Abstract
For a version of the single-server infinite-buffer queuing system which is much more general than M/G/1, the results on the rate of convergence to the stationary mode were reviewed in brief. New sufficient conditions guaranteeing the polynomial estimate of the rate of convergence were established using the “Markovization” method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bambos, N. and Walrand, J., On Stability of State-Dependent Queues and Acyclic Queuing Networks, Adv. Appl. Prob., 1989, vol. 21, no. 3, pp. 681–701.
Browne, S. and Sigman, K., Work-modulated Queues with Applications to Storage Processes, J. Appl. Prob., 1992, vol. 29, pp. 699–712.
Thorisson, H., The Queue GI/G/1: Finite Moments of the Cycle Variables and Uniform Rates of Convergence, Stoch. Process. Appl., 1985, vol. 19, no. 1, pp. 85–99.
Thorisson, H., Coupling, Stationarity, and Regeneration, New York: Springer, 2000.
Tuominen, P. and Tweedie, R.L., Exponential Decay and Ergodicity of General Markov Processes, J. Appl. Prob., 1979, vol. 16, no. 4, pp. 867–880.
Sevast’yanov, B.A., Ergodic Theorem for the Markov Processes and Its Application to the Telephone Systems with Failures, Teor. Veroyat. Primen., 1957, vol. 2, no. 1, pp. 106–116.
Veretennikov, A.Yu., On Ergodicity of the Queuing Systems with Infinite Number of Servers, Mat. Zametki, 1977, vol. 22, no. 4, pp. 561–569.
Kelbert, M. and Veretennikov, A., On the Estimation of Mixing Coefficients for a Multiphase Service System, Queuing Syst., 1997, vol. 25, pp. 325–337.
Dynkin, E.B., Markovskie Protsessy (Markov Processes), Moscow: Fizmatgiz, 1963.
Liptser, R.Sh. and Shiryaev, A.N., Stochastic Calculus on the Probabilistic Spaces with Filtrations, in Stokhasticheskoe ischislenie. Itogi nauki i tekhniki. Sovrem. probl. mat. Fundament. napravleniya (Stochastic Calculus. Resume of Science and Technology. Modern Probl. Mat. Fundamental Lines of Research), Moscow: VINITI, 1989, vol. 49, pp. 114–159.
Ethier, S.N. and Kurtz, T.G., Markov Processes: Characterization and Convergence, New York: Wiley, 1986.
Davis, M.H.A., Piecewise-Deterministic Markov Processes: A General Class of Non-Diffusion Stochastic Models, J. Royal Stat. Soc. Ser. B (Methodological), 1984, vol. 46, no. 3, pp. 353–388.
Etemadi, N., An Elementary Proof of the Strong Law of Large Numbers, Prob. Theory Relat. Fields, 1981, vol. 55, no. 1, pp. 119–122.
Miller, H.D., Geometric Ergodicity in a Class of Denumerable Markov Chains, Z. Wahrsch., 1965, vol. 4, pp. 354–373.
Neuts, M.F. and Teugels, J.L., Exponential Ergodicity of the M/G/1 Queue, SIAM J. Appl. Math., 1969, vol. 17, no. 5, pp. 921–929.
Veretennikov, A.Yu., On Polynomial Mixing and Convergence Rate for the Stochastic Differential and Difference Equations, Teor. Veroyat. Primen., 1999, vol. 44, no. 2, pp. 312–327.
Harris, T.E., The Existence of StationaryMeasures for CertainMarkov Processes, in Proc. Third Berkeley Symp. Math. Statist. Prob., Berkeley: Univ. Calif. Press, 1956, vol. 2, pp. 113–124.
Khas’minskii, R.Z., Ustoichivost’ sistem differentsial’nykh uravnenii pri sluchainykh vozmushcheniyakh ikh parametrov (Stability of Systems of Differential Equations under Random Perturbations of Their Parameters), Moscow: Nauka, 1969.
Author information
Authors and Affiliations
Additional information
Original Russian Text © A.Yu. Veretennikov, 2013, published in Avtomatika i Telemekhanika, 2013, No. 10, pp. 23–35.
Rights and permissions
About this article
Cite this article
Veretennikov, A.Y. On the rate of convergence to the stationary distribution in the single-server queuing systems. Autom Remote Control 74, 1620–1629 (2013). https://doi.org/10.1134/S0005117913100032
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117913100032