Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T21:22:08.618Z Has data issue: false hasContentIssue false
  • English
  • Français

Extremes of nonstationary Gaussian fluid queues

Part of: Stochastic processes Special processes

Published online by Cambridge University Press:  16 November 2018

Krzysztof Dȩbicki  and
Peng Liu
Krzysztof Dȩbicki*
Affiliation:
University of Wrocław
Peng Liu*
Affiliation:
University of Lausanne
*
* Postal address: Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland. Email address: krzysztof.debicki@math.uni.wroc.pl
** Postal address: Department of Actuarial Science, University of Lausanne, UNIL-Dorigny, 1015 Lausanne, Switzerland. Email address: peng.liu@unil.ch
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the asymptotic properties of the transient queue length process Q(t)=max(Q(0)+X(t)-ct, sup0≤s≤t(X(t)-X(s)-c(t-s))), t≥0, in the Gaussian fluid queueing model, where the input process X is modeled by a centered Gaussian process with stationary increments and c>0 is the output rate. More specifically, under some mild conditions on X and Q(0)=x≥0, we derive the exact asymptotics of πx,Tu(u)=ℙ(Q(Tu)>u) as u→∞. The interplay between u and Tu leads to two qualitatively different regimes: short-time horizon when Tu is relatively small with respect to u, and moderate- or long-time horizon when Tu is asymptotically much larger than u. As a by-product, we discuss the implications for the speed of convergence to stationarity of the model studied.

Keywords

Nonstationary queueoverflow probabilityexact asymptoticsGaussian processgeneralized Pickands constantgeneralized Piterbarg constant

MSC classification

Primary: 60G15: Gaussian processes
Secondary: 60G70: Extreme value theory; extremal processes 60K25: Queueing theory
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 3 , September 2018 , pp. 887 - 917
Copyright
Copyright © Applied Probability Trust 2018 

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

[1]Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation. Cambridge University Press.Google Scholar
[2]Burnecki, K. and Michna, Z. (2002). Simulation of Pickands constants. Prob. Math. Statist. 22, 193–199.Google Scholar
[3]Dębicki, K. (2002). Ruin probability for Gaussian integrated processes. Stoch. Process. Appl. 98, 151–174.Google Scholar
[4]Dębicki, K. and Kosiński, K. M. (2014). On the infimum attained by the reflected fractional Brownian motion. Extremes 17, 431–446.Google Scholar
[5]Dębicki, K. and Liu, P. (2016). Extremes of stationary Gaussian storage models. Extremes 19, 273–302.Google Scholar
[6]Dębicki, K. and Mandjes, M. (2003). Exact overflow asymptotics for queues with many Gaussian inputs. J. Appl. Prob. 40, 704–720.Google Scholar
[7]Dębicki, K. and Palmowski, Z. (1999). On-off fluid models in heavy traffic environment. Queueing Systems 33, 327–338.Google Scholar
[8]Dębicki, K. and Rolski, T. (2002). A note on transient Gaussian fluid models. Queueing Systems 41, 321–342.Google Scholar
[9]Dębicki, K., Es-Saghouani, A. and Mandjes, M. (2009). Transient characteristics of Gaussian queues. Queueing Systems 62, 383–409.Google Scholar
[10]Dębicki, K., Hashorva, E. and Liu, P. (2017). Extremes of γ-reflected Gaussian processes with stationary increments. ESAIM Prob. Statist. 21, 495–535.Google Scholar
[11]Dębicki, K., Hashorva, E. and Liu, P. (2017). Uniform tail approximation of homogenous functionals of Gaussian fields. Adv. Appl. Prob. 49, 1037–1066.Google Scholar
[12]Dieker, A. B. (2005). Extremes of Gaussian processes over an infinite horizon. Stoch. Process. Appl. 115, 207–248.Google Scholar
[13]Dieker, A. B. and Mikosch, T. (2015). Exact simulation of Brown-Resnick random fields at a finite number of locations. Extremes 18, 301–314.Google Scholar
[14]Dieker, A. B. and Yakir, B. (2014). On asymptotic constants in the theory of extremes for Gaussian processes. Bernoulli 20, 1600–1619.Google Scholar
[15]Harper, A. J. (2013). Bounds on the suprema of Gaussian processes, and omega results for the sum of a random multiplicative function. Ann. Appl. Prob. 23, 584–616.Google Scholar
[16]Harper, A. J. (2017). Pickands' constant H α does not equal 1/Γ(1/α), for small α. Bernoulli 23, 582–602.Google Scholar
[17]Hashorva, E. (2018). Representations of max-stable processes via exponential tilting. To appear in Stoch. Process. Appl.Google Scholar
[18]Hashorva, E., Ji, L. and Piterbarg, V. I. (2013). On the supremum of γ-reflected processes with fractional Brownian motion as input. Stoch. Process. Appl. 123, 4111–4127.Google Scholar
[19]Hüsler, J. and Piterbarg, V. (1999). Extremes of a certain class of Gaussian processes. Stoch. Process. Appl. 83, 257–271.Google Scholar
[20]Hüsler, J. and Piterbarg, V. (2004). On the ruin probability for physical fractional Brownian motion. Stoch. Process. Appl. 113, 315–332.Google Scholar
[21]Mandjes, M. (2007). Large Deviations for Gaussian Queues. John Wiley, Chichester.Google Scholar
[22]Mandjes, M., Norros, I. and Glynn, P. (2009). On convergence to stationarity of fractional Brownian storage. Ann. Appl. Prob. 19, 1385–1403.Google Scholar
[23]Michna, Z. (1999). On tail probabilities and first passage times for fractional Brownian motion. Math. Meth. Operat. Res. 49, 335–354.Google Scholar
[24]Mikosch, T. and Samorodnitsky, G. (2007). Scaling limits for cumulative input processes. Math. Operat. Res. 32, 890–918.Google Scholar
[25]Norros, I. (1994). A storage model with self-similar input. Queueing Systems 16, 387–396.Google Scholar
[26]PickandsJ., III J., III (1967). Maxima of stationary Gaussian processes. Z. Wahrscheinlichkeitsth. 7, 190–223.Google Scholar
[27]PickandsJ., III J., III (1969). Upcrossing probabilities for stationary Gaussian processes. Trans. Amer. Math. Soc. 145, 51–73.Google Scholar
[28]Piterbarg, V. I. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields. American Mathematical Society, Providence, RI.Google Scholar
[29]Piterbarg, V. I. (2015). Twenty Lectures About Gaussian Processes. Atlantic Financial Press, London.Google Scholar
[30]Piterbarg, V. I. and Prisjažnjuk, V. P. (1978). Asymptotic behavior of the probability of a large excursion for a nonstationary Gaussian process. Teor. Verojatnost. Mat. Statist. 18, 121–134, 183.Google Scholar
[31]Shao, Q.-M. (1996). Bounds and estimators of a basic constant in extreme value theory of Gaussian processes. Statistica Sinica 6, 245–258.Google Scholar
[32]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.Google Scholar
[33]Willinger, W., Taqqu, M. S., Sherman, R. and Wilson, D. V. (1995). Self-similarity through high variability: statistical analysis of ethernet LAN traffic at the source level. Comput. Commun. Rev. 25, 100–113.Google Scholar