Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-05T12:38:51.081Z Has data issue: false hasContentIssue false
  • English
  • Français

On the supremum distribution of integrated stationary Gaussian processes with negative linear drift

Part of: Stochastic processes Special processes

Published online by Cambridge University Press:  01 July 2016

Jinwoo Choe  and
Ness B. Shroff
Jinwoo Choe*
Affiliation:
University of Toronto
Ness B. Shroff*
Affiliation:
Purdue University
*
∗ Postal address: Dept. of Electrical & Computer Engneering, University of Toronto, Ontario, M5S 3GA, Canada. Email address: [email protected]
∗∗ Postal address: School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907-1285, USA. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we study the supremum distribution of a class of Gaussian processes having stationary increments and negative drift using key results from Extreme Value Theory. We focus on deriving an asymptotic upper bound to the tail of the supremum distribution of such processes. Our bound is valid for both discrete- and continuous-time processes. We discuss the importance of the bound, its applicability to queueing problems, and show numerical examples to illustrate its performance.

Keywords

Supremum distributionGaussian processstationary increment with linear driftqueue length distributionextreme value theoryasymptotic upper boundtail probability

MSC classification

Primary: 60G15: Gaussian processes
Secondary: 60G70: Extreme value theory; extremal processes 60K25: Queueing theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 31 , Issue 1 , March 1999 , pp. 135 - 157
Copyright
Copyright © Applied Probability Trust 1999 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Addie, R. G. and Zukerman, M. (1994). An approximation for performance evaluation of stationary single server queues. IEEE Trans. Commun. 42, 3150–3160.Google Scholar
Adler, R. J. (1990). An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes. Institute of Mathematical Statistics, Hayward, CA.Google Scholar
Berman, S. M. and Kono, N. (1989). The maximum of a Gaussian process with non-constant variance: a sharp bound for the distribution tail. Ann. Prob. 17, 632–650.Google Scholar
Bratley, P., Fox, B. L. and Schrage, L. E. (1987). A Guide to Simulation, 2nd edn. Springer, New York.Google Scholar
Chang, C. -S., Heidelberger, P., Juneja, S. and Shahabuddin, P. (1994). Effective bandwidth and fast simulation of ATM intree networks. Performance Evaluation 20, 45–65.Google Scholar
Choe, J. and Shroff, N. B. (1997). A central limit theorem based approach to analyse queue behavior in ATM networks. In Proc. 15th Internat. Teletraffic Congress, Washington DC, pp. 1129–1138.Google Scholar
Choe, J. and Shroff, N. B. (1997). A new method to determine the queue length distribution at an ATM multiplexer. In Proc. IEEE INFOCOM 97, Kobe, Japan. pp. 550–557.Google Scholar
Choe, J. and Shroff, N. B. (1998). A central limit theorem based approach for analyzing queue behavior in high-speed networks. {Technical report}, Purdue University, West Lafayette, IN.Google Scholar
Choudhury, G. L., Lucantoni, D. M. and Whitt, W. (1996). Squeezing the most out of ATM. IEEE Trans. Commun. 44, 203–217.Google Scholar
Elwalid, A. I. (1991). Markov modulated rate processes for modeling, analysis and control of communication networks. , Graduate School of Arts and Sciences, Columbia University.Google Scholar
Gihman, I. I. and Skorohod, A. V. (1974). The Theory of Stochastic Processes. I. Springer, New York.Google Scholar
Glynn, P. W. and Whitt, W. (1994). Logarithmic asymptotics for steady-state tail probabilities in a single-server queue. In Studies in Applied Probability, ed. Galambos, J. and Gani, J. (J. Appl. Prob. 31A). Applied Probability Trust, Sheffield, UK, pp. 131–155.Google Scholar
Kesidis, G., Walrand, J., and Chang, C.-S. (1993). Effective bandwidth for multiclass Markov fluid and other ATM sources. IEEE/ACM Trans. Networking 1, 424–428.CrossRefGoogle Scholar
Loynes, R. M. (1962). The stability of a queue with non-independent inter-arrival and service times. Proc. Cam. Philos. Soc. 58, 497–520.Google Scholar
Maglaris, B., Anastassiou, D., Sen, P., Karlsson, G. and Robbins, J. D. (1988). Performance models of statistical multiplexing in packet video communication. IEEE Trans. Commun. 36, 834–843.Google Scholar
Ross, S. M. (1983). Stochastic Processes. Wiley, New York.Google Scholar
Samorodnitsky, G. (1991). Probability tails of Gaussian extrema. Stochast. Proc. Appl. 38, 55–84.Google Scholar
Shroff, N. B. and Schwartz, M. (1998). Improved loss calculations at an ATM multiplexer. IEEE/ACM Trans. Networking 6, 411–422. In Proc. IEEE INFOCOM, vol. 2. IEEE Computer Society Press, Washington DC. pp. 561–568. An extended version of this paper will appear in IEEE/ACM Transactions on Networking, Aug. 1998.Google Scholar
Simonian, A. (1991). Stationary analysis of a fluid queue with input rate varying as an Ornstein-Uhlenbeck process. SIAM J. Appl. Math. 51, 828–842.Google Scholar
Talagrand, M. (1988). Small tails for the supremum of a Gaussian process. Ann. Inst. Henri Poincaré 24, 307–315.Google Scholar