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Exact overflow asymptotics for queues with many Gaussian inputs

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Krzysztof Dębicki  and
Michel Mandjes
Krzysztof Dębicki*
Affiliation:
CWI, Amsterdam, and University of Wrocław
Michel Mandjes*
Affiliation:
CWI, Amsterdam, and University of Twente
*
∗Postal address: Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland. Email address: [email protected]
∗∗Postal address: CWI, PO Box 94079, 1090 GB Amsterdam, The Netherlands
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Abstract

In this paper we consider a queue fed by a large number of independent continuous-time Gaussian processes with stationary increments. After scaling the buffer exceedance threshold and the (constant) service capacity by the number of sources, we present asymptotically exact results for the probability that the buffer threshold is exceeded. We consider both the stationary overflow probability and the (transient) probability of overflow at a finite time horizon. We give detailed results for the practically important cases in which the inputs are fractional Brownian motion processes or integrated Gaussian processes.

Keywords

Gaussian processfluid queueextremesoverflow probabilityexact asymptoticsintegrated Gaussian processfractional Brownian motion

MSC classification

Primary: 60G15: Gaussian processes
Secondary: 60G70: Extreme value theory; extremal processes 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 3 , September 2003 , pp. 704 - 720
Copyright
Copyright © Applied Probability Trust 2003 

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References

Addie, R., Mannersalo, P., and Norros, I. (1999). Performance formulae for queues with Gaussian input. In Teletraffic Engineering in a Competitive World (Proc. 16th Internat. Teletraffic Cong., Edinburgh, June 1999), eds Key, P. and Smith, D., Elsevier, Amsterdam, pp. 1169–1178.Google 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
Bahadur, R. R., and Rao, R. R. (1960). On deviations of the sample mean. Ann. Math. Statist. 31, 1015–1027.CrossRefGoogle Scholar
Borodin, A., and Salminen, P. (1996). Handbook of Brownian Motionbn-Facts and Formulae. Birkhäuser, Basel.Google Scholar
Botvich, D., and Duffield, N. G. (1995). Large deviations, the shape of the loss curve, and economies of scale in large multiplexers. Queueing Systems 20, 293–320.CrossRefGoogle Scholar
Courcoubetis, C., and Weber, R. (1996). Buffer overflow asymptotics for a buffer handling many traffic sources. J. Appl. Prob. 33, 886–903.Google Scholar
Dębicki, K. (2002). Ruin probabilities for Gaussian integrated processes. Stoch. Process. Appl. 98, 151–174.Google Scholar
Dębicki, K., and Palmowski, Z. (1999). Heavy traffic asymptotics of on-off fluid model. Queuing Systems 33, 327–338.CrossRefGoogle Scholar
Dębicki, K., and Rolski, T. (1995). A Gaussian fluid model. Queueing Systems 20, 433–452.CrossRefGoogle Scholar
Dębicki, K., and Rolski, T. (2000). Gaussian fluid models; a survey. In Proc. Symp. Performance Models for Information Communication Networks (Sendai, Japan, January 2000). Available at http://www.math.uni.wroc.pl/~rolski/.Google Scholar
Dębicki, K., and Rolski, T. (2002). A note on transient Gaussian fluid models. Queueing Systems 41, 321–342.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.Google Scholar
Hüsler, J., and Piterbarg, V. I. (1999). Extremes of a certain class of Gaussian processes. Stoch. Process. Appl. 83, 257–271.Google Scholar
Konstant, D. G., and Piterbarg, V. (1993). Extreme values of the cyclostationary Gaussian random processes. J. Appl. Prob. 30, 82–97.CrossRefGoogle Scholar
Kulkarni, V., and Rolski, T. (1994). Fluid model driven by an Ornstein-Uhlenbeck process. Prob. Eng. Inf. Sci. 8, 403–417.Google Scholar
Leland, W. E., Taqqu, M. S., Willinger, W., and Wilson, D. V. (1994). On the self-similar nature of ethernet traffic. IEEE/ACM Trans. Networking 2, 1–15.CrossRefGoogle Scholar
Likhanov, N., and Mazumdar, R. (1999). Cell loss asymptotics in buffers fed with a large number of independent stationary sources. J. Appl. Prob. 36, 86–96.Google Scholar
Mandjes, M., and Borst, S. (2000). Overflow behavior in queues with many long-tailed inputs. Adv. Appl. Prob. 32, 1150–1167.Google Scholar
Mandjes, M., and Kim, J. H. (2001). Large deviations for small buffers: an insensitivity result. Queueing Systems 37, 349–362.Google Scholar
Massoulié, L., and Simonian, A. (1999). Large buffer asymptotics for the queue with FBM input. J. Appl. Prob. 36, 894–906.Google Scholar
Narayan, O. (1998). Exact asymptotic queue length distribution for fractional Brownian traffic. Adv. Perf. Anal. 1, 39–63.Google Scholar
Norros, I. (1994). A storage model with self-similar input. Queuing Systems 16, 387–396.CrossRefGoogle Scholar
Piterbarg, V. I. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields (Translations Math. Monogr. 148). American Mathematical Society, Providence, RI.Google Scholar
Piterbarg, V. I., and Prisyazhnyuk, V. P. (1979). Asymptotics of the probability of large excursions for a nonstationary Gaussian process. Theory Prob. Math. Statist. 18, 131–144.Google Scholar
Shorack, G. R., and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. John Wiley, New York.Google Scholar
Simonian, A., and Guibert, J. (1995). Large deviations approximation for fluid queues fed by a large number of on/off sources. IEEE J. Selected Areas Commun. 13, 1017–1027.Google Scholar
Simonian, A., and Virtamo, J. (1991). Transient and stationary distributions for fluid queue and input processes with density. SIAM J. Appl. Math. 51, 1731–1739.Google Scholar
Weiss, A. (1986). A new technique of analyzing large traffic systems. Adv. Appl. Prob. 18, 506–532.Google Scholar