Journal of Statistical Physics

, Volume 66, Issue 3–4, pp 803–825 | Cite as

A mean-field limit for a class of queueing networks

  • F. Baccelli
  • F. I. Karpelevich
  • M. Ya. Kelbert
  • A. A. Puhalskii
  • A. N. Rybko
  • Yu. M. Suhov


A model of centralized symmetric message-switched networks is considered, where the messages having a common address must be served in the central node in the order which corresponds to their epochs of arrival to the network. The limitN → ∞ is discussed, whereN is the branching number of the network graph. This procedure is inspired by an analogy with statistical mechanics (the mean-field approximation). The corresponding limit theorems are established and the limiting probability distribution for the network response time is obtained. Properties of this distribution are discussed in terms of an associated boundary problem.

Key words

Message-switched network synchronization constraint discipline (FEFS) starlike configuration graph infinite branching limit mean-field or Poisson approximation generalized Lindley equation and its stationary solution associated boundary problem 


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  1. 1.
    F. Baccelli and P. Brémaud,Palm Probabilities and Stationary Queues (Lecture Notes in Statistics No. 43, Springer-Verlag, 1987).Google Scholar
  2. 2.
    F. Baccelli, E. Gelenbe, and B. Plateau, An end-to-end approach to the resequencing problem,J. Assoc. Comput. Machinery 31(3):474–485 (1984).Google Scholar
  3. 3.
    F. Baccelli and A. M. Makowski, Queuing models for systems with synchronization constraints,Proc. IEEE 77(1):138–161 (1989).Google Scholar
  4. 4.
    F. Baccelli and Z. Liu, On a class of stochastic recursive sequences arising in queueing theory, INRIA report No 984 (March 1989);Ann. Prob., to appear.Google Scholar
  5. 5.
    F. Baccelli, W. A. Massey, and D. Towsley, Acyclic fork-join queuing networks,J. Assoc. Comput. Machinery 36(3):615–642 (1989).Google Scholar
  6. 6.
    A. A. Borovkov,Stochastic Process in Queuing Theory (Springer-Verlag, New York, 1976).Google Scholar
  7. 7.
    A. A. Borovkov,Asymptotic Methods in Queuing Theory (Wiley, Chichester, 1984).Google Scholar
  8. 8.
    A. Brandt, On stationary waiting times and limiting behavior of queues with many servers, I. The general G/G/m/case,Elektron. Informations Verarb. Kybern. 21(1):47–64 (1985); II. The G/GI/m/case,Elektron. Informations Verarb. Kybern. 1(3):151–162 (1985).Google Scholar
  9. 9.
    T. C. Brown and P. K. Pollett, Some distributional approximations in Markovian queueing networks,Adv. Appl. Prob. 14(3):654–671 (1982).Google Scholar
  10. 10.
    S. W. Dharmadhikari and K. Jodgeo, Bounds on moments of certain random variables,Ann. Math. Stat. 40(4):1506–1508 (1969).Google Scholar
  11. 11.
    R. L. Dobrushin and Yu. M. Sukhov, Asymptotical investigation of starlike message switched networks with a large number of radial rays,Problems Information Transmission 12(1):70–94 (1976)[in Russian].Google Scholar
  12. 12.
    D. H. Fook and S. V. Nagajev, Probabilistic inequalities for sums of independent random variables,Theory Prob. Appl. 16(4):660–675 (1971)[in Russian].Google Scholar
  13. 13.
    P. Franken, D. Koenig, U. Arndt, and V. Schmidt,Queues and Point Processes (Chichester, Wiley, 1982).Google Scholar
  14. 14.
    M. Ya. Kelbert and Yu. M. Sukhov, Mathematical problems in queueing network theory, inProbability Theory, Mathematical Statistics, Theoretical Cybernetics, Vol. 26 (VINITI AN SSSR, Moscow, 1988), pp. 3–96 [in Russian].Google Scholar
  15. 15.
    L. Kleinrock,Communication Nets: Stochastic Message Flow and Delay (McGraw-Hill, New York, 1964; reprinted Dover, New York, 1972).Google Scholar
  16. 16.
    V. A. Malyshev and S. A. Berezner, The stability of infinite-server networks with random routing,J. Appl. Prob. 26:363–371 (1989).Google Scholar
  17. 17.
    V. V. Petrov,Sums of Independent Random Variables (Springer-Verlag, Berlin, 1975).Google Scholar
  18. 18.
    H. Thorisson, The queue GI/GI/k: Finite moments of the cyclic variables and uniform rates of convergence,Stoch. Proc. Appl. 19(1):85–99 (1985).Google Scholar

Copyright information

© Plenum Publishing Corporation 1992

Authors and Affiliations

  • F. Baccelli
    • 1
  • F. I. Karpelevich
    • 2
  • M. Ya. Kelbert
    • 3
    • 4
  • A. A. Puhalskii
    • 4
  • A. N. Rybko
    • 4
  • Yu. M. Suhov
    • 4
    • 5
  1. 1.Institut National de Recherche en Informatique et AutomatiqueValbonneFrance
  2. 2.Institute of Transport EngineeringUSSR Ministry of Rail TransportMoscowUSSR
  3. 3.International Institute of Earthquake Prediction Theory and Mathematical GeophysicsUSSR Academy of SciencesMoscowUSSR
  4. 4.Institute for Problems of Information TransmissionUSSR Academy of SciencesMoscowUSSR
  5. 5.Statistical Laboratory, DPMMSUniversity of CambridgeCambridgeEngland

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