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

Journal:
Theor. Probability and Math. Statist. **73** (2006), 181-194

MSC (2000):
Primary 60G07; Secondary 60K25

DOI:
https://doi.org/10.1090/S0094-9000-07-00691-6

Published electronically:
January 19, 2007

MathSciNet review:
2213851

Full-text PDF Free Access

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Abstract: We consider the process \[ A(t)=mt+\sigma \int _0^t X(u) d u,\qquad t\geq 0, \] describing the queue length, where $m$ and $\sigma$ are positive constants, $X(u)$ is a $\varphi$-sub-Gaussian generalized Ornstein–Uhlenbeck stochastic process, and \[ \varphi (u)= \begin {cases} u^r, & |u| >1,\\ u^2, & |u|\le 1, \end {cases}\] $r\geq 2$. 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, \[ \mathsf {P}\left \{\sup _{t\geq 0}\left (A(t) -c t \right )>x \right \}\le L(\gamma ) x^{r/(r-1)}\exp \left \{-\kappa (\gamma )x^{r/(2(r-1))}\right \}, \] where $c>m$ is the service intensity, $x>0$ is the maximum queue length, and $L(\gamma )$ and $\kappa (\gamma )$ are some finite constants.

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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

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