Remote Access Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

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

 

 

An estimate of the probability that the queue length exceeds the maximum for a queue that is a generalized Ornstein-Uhlenbeck stochastic process


Author: R. E. Yamnenko
Translated by: S. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 73 (2005).
Journal: Theor. Probability and Math. Statist. 73 (2006), 181-194
MSC (2000): Primary 60G07; Secondary 60K25
Published electronically: January 19, 2007
MathSciNet review: 2213851
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the process

$\displaystyle A(t)=mt+\sigma \int_0^t X(u)\, d u,\qquad t\geq 0, $

describing the queue length, where $ m$ and $ \sigma$ are positive constants, $ X(u)$ is a $ \varphi$-sub-Gaussian generalized Ornstein-Uhlenbeck stochastic process, and

\begin{displaymath} \varphi(u)= \begin{cases} u^r, & \vert u\vert >1,\\ u^2, & \vert u\vert\le1, \end{cases}\end{displaymath}

$ r\geq2$. The classes of $ \varphi$-sub-Gaussian and strictly $ \varphi$-sub-Gaussian stochastic processes are wider than the class of Gaussian processes and are of interest for modeling stochastic processes appearing in queueing theory and in the mathematics of finance. We obtain an estimate of the probability that the queue length exceeds the maximum allowed for it, namely,

$\displaystyle \mathsf{P}\left\{\sup_{t\geq 0}\left(A(t) -c t \right)>x \right\}\le L(\gamma) x^{r/(r-1)}\exp\left \{-\kappa(\gamma)x^{r/(2(r-1))}\right \}, $

where $ c>m$ is the service intensity, $ x>0$ is the maximum queue length, and $ L(\gamma)$ and $ \kappa(\gamma)$ are some finite constants.


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

  • 1. D. D. Botvich and N. G. Duffield, Large deviations, the shape of the loss curve, and economies of scale in large multiplexers, Queueing Systems Theory Appl. 20 (1995), no. 3-4, 293–320. MR 1356879, 10.1007/BF01245322
  • 2. V. V. Buldygin and Yu. V. Kozachenko, Metric characterization of random variables and random processes, Translations of Mathematical Monographs, vol. 188, American Mathematical Society, Providence, RI, 2000. Translated from the 1998 Russian original by V. Zaiats. MR 1743716
  • 3. Yu. V. Kozachenko and E. I. Ostrovskiĭ, Banach spaces of random variables of sub-Gaussian type, Teor. Veroyatnost. i Mat. Statist. 32 (1985), 42–53, 134 (Russian). MR 882158
  • 4. Yu. V. Kozachenko and Yu. A. Koval′chuk, Boundary value problems with random initial conditions, and functional series from 𝑠𝑢𝑏ᵩ(Ω). I, Ukraïn. Mat. Zh. 50 (1998), no. 4, 504–515 (Russian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 50 (1998), no. 4, 572–585 (1999). MR 1698149, 10.1007/BF02487389
  • 5. Yu. Kozachenko, T. Sottīnen, and O. Vasilik, Self-similar processes with stationary increments in the spaces 𝑆𝑆𝑢𝑏ᵩ(Ω), Teor. Ĭmovīr. Mat. Stat. 65 (2001), 67–78 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 65 (2002), 77–88. MR 1936131
  • 6. Yurii V. Kozachenko and Oksana I. Vasilik, On the distribution of suprema of 𝑆𝑢𝑏ᵩ(Ω) random processes, Proceedings of the Donetsk Colloquium on Probability Theory and Mathematical Statistics (1998), 1998, pp. 147–160. MR 2026624
  • 7. Yu. Kozachenko, O. Vasylyk, and R. Yamnenko, On the probability of exceeding some curve by $ \varphi$-sub-Gaussian random process, Theory Stoch. Process. 9(25) (2003), no. 3-4, 70-80.
  • 8. Olga Vasylyk, Yuriy Kozachenko, and Rostyslav Yamnenko, Upper estimate of overrunning by 𝑆𝑢𝑏ᵩ(Ω) random process the level specified by continuous function, Random Oper. Stochastic Equations 13 (2005), no. 2, 111–128. MR 2152102, 10.1163/156939705323383832
  • 9. M. A. Krasnosel′skiĭ and Ja. B. Rutickiĭ, Convex functions and Orlicz spaces, Translated from the first Russian edition by Leo F. Boron, P. Noordhoff Ltd., Groningen, 1961. MR 0126722
  • 10. Q. Ren and H. Kobayashi, Diffusion approximation modeling for Markov modulated bursty traffic and its applications to bandwidth allocation in ATM networks, IEEE Journal on Selected Areas in Communications 16 (1998), no. 5, 679-691.
  • 11. R. E. Yamnenko, On the probability of exceeding some curve by trajectories of $ \varphi$-sub-Gaussian stochastic process, Visnyk Kyiv. Univ. Ser. Fiz.-Mat. Nauk 3 (2004), 82-86.

Similar Articles

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2000): 60G07, 60K25

Retrieve articles in all journals with MSC (2000): 60G07, 60K25


Additional Information

R. E. Yamnenko
Affiliation: Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kiev 03127, Ukraine
Email: rostyslav_yamnenko@yahoo.com

DOI: https://doi.org/10.1090/S0094-9000-07-00691-6
Keywords: $\varphi$-sub-Gaussian stochastic process, generalized Ornstein--Uhlenbeck process, the distribution of the supremum
Received by editor(s): December 26, 2004
Published electronically: January 19, 2007
Article copyright: © Copyright 2007 American Mathematical Society