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

**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 -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 -sub-Gaussian stochastic process*, Visnyk Kyiv. Univ. Ser. Fiz.-Mat. Nauk**3**(2004), 82-86.

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