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 -

References

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  1. C. Ané and M. Ledoux, On logarithmic Sobolev inequalities for continuous time random walks on graphs. Probab. Theory Related Fields116 (2000) 573–602.  Zbl0964.60063
  2. C. Ané, Clark-Ocone formulas and Poincaré inequalities on the discrete cube. Ann. Inst. H. Poincaré Probab. Statist.37 (2001) 101–137.  Zbl0978.60084
  3. D. Bakry, L'hypercontractivité et son utilisation en théorie des semigroupes. Lectures on probability theory (Saint-Flour, 1992), Lect. Notes Math.1581 (1994) 1–114.  
  4. S. Boucheron, O. Bousquet, G. Lugosi and P. Massart, Moment inequalities for functions of independent random variables. Ann. Probab.33 (2005) 514–560.  Zbl1074.60018
  5. A.-S. Boudou, P. Caputo, P. Dai Pra and G. Posta, Spectral gap estimates for interacting particle systems via a Bochner type inequality. J. Funct. Anal.232 (2006) 222–258.  Zbl1087.60071
  6. S.G. Bobkov and M. Ledoux, On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal.156 (1998) 347–365.  Zbl0920.60002
  7. A.A. Borovkov, Limit laws for queueing processes in multichannel systems. Sibirsk. Mat. Ž.8 (1967) 983–1004.  
  8. S. Bobkov and P. Tetali, Modified Log-Sobolev Inequalities in Discrete Settings, Preliminary version appeared in Proc. of the ACM STOC 2003, pp. 287–296. Cf. , 2003.  Zbl1192.60020URIhttp://www.math.gatech.edu/~tetali/
  9. P. Brémaud, Markov chains, Gibbs fields, Monte Carlo simulation, and queues. Texts Appl. Math.31 (1999) xviii+444.  
  10. D. Chafaï and D. Concordet, A continuous stochastic maturation model, preprint arXiv math. or CNRS HAL ccsd-00003498, 2004.  URIPR/0412193
  11. D. Chafaï, Entropies, convexity, and functional inequalities: on Φ -entropies and Φ -Sobolev inequalities. J. Math. Kyoto Univ.44 (2004) 325–363.  Zbl1079.26009
  12. M.F. Chen, Variational formulas of Poincaré-type inequalities for birth-death processes. Acta Math. Sin. (Engl. Ser.) 19 (2003) 625–644.  Zbl1040.60064
  13. P. Caputo and G. Posta, Entropy dissipation estimates in a Zero-Range dynamics, preprint arXiv math., 2004.  URIPR/0405455
  14. P. Dai Pra and G. Posta, Logarithmic Sobolev inequality for zero-range dynamics: independence of the number of particles. Ann. Probab.33 (2005) 2355–2401.  Zbl1099.60068
  15. P. Dai Pra and G. Posta, Logarithmic Sobolev inequality for zero-range dynamics. Electron. J. Probab.10 (2005) 525–576.  Zbl1109.60080
  16. P. Dai Pra, A.M. Paganoni and G. Posta, Entropy inequalities for unbounded spin systems. Ann. Probab.30 (2002), 1959–1976.  Zbl1013.60076
  17. P. Diaconis and L. Saloff-Coste, Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab.6 (1996) 695–750.  Zbl0867.60043
  18. S.N. Ethier and T.G. Kurtz, Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons Inc., New York, 1986, Characterization and convergence.  Zbl0592.60049
  19. S. Goel, Modified logarithmic Sobolev inequalities for some models of random walk. Stochastic Process. Appl.114 (2004) 51–79.  Zbl1074.60080
  20. O. Johnson and C. Goldschmidt, Preservation of log-concavity on summation, preprint arXiv math., 2005.  URIPR/0502548
  21. A. Joulin, On local Poisson-type deviation inequalities for curved continuous time Markov chains, with applications to birth-death processes, personal communication, preprint 2006.  
  22. A. Joulin and N. Privault, Functional inequalities for discrete gradients and application to the geometric distribution. ESAIM Probab. Stat.8 (2004) 87–101 (electronic).  Zbl1147.60306
  23. S. Karlin and J. McGregor, Linear growth birth and death processes. J. Math. Mech.7 (1958) 643–662.  Zbl0091.13804
  24. F.P. Kelly, Blocking probabilities in large circuit-switched networks. Adv. in Appl. Probab.18 (1986) 473–505.  Zbl0597.60092
  25. F.P. Kelly, Loss networks. Ann. Appl. Probab.1 (1991) 319–378.  Zbl0743.60099
  26. C. Kipnis and C. Landim, Scaling limits of interacting particle systems. Fundamental Principles of Mathematical Sciences 320, Springer-Verlag, Berlin (1999).  Zbl0927.60002
  27. R. Latała and K. Oleszkiewicz, Between Sobolev and Poincaré, Geometric aspects of functional analysis. Lect. Notes Math.1745 (2000) 147–168.  
  28. P. Massart, Concentration inequalities and model selection, Lectures on probability theory and statistics (Saint-Flour, 2003), available on the author's web-site .  URIhttp://www.math.u-psud.fr/~massart/stf2003_massart.pdf
  29. Y. Mao, Logarithmic Sobolev inequalities for birth-death process and diffusion process on the line. Chinese J. Appl. Probab. Statist.18 (2002) 94–100.  Zbl1153.60386
  30. L. Miclo, An example of application of discrete Hardy's inequalities. Markov Process. Related Fields5 (1999) 319–330.  Zbl0942.60081
  31. Ph. Robert, Stochastic networks and queues, french ed., Applications of Mathematics (New York) 52, Springer-Verlag, Berlin, 2003, Stochastic Modelling and Applied Probability.  
  32. R.T. Rockafellar, Convex analysis, Princeton Landmarks in Mathematics, Reprint of the 1970 original, Princeton Paperbacks, Princeton University Press (1997) xviii+451.  
  33. L. Saloff-Coste, Lectures on finite Markov chains. Lectures on probability theory and statistics (Saint-Flour, 1996). Lect. Notes Math.1665 (1997) 301–413.  Zbl0885.60061
  34. B. Ycart, A characteristic property of linear growth birth and death processes. The Indian J. Statist. Ser. A50 (1988) 184–189.  Zbl0662.60093
  35. L. Wu, A new modified logarithmic Sobolev inequality for Poisson point processes and several applications. Probab. Theory Related Fields118 (2000) 427–438.  Zbl0970.60093

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