Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-29T08:50:38.590Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotics of Hybrid Fluid Queues with Lévy Input

Part of: Special processes Limit theorems Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Krzysztof Dębicki ,
Iwona Sierpińska  and
Bert Zwart
Krzysztof Dębicki*
Affiliation:
University of Wrocław
Iwona Sierpińska*
Affiliation:
University of Wrocław
Bert Zwart*
Affiliation:
CWI, VU University Amsterdam, EURANDOM, and Georgia Institute of Technology
*
∗ Postal address: Instytut Matematyczyny, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.
∗ Postal address: Instytut Matematyczyny, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.
∗∗∗∗ Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let {X(t):t∈ℝ} be the integrated on–off process with regularly varying on-periods, and let {Y(t):t∈ℝ} be a centered Lévy process with regularly varying positive jumps (independent of X(·)). We study the exact asymptotics of ℙ(supt≥0{X(t)+Y(t)-ct}>u) as u→∞, with special attention to the case r=c, where r is the increase rate of the on–off process during the on-periods.

Keywords

Exact asymptoticsLévy processfluid modelqueue

MSC classification

Primary: 60G15: Gaussian processes
Secondary: 60G70: Extreme value theory; extremal processes 60F10: Large deviations 60K25: Queueing theory
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 1 , March 2013 , pp. 103 - 113
Copyright
Copyright © Applied Probability Trust 2013 

Footnotes

Supported by MNiSW grant N N2014079 33 (2007–2009) and by a Marie Curie Transfer of Knowledge Fellowship of the European Community's Sixth Framework Programme under contract MTKD-CT-2004-013389.

Partly supported by NSF grants 0727400 and 0805979, an IBM facility award, and a VIDI grant from NWO.

References

Agrawal, R., Makowski, A. and Nain, P. (1999). On a reduced load equivalence for fluid queues under subexponentiality. Queueing Systems 33, 5–41.Google Scholar
Asmussen, S., Schmidli, H. and Schmidt, V. (1999). Tail probability for non-standard risk and queueing processes with subexponential Jumps. Adv. Appl. Prob. 31, 422–447.Google Scholar
Bertoin, J. and Doney, R. A. (1994). Cramer's estimate for Lévy processes. Statist. Prob. Lett. 21, 363–365.Google Scholar
Borst, S. and Zwart, B. (2005). Fluid queues with heavy-tailed M/G/∞ input. Math. Operat. Res. 30, 852–879.Google Scholar
Breiman, L. (1965). On some limit theorems similar to the arc-sin law. Theory Prob. Appl. 10, 323–331.Google Scholar
Cline, D. B. H. and Samorodnitsky, G. (1994). Subexponentiality of the product of independent random variables. Stoch. Process. Appl. 49, 75–98.Google Scholar
Denisov, D. and Zwart, B. (2007) On a theorem of Breiman and a class of random difference equations. J. Appl. Prob. 44, 1031–1046.Google Scholar
Embrechts, P. and Goldie, C. M. (1980). On closure and factorization properties of subexponential and related distributions. J. Austral. Math. Soc. Ser. A 29, 243–256.Google Scholar
Foss, S., Konstantopoulos, T. and Zachary, S. (2007). Discrete and continuous time modulated random walks with heavy-tailed increments. J. Theoret. Prob. 20, 581–612.Google Scholar
Heath, D., Resnick, S. and Samorodnitsky, G. (1998). Heavy tails and long range dependence in ON/OFF processes and associated fluid models. Math. Operat. Res. 23, 145–165.CrossRefGoogle Scholar
Huzak, M., Perman, M., Sikić, H. and Vondracek, Z. (2004). Ruin probabilities and decompositions for general perturbed risk processes. Ann. Appl. Prob. 14, 1378–1397.Google Scholar
Jelenković, P. R. and Lazar, A. A. (1999). Asymptotic results for multiplexing subexponential on-off processes. Adv. Appl. Prob. 31, 394–421.Google Scholar
Jelenković, P. and Momčilović, P. (2003). Asymptotic loss probability in a finite buffer fluid queue with heterogeneous heavy-tailed on-off processes. Ann. Appl. Prob. 13, 576–603.Google Scholar
Jelenković, P., Momčilović, P. and Zwart, B. (2004). Reduced load equivalence under subexponentiality. Queueing Systems 46, 97–112.CrossRefGoogle Scholar
Lelarge, M. (2009). Asymptotic behavior of generalized processor sharing queues under subexponential assumptions. Queueing Systems 62, 51–73.Google Scholar
Maulik, K. and Zwart, B. (2006). Tail asymptotics for exponential functionals of Levy processes. Stoch. Process. Appl. 116, 156–177.Google Scholar
Palmowski, Z. and Zwart, B. (2007). Tail asymptotics of the supremum of a regenerative process. J. Appl. Prob. 44, 349–365.Google Scholar
Port, S. C. (1989). Stable processes with drift on the line. Trans. Amer. Math. Soc. 313, 805–841.Google Scholar
Reich, E. (1958). On the integrodifferential equation of Takács. I. Ann. Math. Statist. 29, 563–570.Google Scholar
Rolski, T., Schlegel, S. and Schmidt, V. (1999). Asymptotics of Palm-stationary buffer content distributions in fluid flow queues. Adv. Appl. Prob. 31, 235–253.Google Scholar
Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall, New York.Google Scholar
Schmidli, H. (1999). Perturbed risk processes: a review. Theory Stoch. Process. 5, 145–165.Google Scholar
Willekens, E. (1987). On the supremum of an infinitely divisible process. Stoch. Process. Appl. 26, 173–175.Google Scholar
Zwart, B., Borst, S. and Dȩbicki, K. (2005). Subexponential asymptotics of hybrid fluid and ruin models. Ann. Appl. Prob. 15, 500–517.Google Scholar
Zwart, B., Borst, S. and Dȩbicki, K. (2005). Reduced load equivalence for Gaussian processes. Operat. Res. Lett. 33, 502–510.CrossRefGoogle Scholar
Zwart, B., Borst, S. and Mandjes, M. (2004). Exact asymptotics for fluid queues fed by heavy-tailed on–off flows. Ann. Appl. Prob. 14, 903–957.Google Scholar