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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