Abstract
A multi-class single server queue under non-preemptive static buffer priority (SBP) service discipline is considered in this paper. Using a bounding technique, we obtain the fluid approximation for the queue length and busy time processes. Furthermore, we prove that the convergence rate of the fluid approximation for the queue length and busy time processes is exponential for large N. Additionally, a sufficient condition for stability is obtained.
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Harrison, J. M.: Brownian models of queueing networks with heterogeneous customer populations. In: Stochastic Differential Systems, Stochastic Control Theory and Applications (W. Fleming and P. L. Lions eds.), Vol. 10 of Proceedings of the IMA, Springer-Verlag, New York, 1988, 147–186
Chen, H.: Fluid approximations and stability of multiclass queueing networks I: Work-conserving disciplines. Annals of Applied Probability, 5, 637–665 (1995)
Chen, H.: Rate of convergence of fluid approximation for generalized Jackson network. Journal of Applied Probability, 33, 804–814 (1996)
Chen, H., Mandelbaum, A.: Leontief systems, RBV’s and RBM’s. In: the Proceedings of the Imperial College Workshop on Applied Stochastic Processes (Davis, M. H. A. and Elliotte, R. J. eds.), Fordon and Breach Science Publisher, 1991
Chen, H., Shanthikumar, J. G.: Fluid limits and diffusion approximations for networks of multi-server queues in heavy traffic. Discrete Event Dynamic Systems, 4, 269–291 (1994)
Chen, H., Yao, D. D.: Fundamentals of Queueing Networks, Springer-Verlag, New York, 2001
Chen, H., Zhang, H.: Stability of multiclass queueing networks under priority service disciplines. Operation Research, 48, 26–37 (2000)
Dai, J. G.: On the positive Harris recurrence for multiclass queueing networks. Annals of Applied Probability, 5, 49–77 (1995)
Dai, J. G.: Stability of Fluid and Stochastic Processing Networks, Miscellanea Publication, No. 9, Centre for Mathematical Physics and Stochastic, Denmark (http://www.maphysto.dk/), January, 1999
Dai, J. G., Weiss, G.: Stablity and instability of fluid models for re-entrant lines. Mathematics of Operations Research, 21, 115–134 (1996)
Guo, Y. J., Yang, Y., Yu, J.: Rate of convergence of fluid approximation for a multi-class single-server queue. Acta Mathematicae Applicatae Sinica, Chinese Series, 29, 1125–1138 (2006)
Guo, Y. J.: Convergence rate of fluid approximation for Jackson network with multi-server in heavy traffic. J. Sys. Sci. & Math. Scis., Chinese Series, 28, 1118–1133 (2008)
Guo, Y. J.: Fluid approximation and its convergence rate for GI/G/1 queue with vacations. Acta Mathematicae Applicatae Sinica, 27, 43–58 (2011)
Newell, G. F.: Applications of Queueing Theory, Chapman and Hall, New York, 1982
Whitt, W.: Preservation of rates of convergence under mappings. Zeitschrift Wahrescheinlichkeitstheorie, 29, 39–44 (1974)
Skorohod, A. V.: Stochastic differential equations for a bounded region. Theor. Prob. Appl., 6, 261–274 (1961)
Harrison, J. M., Reiman, M. I.: Reflected Brownian motion in an orthant. The Annals of Probability, 9, 302–308 (1981)
Meyn, S. P., Tweedie, R. L.: Stability of Markovian processes II: Continuous time processes and sample chains. Adv. Appl. Probab., 25, 487–517 (1993)
Reiman, M. I.: Open queueing networks in heavy traffic. Mathematics of Operations Research, 9, 441–458 (1984)
Ross, S. M.: Stochastic Processes, second edition, Wiley, New York, 1996
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Supported by National Natural Science Foundation of China (Grant No. 10901023), and the Fundamental Research Funds for the Central Universities (Grant Nos. BUPT2009RC0707 and BUPT2011RC0704)
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Guo, Y.J. On the fluid approximation for a multiclass queue under non-preemptive SBP service discipline. Acta. Math. Sin.-English Ser. 28, 379–404 (2012). https://doi.org/10.1007/s10114-012-9643-z
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DOI: https://doi.org/10.1007/s10114-012-9643-z
Keywords
- Multi-class queue
- non-preemptive SBP service discipline
- fluid approximation
- stability
- exponential rate of convergence