Steady state and scaling limit for a traffic congestion model

Ilie Grigorescu; Min Kang

ESAIM: Probability and Statistics (2010)

  • Volume: 14, page 271-285
  • ISSN: 1292-8100

Abstract

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In a general model (AIMD) of transmission control protocol (TCP) used in internet traffic congestion management, the time dependent data flow vector x(t) > 0 undergoes a biased random walk on two distinct scales. The amount of data of each component xi(t) goes up to xi(t)+a with probability 1-ζi(x) on a unit scale or down to γxi(t), 0 < γ < 1 with probability ζi(x) on a logarithmic scale, where ζi depends on the joint state of the system x. We investigate the long time behavior, mean field limit, and the one particle case. According to c = lim inf|x|→∞ |x|ζi(x) , the process drifts to ∞ in the subcritical c < c+(n, γ) case and has an invariant probability measure in the supercritical case c > c+(n, γ). Additionally, a scaling limit is proved when ζi(x) and a are of order N–1 and t → Nt, in the form of a continuum model with jump rate α(x).

How to cite

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Grigorescu, Ilie, and Kang, Min. "Steady state and scaling limit for a traffic congestion model." ESAIM: Probability and Statistics 14 (2010): 271-285. <http://eudml.org/doc/250828>.

@article{Grigorescu2010,
abstract = { In a general model (AIMD) of transmission control protocol (TCP) used in internet traffic congestion management, the time dependent data flow vector x(t) > 0 undergoes a biased random walk on two distinct scales. The amount of data of each component xi(t) goes up to xi(t)+a with probability 1-ζi(x) on a unit scale or down to γxi(t), 0 < γ < 1 with probability ζi(x) on a logarithmic scale, where ζi depends on the joint state of the system x. We investigate the long time behavior, mean field limit, and the one particle case. According to c = lim inf|x|→∞ |x|ζi(x) , the process drifts to ∞ in the subcritical c < c+(n, γ) case and has an invariant probability measure in the supercritical case c > c+(n, γ). Additionally, a scaling limit is proved when ζi(x) and a are of order N–1 and t → Nt, in the form of a continuum model with jump rate α(x). },
author = {Grigorescu, Ilie, Kang, Min},
journal = {ESAIM: Probability and Statistics},
keywords = {TCP; AIMD; fluid limit; mean field interaction; invariant measures; scaling limit},
language = {eng},
month = {10},
pages = {271-285},
publisher = {EDP Sciences},
title = {Steady state and scaling limit for a traffic congestion model},
url = {http://eudml.org/doc/250828},
volume = {14},
year = {2010},
}

TY - JOUR
AU - Grigorescu, Ilie
AU - Kang, Min
TI - Steady state and scaling limit for a traffic congestion model
JO - ESAIM: Probability and Statistics
DA - 2010/10//
PB - EDP Sciences
VL - 14
SP - 271
EP - 285
AB - In a general model (AIMD) of transmission control protocol (TCP) used in internet traffic congestion management, the time dependent data flow vector x(t) > 0 undergoes a biased random walk on two distinct scales. The amount of data of each component xi(t) goes up to xi(t)+a with probability 1-ζi(x) on a unit scale or down to γxi(t), 0 < γ < 1 with probability ζi(x) on a logarithmic scale, where ζi depends on the joint state of the system x. We investigate the long time behavior, mean field limit, and the one particle case. According to c = lim inf|x|→∞ |x|ζi(x) , the process drifts to ∞ in the subcritical c < c+(n, γ) case and has an invariant probability measure in the supercritical case c > c+(n, γ). Additionally, a scaling limit is proved when ζi(x) and a are of order N–1 and t → Nt, in the form of a continuum model with jump rate α(x).
LA - eng
KW - TCP; AIMD; fluid limit; mean field interaction; invariant measures; scaling limit
UR - http://eudml.org/doc/250828
ER -

References

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  1. 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.  
  2. F. Baccelli, D. McDonald and J. Reynier, A mean-field model for multiple TCP connections through a buffer implementing RED. Performance Evaluation Archive49 (2002) 77-97.  
  3. 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.  
  4. 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.  
  5. H. Cai and D.Y. Eun, Stability of Network Congestion Control with Asynchronous Updates. In Proc. IEEE CDC 2006, San Diego, CA (2006).  
  6. A. Dhamdere and C. Dovrolis, Open issues in router-buffer sizing. ACM SIGCOMM Comput. Commun. Rev.36. ACM New York, NY (2006) 87-92.  
  7. M. Duflo, Random iterative models. Volume 34 of Applications of Mathematics (New York). Springer-Verlag, Berlin (1997).  
  8. V. Dumas, F. Guilleaumin and P. Robert, A Markovian analysis of Additive-Increase Multiplicative-Decrease (AIMD) algorithms. Adv. Appl. Probab.34 (2002) 85–111.  
  9. 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.  
  10. D.Y. Eun, Fluid approximation of a Markov chain for TCP/AQM with many flows. Preprint.  
  11. I. Grigorescu and M. Kang, Hydrodynamic Limit for a Fleming-Viot Type System. Stoch. Process. Appl.110 (2004) 111–143.  
  12. I. Grigorescu and M. Kang, Tagged particle limit for a Fleming-Viot type system. Electron. J. Probab.11 (2006) 311–331 (electronic).  
  13. I. Grigorescu and M. Kang, Recurrence and ergodicity for a continuous AIMD model. Preprint.  
  14. F. Guillemin, P. Robert and B. Zwart, AIMD algorithms and exponential functionals. Ann. Appl. Probab.14 (2004) 90–117.  
  15. J.K. Hale, Ordinary Differential Equations. Wiley-Interscience, New York (1969).  
  16. C. Kipnis and C. Landim, Scaling Limits of Interacting Particle Systems. Springer-Verlag, New York (1999).  
  17. K. Maulik and B. Zwart, An extension of the square root law of TCP. Ann. Oper. Res.170 (2009) 217-232.  
  18. S.P. Meyn and R.L. Tweedie, Markov chains and stochastic stability. Communications and Control Engineering Series. Springer-Verlag, London, Ltd. (1993).  
  19. T.J. Ott and J. Swanson, Asymptotic behavior of a generalized TCP congestion avoidance algorithm. J. Appl. Probab.44 (2007) 618–635.  

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