Binomial-Poisson entropic inequalities and the M/M/∞ queue

Djalil Chafaï

ESAIM: Probability and Statistics (2006)

  • Volume: 10, page 317-339
  • ISSN: 1292-8100

Abstract

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This article provides entropic inequalities for binomial-Poisson distributions, derived from the two point space. They appear as local inequalities of the M/M/∞ queue. They describe in particular the exponential dissipation of Φ-entropies along this process. This simple queueing process appears as a model of “constant curvature”, and plays for the simple Poisson process the role played by the Ornstein-Uhlenbeck process for Brownian Motion. Some of the inequalities are recovered by semi-group interpolation. Additionally, we explore the behaviour of these entropic inequalities under a particular scaling, which sees the Ornstein-Uhlenbeck process as a fluid limit of M/M/∞ queues. Proofs are elementary and rely essentially on the development of a “Φ-calculus”.

How to cite

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Chafaï, Djalil. "Binomial-Poisson entropic inequalities and the M/M/∞ queue." ESAIM: Probability and Statistics 10 (2006): 317-339. <http://eudml.org/doc/249753>.

@article{Chafaï2006,
abstract = { This article provides entropic inequalities for binomial-Poisson distributions, derived from the two point space. They appear as local inequalities of the M/M/∞ queue. They describe in particular the exponential dissipation of Φ-entropies along this process. This simple queueing process appears as a model of “constant curvature”, and plays for the simple Poisson process the role played by the Ornstein-Uhlenbeck process for Brownian Motion. Some of the inequalities are recovered by semi-group interpolation. Additionally, we explore the behaviour of these entropic inequalities under a particular scaling, which sees the Ornstein-Uhlenbeck process as a fluid limit of M/M/∞ queues. Proofs are elementary and rely essentially on the development of a “Φ-calculus”. },
author = {Chafaï, Djalil},
journal = {ESAIM: Probability and Statistics},
keywords = {Functional inequalities; Markov processes; entropy; birth and death processes; queues.; functional inequalities; birth and death processes; queues},
language = {eng},
month = {9},
pages = {317-339},
publisher = {EDP Sciences},
title = {Binomial-Poisson entropic inequalities and the M/M/∞ queue},
url = {http://eudml.org/doc/249753},
volume = {10},
year = {2006},
}

TY - JOUR
AU - Chafaï, Djalil
TI - Binomial-Poisson entropic inequalities and the M/M/∞ queue
JO - ESAIM: Probability and Statistics
DA - 2006/9//
PB - EDP Sciences
VL - 10
SP - 317
EP - 339
AB - This article provides entropic inequalities for binomial-Poisson distributions, derived from the two point space. They appear as local inequalities of the M/M/∞ queue. They describe in particular the exponential dissipation of Φ-entropies along this process. This simple queueing process appears as a model of “constant curvature”, and plays for the simple Poisson process the role played by the Ornstein-Uhlenbeck process for Brownian Motion. Some of the inequalities are recovered by semi-group interpolation. Additionally, we explore the behaviour of these entropic inequalities under a particular scaling, which sees the Ornstein-Uhlenbeck process as a fluid limit of M/M/∞ queues. Proofs are elementary and rely essentially on the development of a “Φ-calculus”.
LA - eng
KW - Functional inequalities; Markov processes; entropy; birth and death processes; queues.; functional inequalities; birth and death processes; queues
UR - http://eudml.org/doc/249753
ER -

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