×

zbMATH — the first resource for mathematics

Efficient algorithms for transient analysis of stochastic fluid flow models. (English) Zbl 1085.60065
For the canonical Markov-modulated fluid flow model, efficient algorithms for the transient busy period analysis are developed. In particular, an algorithm for the fluid model that is similar to Latouche-Ramaswami algorithm for quasi-birth-death processes and has quadratic convergence is presented. The proofs are based on matrix-geometric method.

MSC:
60K25 Queueing theory (aspects of probability theory)
90B05 Inventory, storage, reservoirs
68M20 Performance evaluation, queueing, and scheduling in the context of computer systems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ahn, S. and Ramaswami, V. (2003). Fluid flow models and queues—a connection by stochastic coupling. Stoch. Models 19, 325–348. · Zbl 1021.60073 · doi:10.1081/STM-120023564
[2] Ahn, S. and Ramaswami, V. (2004). Transient analysis of fluid flow models via stochastic coupling to a queue. Stoch. Models 20, 71–101. · Zbl 1038.60086 · doi:10.1081/STM-120028392
[3] Anick, D., Mitra, D. and Sondhi, M. M. (1982). Stochastic theory of a data-handling system with multiple sources. Bell System Tech. J. 61, 1871–1894.
[4] Asmussen, S. (1994). Busy period analysis, rare events and transient behavior in fluid flow models. J. Appl. Math. Stoch. Anal. 7, 269–299. · Zbl 0826.60086 · doi:10.1155/S1048953394000262 · eudml:47296
[5] Asmussen, S. (1995). Stationary distributions for fluid flow models with or without Brownian noise. Stoch. Models 11, 1–20. · Zbl 0817.60086 · doi:10.1080/15326349508807330
[6] Çinlar, E. (1975). Introduction to Stochastic Processes . Prentice Hall, Englewood Cliffs, NJ.
[7] Gaver, D. P. and Lehoczky, J. P. (1982). Channels that cooperatively service a data stream and voice messages. IEEE Trans. Commun. 30, 1153–1161. · Zbl 0478.94001 · doi:10.1109/TCOM.1982.1095563
[8] Graham, A. (1981). Kronecker Products and Matrix Calculus: with Applications . John Wiley, New York. · Zbl 0497.26005
[9] Kobayashi, H. and Ren, Q. (1992). A mathematical theory for transient analysis of communication networks. IEICE Trans. Commun. 12, 1266–1276.
[10] Latouche, G. (1993). Algorithms for infinite Markov chains with repeating columns. In Linear Algebra, Markov Chains and Queueing Models (IMA Vol. Math. Appl. 48 ), eds C. D. Meyer and R. J. Plemmons, Springer, New York, pp. 231–265. · Zbl 0790.65121
[11] Latouche, G. and Ramaswami, V. (1993). A logarithmic reduction algorithm for quasi-birth-and-death processes. J. Appl. Prob. 30, 650–674. · Zbl 0789.60055 · doi:10.2307/3214773
[12] Latouche, G. and Ramaswami, V. (1999). Introduction to Matrix Analytic Methods in Stochastic Modeling . Society for Industrial and Applied Mathematics, Philadelphia, PA. · Zbl 0922.60001 · doi:10.1137/1.9780898719734
[13] Neuts, M. F. (1981). Matrix-Geometric Solutions in Stochastic Models. An Algorithmic Approach . John Hopkins University Press, Baltimore, MD. · Zbl 0469.60002
[14] Ortega, J. M. and Rheinboldt, W. C. (1970). Iterative Solution of Nonlinear Equations in Several Variables . Academic Press, New York. · Zbl 0241.65046
[15] Ramaswami, V. (1999). Matrix analytic methods for stochastic fluid flows. In Teletraffic Engineering in a Competitive World . (Proc. 16th Internat. Teletraffic Congress), eds D. Smith and P. Key, Elsevier, New York, pp. 1019–1030.
[16] Rogers, L. C. G. (1994). Fluid models in queueing theory and Wiener–Hopf factorization of Markov chains. Ann. Appl. Prob. 4, 390–413. · Zbl 0806.60052 · doi:10.1214/aoap/1177005065
[17] Sericola, B. (1998). Transient analysis of stochastic fluid models. Performance Evaluation 32, 245–263.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.