Binomial-Poisson entropic inequalities and the M/M/ queue
ESAIM: Probability and Statistics, Volume 10  (2006), p. 317-339

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”.

DOI : https://doi.org/10.1051/ps:2006013
Classification:  26D15,  46E99,  47D07,  60J27,  60J60,  60J75,  94A17
Keywords: functional inequalities, Markov processes, entropy, birth and death processes, queues
@article{PS_2006__10__317_0,
     author = {Chafa\"\i , Djalil},
     title = {Binomial-Poisson entropic inequalities and the $M/M/\infty $ queue},
     journal = {ESAIM: Probability and Statistics},
     publisher = {EDP-Sciences},
     volume = {10},
     year = {2006},
     pages = {317-339},
     doi = {10.1051/ps:2006013},
     zbl = {pre05216861},
     mrnumber = {2247924},
     language = {en},
     url = {http://www.numdam.org/item/PS_2006__10__317_0}
}
Chafaï, Djalil. Binomial-Poisson entropic inequalities and the $M/M/\infty $ queue. ESAIM: Probability and Statistics, Volume 10 (2006) , pp. 317-339. doi : 10.1051/ps:2006013. http://www.numdam.org/item/PS_2006__10__317_0/

[1] C. Ané and M. Ledoux, On logarithmic Sobolev inequalities for continuous time random walks on graphs. Probab. Theory Related Fields 116 (2000) 573-602. | Zbl 0964.60063

[2] C. Ané, Clark-Ocone formulas and Poincaré inequalities on the discrete cube. Ann. Inst. H. Poincaré Probab. Statist. 37 (2001) 101-137. | Numdam | Zbl 0978.60084

[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. | Zbl 0856.47026

[4] S. Boucheron, O. Bousquet, G. Lugosi and P. Massart, Moment inequalities for functions of independent random variables. Ann. Probab. 33 (2005) 514-560. | Zbl 1074.60018

[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. | Zbl 1087.60071

[6] S.G. Bobkov and M. Ledoux, On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal. 156 (1998) 347-365. | Zbl 0920.60002

[7] A.A. Borovkov, Limit laws for queueing processes in multichannel systems. Sibirsk. Mat. Ž. 8 (1967) 983-1004. | Zbl 0182.53401

[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. http://www.math.gatech.edu/~tetali/, 2003.

[9] P. Brémaud, Markov chains, Gibbs fields, Monte Carlo simulation, and queues. Texts Appl. Math. 31 (1999) xviii+444. | MR 1689633 | Zbl 0949.60009

[10] D. Chafaï and D. Concordet, A continuous stochastic maturation model, preprint arXiv math.PR/0412193 or CNRS HAL ccsd-00003498, 2004.

[11] D. Chafaï, Entropies, convexity, and functional inequalities: on Φ-entropies and Φ-Sobolev inequalities. J. Math. Kyoto Univ. 44 (2004) 325-363. | Zbl 1079.26009

[12] M.F. Chen, Variational formulas of Poincaré-type inequalities for birth-death processes. Acta Math. Sin. (Engl. Ser.) 19 (2003) 625-644. | Zbl 1040.60064

[13] P. Caputo and G. Posta, Entropy dissipation estimates in a Zero-Range dynamics, preprint arXiv math.PR/0405455, 2004. | MR 2322692 | Zbl 1126.60082

[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. | Zbl 1099.60068

[15] P. Dai Pra and G. Posta, Logarithmic Sobolev inequality for zero-range dynamics. Electron. J. Probab. 10 (2005) 525-576. | Zbl 1109.60080

[16] P. Dai Pra, A.M. Paganoni and G. Posta, Entropy inequalities for unbounded spin systems. Ann. Probab. 30 (2002), 1959-1976. | Zbl 1013.60076

[17] P. Diaconis and L. Saloff-Coste, Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 (1996) 695-750. | Zbl 0867.60043

[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. | MR 838085 | Zbl 0592.60049

[19] S. Goel, Modified logarithmic Sobolev inequalities for some models of random walk. Stochastic Process. Appl. 114 (2004) 51-79. | Zbl 1074.60080

[20] O. Johnson and C. Goldschmidt, Preservation of log-concavity on summation, preprint arXiv math.PR/0502548, 2005. | MR 2219340

[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. | MR 2348750

[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). | Numdam | Zbl pre02161877

[23] S. Karlin and J. Mcgregor, Linear growth birth and death processes. J. Math. Mech. 7 (1958) 643-662. | Zbl 0091.13804

[24] F.P. Kelly, Blocking probabilities in large circuit-switched networks. Adv. in Appl. Probab. 18 (1986) 473-505. | Zbl 0597.60092

[25] F.P. Kelly, Loss networks. Ann. Appl. Probab. 1 (1991) 319-378. | Zbl 0743.60099

[26] C. Kipnis and C. Landim, Scaling limits of interacting particle systems. Fundamental Principles of Mathematical Sciences 320, Springer-Verlag, Berlin (1999). | MR 1707314 | Zbl 0927.60002

[27] R. Latała and K. Oleszkiewicz, Between Sobolev and Poincaré, Geometric aspects of functional analysis. Lect. Notes Math. 1745 (2000) 147-168. | Zbl 0986.60017

[28] P. Massart, Concentration inequalities and model selection, Lectures on probability theory and statistics (Saint-Flour, 2003), available on the author's web-site http://www.math.u-psud.fr/~massart/stf2003_massart.pdf. | Zbl pre05150953

[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.

[30] L. Miclo, An example of application of discrete Hardy's inequalities. Markov Process. Related Fields 5 (1999) 319-330. | Zbl 0942.60081

[31] Ph. Robert, Stochastic networks and queues, french ed., Applications of Mathematics (New York) 52, Springer-Verlag, Berlin, 2003, Stochastic Modelling and Applied Probability. | MR 1996883 | Zbl 1038.60091

[32] R.T. Rockafellar, Convex analysis, Princeton Landmarks in Mathematics, Reprint of the 1970 original, Princeton Paperbacks, Princeton University Press (1997) xviii+451. | MR 1451876 | Zbl 0932.90001

[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. | Zbl 0885.60061

[34] B. Ycart, A characteristic property of linear growth birth and death processes. The Indian J. Statist. Ser. A 50 (1988) 184-189. | Zbl 0662.60093

[35] L. Wu, A new modified logarithmic Sobolev inequality for Poisson point processes and several applications. Probab. Theory Related Fields 118 (2000) 427-438. | Zbl 0970.60093