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Theory of Probability and Mathematical Statistics

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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
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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?)

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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

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