# ANALYSIS OF H2/E2/1 SYSTEM AND HER OF THE ANALOG WITH SHIFTED INPUT DISTRIBUTIONS

## Authors

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

## Keywords:

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

## Abstract

Context. In the queuing theory of a research of the G/G/1 systems are relevant because it is impossible to receive decisions for the average waiting time in queue in a final form in case of arbitrary laws of distributions of an input flow and service time. Therefore, the study of such systems for particular cases of input distributions is important. The problem of finding a solution for the average waiting time in queue in a closed form for two systems with ordinary and shifted hyperexponential and erlangian input distributions is considered.

Objective. Obtaining a solution for the main system characteristic – the average waiting time in queue for two queuing systems of type G/G/1 with ordinary and shifted hyperexponential 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 waiting time 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. The spectral decompositions of the solution of the Lindley integral equation for a pair of dual systems are for the first time received, with the help of which the formulas for the average waiting time in a closed form are derived.

Conclusions. The spectral expansions of the solution of the Lindley integral equation for the systems under consideration are obtained and with their help the formulas for the average waiting time in the queue for these systems in a closed form are derived. It is shown that in systems with a time lag, the average waiting time is less than in conventional systems The obtained formula for the average waiting time expands and complements the well-known queuing theory incomplete formula for the average waiting time for G/G/1 systems with arbitrary laws of the input flow distribution and service time. This approach allows us to calculate the average latency for these systems in mathematical packages for a wide range of traffic parameters. All other characteristics of the systems are derived from the waiting time. In addition to the average waiting time, 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 waiting time from its average value, the jitter can be determined through the variance of the waiting time. The results are published for the first time.

## Author Biography

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

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

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