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Exact asymptotics for a queue with fractional Brownian input and applications in ATM networks

Part of: Stochastic processes Special processes

Published online by Cambridge University Press:  14 July 2016

Tyrone E. Duncan ,
Yi Yan  and
Peng Yan
Tyrone E. Duncan*
Affiliation:
University of Kansas
Yi Yan*
Affiliation:
Sprint
Peng Yan*
Affiliation:
Ohio State University
*
∗ Postal address: Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA.
∗∗ Postal address: Sprint, Network Design, 7171 W. 95th Street, Overland Park, KS 66212, USA. Email address: [email protected]
∗∗∗ Postal address: Department of Electrical Engineering, Ohio State University, Columbus, OH 43210, USA.
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Abstract

In this paper, a single channel FIFO fluid queue with an infinite buffer space and a long-range dependent input is studied. The input traffic is modeled by an average input rate plus a standard fractional Brownian motion as the fluctuation. Lower and upper bounds are derived for the tail distribution of the transient queue length at time T, which result in a logarithmic characterization of the asymptotic behavior of the tail distribution. Furthermore, the exact asymptotic is also obtained. It is observed that the transient queue length under fractional Brownian input does not suffer from the heavy-tail property as does the steady-state queue length. The results are used to compute the equivalent bandwidth requirement for ATM broadband connections with fractional Brownian traffic feed and finite connection holding time.

Keywords

Fractional Brownian motionlong-range dependent propertyasymptotic behaviorheavy tailequivalent bandwidth

MSC classification

Primary: 60G18: Self-similar processes
Secondary: 60G70: Extreme value theory; extremal processes 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 4 , December 2001 , pp. 932 - 945
Copyright
Copyright © by the Applied Probability Trust 2001 

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References

Adler, R. J. (1990). An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes. (Inst. Math. Statist. Lecture Notes Monogr. 12). Institute of Mathematical Studies, Hayward, CA.CrossRefGoogle Scholar
Beran, J., Sherman, R., Taqqu, M. S., and Willinger, W. (1995). Long-range dependence in variable-bit-rate video traffic. IEEE Trans. Commun. 43, 1566–1579.CrossRefGoogle Scholar
Chang, C.-S., Yao, D. D., and Zajic, T. (1999). Large deviations, moderate deviations, and queues with long-range dependent input. Adv. Appl. Prob. 31, 254–277.CrossRefGoogle Scholar
Choe, J., and Shroff, N. B. (1999). Supremum distribution of Gaussian processes and queuing analysis including long-range dependent and self-similar processes. Tech. Rep., Department of Electrical and Computer Engineering, University of Toronto.Google Scholar
Duffield, N. G., and O'Connell, N. (1995). Large deviations and overflow probabilities for the general single-server queue, with applications. Math. Proc. Camb. Phil. Soc. 118, 363–374.CrossRefGoogle Scholar
Elwalid, A. I., and Mitra, D. (1993). Effective bandwidth of general Markovian traffic sources and admission control of high-speed networks. IEEE/ACM Trans. Networking 1, 329–343.CrossRefGoogle Scholar
Erramilli, A., Narayan, O., and Willinger, W. (1996). Experimental studies on the performance impacts of long range dependence. IEEE/ACM Trans. Networking 4, 209–223.CrossRefGoogle Scholar
Guerin, R., Ahmadi, H., and Naghshineh, M. (1991). Equivalent capacity and its application to bandwidth allocation in high-speed networks. IEEE J. Select. Areas Commun. 9, 968–981.CrossRefGoogle Scholar
Harrison, J. M. (1985). Brownian Motion and Stochastic Flow Systems. John Wiley, New York.Google Scholar
Husler, J., and Piterbarg, V. I. (1999). Extremes of a certain class of Gaussian processes. Stoch. Process. Appl. 83, 257–271.CrossRefGoogle Scholar
Leland, W. E., Taqqu, M. S., Willinger, W., and Wilson, D. V. (1994). On the self-similar nature of ethernet traffic (extended version). IEEE/ACM Trans. Networking 2, 1–15.CrossRefGoogle Scholar
Narayan, O. (1998). Exact asymptotic queue-length distribution for fractional Brownian traffic. Adv. Performance Anal. 1, 39–54.Google Scholar
Norros, I. (1994). A storage model with self-similar input. Queueing Systems 16, 387–396.CrossRefGoogle Scholar
Piterbarg, V. I. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields. (Transl. Math. Monogr. 148). American Mathematical Society, Providence, RI.Google Scholar
Taqqu, M. S., Willinger, W., and Sherman, R. (1997). Proof of a fundamental result in self-similar traffic modeling. Comput. Commun. Rev. 27, 5–23.CrossRefGoogle Scholar
Willinger, W., Taqqu, M. S., Leland, W. E., and Wilson, D. V. (1995). Self-similarity in high-speed packet traffic: analysis and modeling of ethernet traffic measurements. Statist. Sci. 10, 67–85.CrossRefGoogle Scholar