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Asymptotic analysis of a fluid model modulated by an M/M/1 queue

Part of: Markov processes Asymptotic theory Special processes

Published online by Cambridge University Press:  01 July 2016

Charles Knessl  and
Diego Ernesto Dominici
Charles Knessl*
Affiliation:
University of Illinois at Chicago
Diego Ernesto Dominici*
Affiliation:
State University of New York at New Paltz
*
∗ Postal address: Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago (M/C 249), 851 South Morgan Street, Chicago, IL 60607-7045, USA. Email address: [email protected]
∗∗ Postal address: Department of Mathematics, State University of New York at New Paltz, 1 Hawk Dr. Suite 9, New Paltz, NY 12561-2443, USA. Email address: [email protected]
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Abstract

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We analyze asymptotically a differential-difference equation that arises in a Markov-modulated fluid model. We use singular perturbation methods to analyze the problem with appropriate scalings of the two state variables. In particular, the ray method and asymptotic matching are used.

Keywords

Fluid queueasymptoticsM/M/1 queueray method

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60J25: Continuous-time Markov processes on general state spaces 34E10: Perturbations, asymptotics
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 40 , Issue 3 , September 2008 , pp. 856 - 881
Copyright
Copyright © Applied Probability Trust 2008 

References

Adan, I. and Resing, J. (1996). Simple analysis of a fluid queue driven by an M/M/1 queue. Queueing Systems Theory Appl. 22, 171–174.CrossRefGoogle Scholar
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.Google Scholar
Barbot, N. and Sericola, B. (2002). Stationary solution to the fluid queue fed by an M/M/1 queue. J. Appl. Prob. 39, 359–369.Google Scholar
Dominici, D. and Knessl, C. (2005). Geometrical optics approach to Markov-modulated fluid models. Stud. Appl. Math. 114, 45–93.Google Scholar
Hashida, O. and Fujika, M. (1973). Queueing models for buffer memory in store-and-forward systems. In Proc. Seventh Internat. Teletraffic Congress (Stockholm, June 1973), Paper 323, 7 pp.Google Scholar
Keller, J. B. (1978). Rays, waves and asymptotics. Bull. Amer. Math. Soc. 84, 727–750.Google Scholar
Lenin, R. B. and Parthasarathy, P. R. (2000). A computational approach for fluid queues driven by truncated birth–death processes. Methodol. Comput. Appl. Prob. 2, 373–392.CrossRefGoogle Scholar
McDonald, D. and Qian, K. (1998). An approximation method for complete solutions of Markov-modulated fluid models. Queueing Systems Theory Appl. 30, 365–384.Google Scholar
Miyazawa, M. (1994). Palm calculus for a process with a stationary random measure and its applications to fluid queues. Queueing Systems Theory Appl. 17, 183–211.CrossRefGoogle Scholar
Parthasarathy, P. R., Vijayashree, K. V. and Lenin, R. B. (2002). An M/M/1 driven fluid queue—continued fraction approach. Queueing Systems 42, 189–199.Google Scholar
Ren, Q. and Kobayashi, H. (1992). A mathematical theory for transient analysis of communications networks. IEICE Trans. Commun. 12, 1266–1276.Google Scholar
Ren, Q. and Kobayashi, H. (1995). Transient solutions for the buffer behavior in statistical multiplexing. Performance Evaluation 23, 65–87.Google Scholar
Sericola, B. (2001). A finite buffer fluid queue driven by a Markovian queue. Queueing Systems Theory Appl. 38, 213–220.Google Scholar
Sericola, B., Parthasarathy, P. R. and Vijayashree, K. V. (2005). Exact transient solution of an M/M/1 driven fluid queue. Internat. J. Comput. Math. 82, 659–671.Google Scholar
Tanaka, T., Hashida, O. and Takahashi, Y. (1995). Transient analysis of fluid model for ATM statistical multiplexer. Performance Evaluation 23, 145–162.Google Scholar
Tucker, R. C. F. (1988). Accurate method for analysis of a packet-speech multiplexer with limited delay. IEEE Trans. Commun. 36, 479–483.Google Scholar
Van Doorn, E. A. and Scheinhardt, W. R. W. (1997). A fluid queue driven by an infinite-state birth–death process. In Teletraffic Contributions for the Information Age, eds Ramaswami, V. and Wirth, P., Elsevier, Amsterdam, pp. 465–475.Google Scholar
Virtamo, J. and Norros, I. (1994). Fluid queue driven by an M/M/1 queue. Queueing Systems Theory Appl. 16, 373–386.Google Scholar
Wijngaard, J. (1979). The effect of interstage buffer storage on the output of two unreliable production units in series with different production rates. AIIE Trans. 11, 42–47.CrossRefGoogle Scholar