Remote Access Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)

 
 

 

On the distribution of storage processes from the class $ V(\varphi,\psi)$


Authors: R. E. Yamnenko and O. S. Shramko
Translated by: S. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 83 (2010).
Journal: Theor. Probability and Math. Statist. 83 (2011), 191-206
MSC (2010): Primary 60G07; Secondary 60K25
DOI: https://doi.org/10.1090/S0094-9000-2012-00851-X
Published electronically: February 2, 2012
MathSciNet review: 2768858
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Estimates for the distribution of a storage process

$\displaystyle Q(t)=\sup _{s\le t}\big (X(t)-X(s)-(f(t)-f(s))\big )$

are obtained in the paper, where $ (X(t),t\in T)$ is a stochastic process belonging to the class $ V(\varphi ,\psi )$ and where the service output rate $ f(t)$ is a continuous function. In particular, the results hold if $ (X(t),t\in T)$ is a Gaussian process. Several examples of applications of the results obtained in the paper are given for sub-Gaussian stationary stochastic processes.

References [Enhancements On Off] (What's this?)

  • 1. R. Addie, P. Mannersalo, and I. Norros, Most probable paths and performance formulae for buffers with Gaussian input traffic, Eur. Trans. Telecommun. 13 (3) (2002), 183-196.
  • 2. P. Boulongne, D. Pierre-Loti-Viaud, and V. Piterbarg, On average losses in the ruin problem with fractional Brownian motion as input, Extremes 12 (2009), 77-91. MR 2480724 (2010f:60115)
  • 3. V. V. Buldygin and Yu. V. Kozachenko, Metric Characterization of Random Variables and Random Processes, TViMS, Kiev, 1998; English transl., American Mathematical Society, Providence, RI, 2000. MR 1743716 (2001g:60089)
  • 4. N. G. Duffield and N. O'Connell, Large deviations and overflow probabilities for the general single-server queue, with applications, Math. Proc. Cambridge Phil. Soc. 118 (1995), 363-374. MR 1341797 (96f:60039)
  • 5. Yu. V. Kozachenko and E. I. Ostrovskii, Banach spaces random variables of sub-Gaussian type, Teor. Veroyatnost. i Mat. Statist. 32 (1985), 42-53; English transl. in Theory Probab. Math. Statist. 32 (1986), 45-56. MR 882158 (88e:60009)
  • 6. Yu. V. Kozachenko and Yu. A. Kovalchuk, Boundary value problems with random initial conditions, and functional series from $ \textnormal {Sub}_\varphi (\Omega )$, I, Ukrain. Mat. Zh. 50 (1998), no. 4, 504-515; English transl. in Ukrainian Math. J. 50 (1999), no. 4, 572-585. MR 1698149 (2000f:60029)
  • 7. Yu. Kozachenko, O. Vasylyk, and R. Yamnenko, Upper estimate of overrunning by $ \textnormal {Sub}_\varphi (\Omega )$ random process the level specified by continuous function, Random Oper. and Stoch. Equ. 13 (2005), no. 2, 111-128. MR 2152102 (2006b:60207)
  • 8. L. Massoulie and A. Simonian, Large buffer asymptotics for the queue with fractional Brownian input, J. Appl. Probab. 36 (1999), 894-906. MR 1737061 (2000i:60108)
  • 9. I. Norros, A storage model with self-similar input, Queueing Syst. 16 (1994), 387-396. MR 1278465 (95a:60142)
  • 10. I. Norros, On the use of fractional Brownian motions in the theory of connectionless networks, IEEE Journal on selected areas in communications 13 (1995), no. 6, 953-962.
  • 11. R. Yamnenko, Ruin probability for generalized $ \varphi $-sub-Gaussian fractional Brownian motion, Theory Stoch. Process. 12 (28) (2006), no. 1-2, 261-275. MR 2316577 (2008g:60109)
  • 12. R. Yamnenko and O. Vasylyk, Random process from the class $ V(\varphi ,\psi )$: exceeding a curve, Theory Stoch. Process. 13 (29) (2007), no. 4, 219-232. MR 2482262 (2010a:60125)
  • 13. Yu. V. Kozachenko, O. I. Vasylyk, and R. E. Yamnenko, $ \varphi $-sub-Gaussian Stochastic Processes, Kyiv University, Kyiv, 2008. (Ukrainian)

Similar Articles

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2010): 60G07, 60K25

Retrieve articles in all journals with MSC (2010): 60G07, 60K25


Additional Information

R. E. Yamnenko
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 2, Kiev 03127, Ukraine
Email: yamnenko@univ.kiev.ua

O. S. Shramko
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 2, Kiev 03127, Ukraine
Email: shes21@ukr.net

DOI: https://doi.org/10.1090/S0094-9000-2012-00851-X
Keywords: Metric entropy, queue, storage process, estimate of a distribution, sub-Gaussian process
Received by editor(s): April 21, 2010
Published electronically: February 2, 2012
Article copyright: © Copyright 2012 American Mathematical Society

American Mathematical Society