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 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]: Year: Volume: Issue: Page: Find

 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2018, Volume 22, Number 4, Pages 702–713 (Mi vsgtu1607)

Mathematical Modeling, Numerical Methods and Software Complexes

The average waiting time in a $H_2/H_2/1$ queueing system with delay

V. N. Tarasov, E. Akhmetshina

Povolzhskiy State University of Telecommunications and Informatics, , Samara, 443010, Russian Federation

Abstract: In queueing theory, the study of $G/G/1$ systems is particularly relevant due to the fact that until now there is no solution in the final form in the general case. Here $G$ on Kendall's symbolics means arbitrary distribution law of intervals between requirements of an input flow and service time.
In this article, the task of determination of characteristics of a $H_2/H_2/1$ queueing system with delay of the $G/G/1$ type is considered using the classical method of spectral decomposition of the solution of the Lindley integral equation.
As input distributions for the considered system, probabilistic mixtures of exponential distributions shifted to the right of the zero point are chosen, that is, hyperexponential distributions $H_2$. For such distribution laws, the method of spectral decomposition allows one to obtain a solution in closed form. It is shown that in such systems with a delay, the average waiting time for calls in the queue is less than in conventional systems. This is due to the fact that the operation of time shift reduces the coefficients of variation of the intervals between the receipts and the service time, and as is known from queueing theory, the average wait time of requirements is related to these coefficients of variation by a quadratic dependence. The $H_2/H_2/1$ queueing system with a delay can quite well be used as a mathematical model of modern teletraffic.

Keywords: system with delay, $H_2/H_2/1$ queueing system, Laplace transformation, average waiting time in the queue
Author to whom correspondence should be addressed

DOI: https://doi.org/10.14498/vsgtu1607

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Bibliographic databases:

UDC: 519.872
MSC: 90B22, 60K25
Revised: October 7, 2018
Accepted: November 12, 2018
First online: December 29, 2018

Citation: V. N. Tarasov, E. Akhmetshina, “The average waiting time in a $H_2/H_2/1$ queueing system with delay”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 22:4 (2018), 702–713

Citation in format AMSBIB
\Bibitem{TarAkh18} \by V.~N.~Tarasov, E.~Akhmetshina \paper The average waiting time in a $H_2/H_2/1$ queueing system with delay \jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.] \yr 2018 \vol 22 \issue 4 \pages 702--713 \mathnet{http://mi.mathnet.ru/vsgtu1607} \crossref{https://doi.org/10.14498/vsgtu1607} \elib{http://elibrary.ru/item.asp?id=36681028}