Abstract
It is known that in many queueing systems uid limits are deterministic functions. Models and conditions which lead to random uid limits have not received much attention. This paper is devoted to a study of a queueing network whose uid limits admit a random and uncountable branching at certain points. Stability conditions for this model are investigated by the use of recent results from the theory of branching processes.
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REFERENCES
Rybko, A.N. and Stolyar, A.L., Ergodicity of Stochastic Processes Describing the Operation of Open Queueing Networks, Probl. Peredachi Inf., 1992, vol. 28, no.3, pp. 3–26 [Probl. Inf. Trans. (Engl. Transl.), 1992, vol. 28, no. 3, pp. 199–220].
Dai, J.G., On Positive Harris Recurrence of Multiclass Queueing Networks: A Unified Approach via Fluid Models, Ann. Appl. Probab., 1995, vol. 5, no.1, pp. 49–77.
Stolyar, A.L., On the Stability of Multiclass Queueing Networks: A Relaxed Sufficient Condition via Limiting Fluid Processes, Markov Process. Related Fields, 1995, vol. 1, no.4, pp. 491–512.
Dai, J.G. and Meyn, S.P., Stability and Convergence of Moments for Multiclass Queueing Networks via Fluid Limit Models, IEEE Trans. Automat. Control, 1995, vol. 40, no.11, pp. 1889–1904.
Chen, H., Fluid Approximations and Stability of Multiclass Queueing Networks: Work-Conserving Disciplines, Ann. Appl. Probab., 1995, vol. 5, no.3, pp. 637–665.
Down, D.G., On the Stability of Polling Models with Multiple Servers, CWI Report BS-R 9605, Amsterdam, 1996.
Foss, S.G. and Rybko, A.N., Stability of Multiclass Jakson-Type Networks, Markov Process. Related Fields, 1996, vol. 2, no.3, pp. 461–486.
Meyn, S.P., Transience of Multiclass Queueing Networks and Their Fluid Models, Ann. Appl. Probab., 1995, vol. 5, no.4, pp. 946–957.
Puhalskii, A.A. and Rybko, A.N., Nonergodicity of a Queueing Network under Nonstability of Its Fluid Model, Probl. Peredachi Inf., 2000, vol. 36, no.1, pp. 26–47 [Probl. Inf. Trans. (Engl. Transl.), 2000, vol. 36, no. 1, pp. 23–41].
Kumar, P.R. and Seidman, T.I., Dynamic Instabilities and Stabilization Methods in Distributed Real-Time Scheduling of Manufacturing Systems, IEEE Trans. Automat. Control, 1990, vol. 35, no.3, pp. 289–298.
Chen, H. and Mandelbaum, A., Discrete Flow Networks: Bottleneck Analysis and Fluid Approximations, Math. Oper. Res., 1991, vol. 16, no.2, pp. 408–446.
Bramson, M., Instability of FIFO Queueing Networks, Ann. Appl. Probab., 1994, vol. 4, no.2, pp. 414–431.
Malyshev, V.A., Networks and Dynamical Systems, Adv. Appl. Probab., 1993, vol. 25, no.1, pp. 140–175.
Ignatyik, I. and Malyshev, V., Classification of Random Walks in ℤ +4 , Selecta Math., 1993, vol. 12, no.2, pp. 129–166.
Fayolle, G., Malyshev, V.A., and Menshikov, M.V., Topics in the Constructive Theory of Countable Markov Chains, Cambridge: Cambridge Univ. Press, 1995.
Foss, S. and Kovalevskii, A., A Stability Criterion via Fluid Limits and Its Application to a Polling System, Queueing Syst., 1999, vol. 32, no.1–3, pp. 131–168.
Hunt, P.J., Pathological Behavior in Loss Networks, J. Appl. Probab., 1995, vol. 32, no.2, pp. 519–533.
Harris, T.E., The Theory of Branching Processes, Berlin: Springer, 1963. Translated under the title Teoriya vetvyashchikhsya sluchainykh protsessov, Moscow: Mir, 1966.
Asmussen, S. and Hering, H., Branching Processes, Boston: Birkhauser, 1983.
Athreya, K.B. and Ney, P.F., Branching Processes, Berlin: Springer, 1972.
Roesler, U., Topchii, V.A., and Vatutin, V.A., The Rate of Convergence for Weighted Branching Processes, Siberian Adv. Math., 2002, vol. 12, no.4, pp. 57–82.
Shiryaev, A.N., Veroyatnost', Moscow: Nauka, 1989, 2nd ed. Translated under the title Probability, New York: Springer, 1996.
Kratzer, A. and Franz, W., Transzendente Funktionen, Leipzig: Akademische, 1960. Translated under the title Transtsendentnye funktsii, Moscow: Inostr. Lit., 1963.
Foss, S.G. and Denisov, D.E., On Transience Conditions for Markov Chains, Sibirsk. Mat. Zh., 2001, vol. 42, no.2, pp. 425–433 [Siberian Math. J. (Engl. Transl.), 2001, vol. 42, no. 2, pp. 364–371].
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Translated from Problemy Peredachi Informatsii, No. 3, 2005, pp. 76–104.
Original Russian Text Copyright © 2005 by Kovalevskii, Topchii, Foss.
Supported in part by the Russian Foundation for Basic Research, project no. 02-01-00358, and Grant of the President of the Russian Federation for Leading Scientific Schools, no. 2139.2003.1.
Supported in part by the Russian Foundation for Basic Research, project no. 03-01-00045, Grant of the President of the Russian Federation for Leading Scientific Schools, no. 2139.2003.1, and Program 1.1 of the Branch of Mathematical Sciences of the Russian Academy of Sciences “Modern Problems of Theoretical Mathematics in the Institute of Mathematics of the Siberian Branch of the RAS.”
Supported in part by the Russian Foundation for Basic Research, project no. 02-01-00358, and EPSRC, grant no. T27099/01.
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Kovalevskii, A.P., Topchii, V.A. & Foss, S.G. On the Stability of a Queueing System with Uncountably Branching Fluid Limits. Probl Inf Transm 41, 254–279 (2005). https://doi.org/10.1007/s11122-005-0030-6
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DOI: https://doi.org/10.1007/s11122-005-0030-6