# MATHEMATICAL DELAY MODEL BASED ON SYSTEMS WITH HYPERERLANGIAN AND ERLANGIAN DISTRIBUTIONS

## Keywords:

hypererlangian and erlangian distribution laws, Lindley integral equation, spectral decomposition method, Laplace transform.

## Abstract

Context. Studies of G/G/1 systems in queuing theory are relevant because such systems are of interest for analyzing the delay of data transmission systems. At the same time, it is impossible to obtain solutions for the delay in the final form in the general case for arbitrary laws of distribution of the input flow and service time. Therefore, it is important to study such systems for particular cases of input distributions. We consider the problem of deriving a solution for the average queue delay in a closed form for two systems with ordinary and shifted hypererlangian and erlangian input distributions.

Objective. Obtaining a solution for the main characteristic of the system – the average delay of requests in the queue for two queuing systems of the G/G/1 type with ordinary and with shifted hypererlangian and erlangian input distributions.

Method. To solve this problem, we used the classical method of spectral decomposition of the solution of the Lindley integral equation. This method allows to obtaining a solution for the average delay for systems under consideration in a closed form. The method of spectral decomposition of the solution of the Lindley integral equation plays an important role in the theory of systems G/G/1. For the practical application of the results obtained, the well-known method of moments of probability theory is used.

Results. For the first time, spectral expansions of the solution of the integral Lindley equation for two systems are obtained, with the help of which calculation formulas for the average delay in a queue in a closed form are derived. Thus, mathematical models of queuing delay for these systems have been built.

Conclusions. These formulas expand and supplement the known queuing theory formulas for the average delay G/G/1 systems with arbitrary laws distributions of input flow and service time. This approach allows us to calculate the average delay for these systems in mathematical packages for a wide range of traffic parameters. In addition to the average delay, such an approach makes it possible to determine also moments of higher orders of waiting time. Given the fact that the packet delay variation (jitter) in telecommunications is defined as the spread of the delay from its average value, the jitter can be determined through the variance of the delay.

## Author Biography

### V. N. Tarasov , Volga State University of Telecommunications and Informatics, Samara, Russian Federation.

Dr. Sc., Professor, Head of Department of Software and Management in Technical Systems.

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2021-03-27

## How to Cite

Tarasov , V. N. . (2021). MATHEMATICAL DELAY MODEL BASED ON SYSTEMS WITH HYPERERLANGIAN AND ERLANGIAN DISTRIBUTIONS . Radio Electronics, Computer Science, Control, (1), 87–96. https://doi.org/10.15588/1607-3274-2021-1-9

## Section

Mathematical and computer modelling