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Grands Réseaux Aléatoires: comportement asymptotique et points fixes

Abstract : A cornerstone result in queueing theory due to Burke states that the departure process from a stationary M/M/1 queue is a Poisson process having the same intensity as the arrival process. I proved different extensions of this theorem to the single server queue and to the storage model. Furthermore, I looked at these models in tandem in the (\em transient regime). I showed that these models are dual in a strong fashion. Indeed I proved that the equations governing the dynamics of both models (queue and store) are the same even though the interesting variables are different whether we are interested in the queueing model or in the storage model. I use this duality to give an elegant proof, using analogies with the Robinson-Schensted-Knuth algorithm, of the property of symmetry of both systems. Moreover I explored the correlations between the laws of the services of successive customers belonging to the same busy period. Burke's theorem can be seen as a fixed point result: the Poisson process is a fixed point for the queue with exponential service times. I explored fixed points for the single server queue and the storage model in the context of large deviations where the arrivals and the services are described by means of their rate functions.
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Contributor : Moez Draief <>
Submitted on : Monday, August 8, 2005 - 4:50:17 PM
Last modification on : Saturday, March 28, 2020 - 2:12:34 AM
Long-term archiving on: : Friday, September 14, 2012 - 2:05:23 PM


  • HAL Id : tel-00009919, version 1



Moez Draief. Grands Réseaux Aléatoires: comportement asymptotique et points fixes. Mathématiques [math]. Université Paris-Diderot - Paris VII, 2005. Français. ⟨tel-00009919⟩



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