ABSTRACTS OF ARTICLES OF THE JOURNAL "INFORMATION TECHNOLOGIES".
No. 4. Vol. 26. 2020

DOI: 10.17587/it.26.195-202

V. N. Tarasov, D. Sc., Professor, Head of Chair, e-mail: veniamin_tarasov@mail.ru, N. F. Bakhareva, D. Sc., Professor, Head of Chair, e-mail: nadin1956_04@inbox.ru,
E. G. Akhmetshina, Postgraduate, Povolzhsky State University of Telecommunications and Informatics, Samara, 443010, Russian Federation

Teletrafic Models Based on Dual Systems with Delay with Hyperexponential and Exponential Distributions

In queuing theory, the G/M/1 and M/G/1 systems are widely used, while for the first system there is still no final solution in the general case. Here G in the first system according to Kendall symbolism means an arbitrary law of the distribution of intervals between the requirements of the input flow, M is the exponential law of service time, and in the second system, it is exactly the opposite. The article considers the problem of determining the characteristics of queuing systems (QS) H₂2/1 with delay with hyperexponential (H₂) and exponential (M) distributions. This problem is solved using the classical method of spectral decomposition of the solution of the Lindley integral equation. As input distributions for the systems under consideration, probabilistic mixtures of exponential distributions shifted to the right from the zero point and shifted exponential distributions are selected. For such distribution laws, the spectral decomposition method allows one to obtain a closed-form solution. It is shown that in such systems with delay, the average waiting time for requirements in the queue is shorter than in conventional systems. This is because the time shift operation reduces the coefficient of variation of the intervals between receipts and the service time, and as is known from the queuing theory, the average waiting time for requirements is associated with these coefficients of variation by a quadratic dependence. QS H₂/M/1 and M/H₂/1 with delay can very well be used as a mathematical model of modern teletraffic.
Keywords: Delayed system, dual pair H₂/M/1 and M/H₂/1, Laplace transform, average waiting time in a queue, Lindley integral equation

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