The Annals of Applied Probability

Large deviations problems for star networks: The min policy

Franck Delcoigne and Arnaud de La Fortelle

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Abstract

We are interested in analyzing the effect of bandwidth sharing for telecommunication networks. More precisely, we want to calculate which routes are bottlenecks by means of large deviations techniques. The method is illustrated in this paper on a star network, where the bandwidth is shared between customers according to the so-called min policy. We prove a sample path large deviation principle for a rescaled process n−1Qnt, where Qt represents the joint number of connections at time t. The main result is to compute the rate function explicitly. The major step consists in deriving large deviation bounds for an empirical generator constructed from the join number of customers and arrivals on each route. The rest of the analysis relies on a suitable change of measure together with a localization procedure. An example shows how this can be used practically.

Article information

Source
Ann. Appl. Probab., Volume 14, Number 2 (2004), 1006-1028.

Dates
First available in Project Euclid: 23 April 2004

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1082737120

Digital Object Identifier
doi:10.1214/105051604000000198

Mathematical Reviews number (MathSciNet)
MR2052911

Zentralblatt MATH identifier
1047.60021

Subjects
Primary: 60F10: Large deviations
Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) [See also 90Bxx]

Keywords
Large deviations rate function empirical generator change of measure contraction principle entropy star network bandwidth sharing min protocol

Citation

Delcoigne, Franck; de La Fortelle, Arnaud. Large deviations problems for star networks: The min policy. Ann. Appl. Probab. 14 (2004), no. 2, 1006--1028. doi:10.1214/105051604000000198. https://projecteuclid.org/euclid.aoap/1082737120


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References

  • Atar, R. and Dupuis, P. (1999). Large deviations and queueing networks: Methods for rate function identification. Stochastic Process. Appl. 84 255--296.
  • Baskett, F., Chandy, K. M., Muntz, R. R. and Palacios, F. G. (1975). Open, closed, and mixed networks of queues with different classes of customers. J. Assoc. Comput. Mach. 22 248--260.
  • Delcoigne, F. and de La Fortelle, A. (2001). Large deviations problems for star networks: The min policy. Technical Report 4143, INRIA.
  • Delcoigne, F. and de La Fortelle, A. (2002). Large deviations rate function for polling systems. Queueing Syst. Theory Appl. 41 13--44.
  • Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Springer, New York.
  • Deuschel, J.-D. and Stroock, D. W. (1989). Large Deviations. Academic Press, Boston.
  • Dupuis, P. and Ellis, R. S. (1992). Large deviations for Markov processes with discontinuous statistics. II. Random walks. Probab. Theory Related Fields 91 153--194.
  • Dupuis, P. and Ellis, R. S. (1995). The large deviation principle for a general class of queueing systems. I. Trans. Amer. Math. Soc. 347 2689--2751.
  • Dupuis, P., Ellis, R. S. and Weiss, A. (1991). Large deviations for Markov processes with discontinuous statistics. I. General upper bounds. Ann. Probab. 19 1280--1297.
  • Dupuis, P., Ishii, H. and Soner, H. M. (1990). A viscosity solution approach to the asymptotic analysis of queueing systems. Ann. Probab. 18 226--255.
  • Fayolle, G., de La Fortelle, A., Lasgouttes, J. M., Massoulie, L. and Roberts, J. (2001). Best-effort networks: Modeling and performance analysis via large networks asymptotics. In Proc. of IEEE INFOCOM'01.
  • Fayolle, G. and Lasgouttes, J.-M. (2001). Partage de bande passante dans un réseau: Approches probabilistes. Technical Report 4202, INRIA.
  • Fayolle, G., Mitrani, I. and Iasnogorodski, R. (1980). Sharing a processor among many job classes. J. Assoc. Comput. Mach. 27 519--532.
  • Freidlin, M. I. and Wentzell, A. D. (1984). Random Perturbations of Dynamical Systems. Springer, New York.
  • Ignatiouk-Robert, I. (2000). Large deviations of Jackson networks. Ann. Appl. Probab. 10 962--1001.
  • Ignatyuk, I. A., Malyshev, V. A. and Shcherbakov, V. V. (1994). The influence of boundaries in problems on large deviations. Uspekhi Mat. Nauk. 49 43--102.
  • Ioffe, A. D. and Tihomirov, V. M. (1979). Theory of Extremal Problems. North-Holland, Amsterdam.
  • de La Fortelle, A. (2001). Large deviation principle for Markov chains in continuous time. Problemy Peredachi Informatsii 36 120--140.
  • Shwartz, A. and Weiss, A. (1995). Large Deviations for Performance Analysis. Chapman and Hall, London.