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Steady state and scaling limit for a traffic congestion model

Published online by Cambridge University Press:  29 October 2010

Ilie Grigorescu  and
Min Kang
Ilie Grigorescu*
Affiliation:
Department of Mathematics, University of Miami, Coral Gables, FL, 33124-4250, USA
Min Kang*
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA
*
Corresponding authors: [email protected], [email protected]
Corresponding authors: [email protected], [email protected]
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Abstract

In a general model (AIMD) of transmission control protocol (TCP)used in internet traffic congestion management, the time dependentdata flow vector x(t) > 0 undergoes a biased random walk ontwo distinct scales. The amount of data of each component x i (t)goes up to x i (t)+a with probability 1-ζ i (x) ona unit scale or down to γ x i (t), 0 < γ < 1 withprobability ζ i (x) on a logarithmic scale, where ζ i depends on the joint state of the system x. Weinvestigate the long time behavior, mean field limit, and the oneparticle case. According to c = lim inf|x|→∞ |x|ζ i (x), the process drifts to ∞ in thesubcritical c < c + (n, γ) case and has an invariantprobability measure in the supercritical case c > c + (n, γ).Additionally, a scaling limit is proved when ζ i (x)and a are of order N –1 and t → N t, in the form of acontinuum model with jump rate α(x).

Keywords

TCPAIMDfluid limitmean field interaction
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 14 , 2010 , pp. 271 - 285
Copyright
© EDP Sciences, SMAI, 2010

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References

G. Appenzeller, I. Keslassy and N. McKeown, Sizing router buffer. In Proc. of the 2004 Conf. on Applications, Technologies, Architectures, and Protocols for Computers Communications, Portland, OR, USA. ACM New York, NY (2004), pp. 281-292.
Baccelli, F., McDonald, D. and Reynier, J., A mean-field model for multiple TCP connections through a buffer implementing RED. Performance Evaluation Archive 49 (2002) 77-97. CrossRef
F. Baccelli, A. Chaintreau, D. De Vleeschauwer and D. McDonald, A mean-field analysis of short lived interacting TCP flows. In Proc. of the Joint Int. Conf. on Measurement and Modeling of Computer Systems, New York, NY, USA, June 10–14, 2004 (SIGMETRICS '04/Performance '04). ACM New York, NY (2004), pp. 343–354.
P. Billingsley, Convergence of Probability Measures. Wiley Series in Probability and Statistics, New York (1999). Ch. 3, pp. 109-153 or more precisely, Ch. 3.15, pp. 123-136.
H. Cai and D.Y. Eun, Stability of Network Congestion Control with Asynchronous Updates. In Proc. IEEE CDC 2006, San Diego, CA (2006).
A. Dhamdere and C. Dovrolis, Open issues in router-buffer sizing. ACM SIGCOMM Comput. Commun. Rev. 36. ACM New York, NY (2006) 87-92.
M. Duflo, Random iterative models. Volume 34 of Applications of Mathematics (New York). Springer-Verlag, Berlin (1997).
Dumas, V., Guilleaumin, F. and Robert, P., Markovian, A analysis of Additive-Increase Multiplicative-Decrease (AIMD) algorithms. Adv. Appl. Probab. 34 (2002) 85–111. CrossRef
D.Y. Eun, On the limitation of fluid-based approach for internet congestion control. In Proc. IEEE Int. Conf. on Computer Communications and Networks, ICCCN, San Diego, CA, USA. J. Telecommun. Syst. 34 (2007) 3-11.
D.Y. Eun, Fluid approximation of a Markov chain for TCP/AQM with many flows. Preprint.
Grigorescu, I. and Kang, M., Hydrodynamic Limit for a Fleming-Viot Type System. Stoch. Process. Appl. 110 (2004) 111–143. CrossRef
Grigorescu, I. and Kang, M., Tagged particle limit for a Fleming-Viot type system. Electron. J. Probab. 11 (2006) 311–331 (electronic). CrossRef
I. Grigorescu and M. Kang, Recurrence and ergodicity for a continuous AIMD model. Preprint.
Guillemin, F., Robert, P. and Zwart, B., AIMD algorithms and exponential functionals. Ann. Appl. Probab. 14 (2004) 90–117.
J.K. Hale, Ordinary Differential Equations. Wiley-Interscience, New York (1969).
C. Kipnis and C. Landim, Scaling Limits of Interacting Particle Systems. Springer-Verlag, New York (1999).
Maulik, K. and Zwart, B., An extension of the square root law of TCP. Ann. Oper. Res. 170 (2009) 217-232. CrossRef
S.P. Meyn and R.L. Tweedie, Markov chains and stochastic stability. Communications and Control Engineering Series. Springer-Verlag, London, Ltd. (1993).
Ott, T.J. and Swanson, J., Asymptotic behavior of a generalized TCP congestion avoidance algorithm. J. Appl. Probab. 44 (2007) 618–635. CrossRef