-Wesley, Boston.Google Scholar
Vojnović, M. and Boudec, J.-Y. L. (2001). Some observations on equation-based rate control. In Proc. ITC'17 (Salvador de Bahia, Brazil, 2001).CrossRefGoogle Scholar
Vojnović, M., Boudec, J.-Y. L. and Boutremans, C. (2000). Global fairness of additive-increase and mutliplicative-decrease with heteregeneous round trip times. In Proc. IEEE Infocom (Tel Aviv, Israel, March 2000).Google Scholar