Abstract
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.
Similar content being viewed by others
References
F. Baccelli and P. Brémaud,Palm Probabilities and Stationary Queues (Lecture Notes in Statistics No. 43, Springer-Verlag, 1987).
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).
F. Baccelli and A. M. Makowski, Queuing models for systems with synchronization constraints,Proc. IEEE 77(1):138–161 (1989).
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.
F. Baccelli, W. A. Massey, and D. Towsley, Acyclic fork-join queuing networks,J. Assoc. Comput. Machinery 36(3):615–642 (1989).
A. A. Borovkov,Stochastic Process in Queuing Theory (Springer-Verlag, New York, 1976).
A. A. Borovkov,Asymptotic Methods in Queuing Theory (Wiley, Chichester, 1984).
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).
T. C. Brown and P. K. Pollett, Some distributional approximations in Markovian queueing networks,Adv. Appl. Prob. 14(3):654–671 (1982).
S. W. Dharmadhikari and K. Jodgeo, Bounds on moments of certain random variables,Ann. Math. Stat. 40(4):1506–1508 (1969).
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].
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].
P. Franken, D. Koenig, U. Arndt, and V. Schmidt,Queues and Point Processes (Chichester, Wiley, 1982).
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].
L. Kleinrock,Communication Nets: Stochastic Message Flow and Delay (McGraw-Hill, New York, 1964; reprinted Dover, New York, 1972).
V. A. Malyshev and S. A. Berezner, The stability of infinite-server networks with random routing,J. Appl. Prob. 26:363–371 (1989).
V. V. Petrov,Sums of Independent Random Variables (Springer-Verlag, Berlin, 1975).
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).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Baccelli, F., Karpelevich, F.I., Kelbert, M.Y. et al. A mean-field limit for a class of queueing networks. J Stat Phys 66, 803–825 (1992). https://doi.org/10.1007/BF01055703
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01055703