Abstract
In this paper we study the distribution of the supremum over interval [0,T] of a centered Gaussian process with stationary increments with a general negative drift function. This problem is related to the distribution of the buffer content in a transient Gaussian fluid queue Q(T) at time T, provided that at time 0 the buffer is empty. The general theory is illustrated by detailed considerations of different cases for the integrated Gaussian process and the fractional Brownian motion. We give asymptotic results for P(Q(T)>x) and P(sup 0≤t≤T Q(t)>x) as x→∞.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D. Anick, D. Mitra and M.M. Sondhi, Stochastic theory of a data handling system with multiple sources, Bell System Techn. J. 61 (1982) 1871-1894.
S. Berman, An asymptotic formula for the distribution of the maximum of a Gaussian process with stationary increments, J. Appl. Probab. 22 (1985) 454-460.
K. Dębicki, Supremum of Gaussian stochastic processes and their applications in high speed transfer models, Ph.D. thesis, Wroc?aw (1999).
K. Dębicki, Ruin probabilities for Gaussian integrated processes, Stochastic Process. Appl. 98 (2002) 151-174.
K. Dębicki and Z. Palmowski, Heavy traffic asymptotics of on-off fluid model, Queueing Systems 33 (1999) 327-338.
K. Dębicki and T. Rolski, Gaussian fluid models: A survey, in: Symposium on Performance Models for Information Communication Networks, Sendai, Japan, 23-25 January 2000.
J.M. Harrison, Brownian Motion and Stochastic Flow Systems (Wiley, New York, 1985).
E. Hashorva and J. Hüsler, Extremes of Gaussian processes with maximal variance near the boundary points, Methodology Comput. Appl. Probab. 2 (2000) 255-270.
J. Hüsler and V.I. Piterbarg, Extremes of a certain class of Gaussian processes, Stochastic Process. Appl. 83 (1999) 257-271.
K. Kobayashi and Y. Takahashi, The tail behavior of the stationary distribution of a fluid queue with a Gaussian-type input rate processes, J. Oper. Res. Soc. Japan 40 (1997) 75-89.
K. Kobayashi and Y. Takahashi, Overflow probability for a discrete-time queue with non-stationary multiplexed input, Telecommunication Systems 15 (2000) 157-166.
V. Kulkarni and T. Rolski, Fluid model driven by an Ornstein-Uhlenbeck process, Probab. Engrg. Inform. Sci. 8 (1994) 403-417.
W.E. Leland, M.S. Taqqu, W.Willinger and D.V. Wilson, On the self-similar nature of Ethernet traffic, IEEE/ACM Trans. Network. 2 (1994) 1-15.
Z. Michna, Self-similar processes in collective risk theory, J. Appl. Math. Stochastic Anal. 11 (1998) 429-448.
O. Narayan, Exact asymptotic queue length distribution for fractional Brownian traffic, Adv. in Performance Anal. 1 (1998) 39-63.
I. Norros, A storage model with self-similar input, Queueing Systems 16 (1994) 387-396.
Z. Palmowski and T. Rolski, The superposition of alternating on-off flows and a fluid model, Ann. Appl. Probab. 8 (1998) 524-541.
V.I. Piterbarg, Asymptotic Methods in the Theory of Gaussian Processes and Fields, Translations of Mathematical Monographs, Vol. 148 (Amer. Math. Soc., Providence, RI, 1996).
V.I. Piterbarg and V. Prisyazhnyuk, Asymptotic behavior of the probability of a large excursion for a nonstationary Gaussian processes, Teor. Veroyatnost. Mat. Statist. 18 (1978) 121-133.
A.J. Zeevi and P.W. Glynn, On the maximum workload of a queue fed by fractional Brownian motion, Ann. Appl. Probab. 10(4) (2000) 1084-1099.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dębicki, K., Rolski, T. A Note on Transient Gaussian Fluid Models. Queueing Systems 41, 321–342 (2002). https://doi.org/10.1023/A:1016283330996
Issue Date:
DOI: https://doi.org/10.1023/A:1016283330996