# MODEL OF TELETRAFFIC BASED ON QUEUEING SYSTEMS E2/HE2/1 WITH ORDINARY AND SHIFTED INPUT DISTRIBUTIONS

## Authors

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

## Keywords:

Erlang and Hyper-Erlang distribution laws, Lindley integral equation, spectral decomposition method, Laplace transform.

## Abstract

Context. The study of G/G/1 systems is related to their relevance in the modern theory of teletraffic and, therefore, in the theory of computing systems and networks. In turn, this follows from the fact that it is impossible to obtain solutions for the waiting time in these systems in the final form in the general case with arbitrary laws of the distribution of the input flow and service time. Therefore, the study of such systems for particular cases of input distributions is important.

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 conventional and with shifted second-order Erlang and Hyper-Erlang input distributions.

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

Results. For the first time, spectral expansions of the solution of the Lindley integral equation are obtained for two systems, with the help of which the formulas for the average waiting time in the queue are derived in closed form. The system with shifted Erlang and Hyper-Erlang input distributions provides shorter waiting times for requirements in the queue compared to a conventional system by reducing the coefficients of variation of intervals between requirements and of service time.

Conclusions. Spectral expansions of the solution of the Lindley integral equation for the systems under consideration are obtained and their complete coincidence is proved. Consequently, the formulas for the average waiting time in the queue for these systems are the same, but with modified parameters. These formulas expand and supplement the known queuing theory incomplete formula for the average waiting time for G/G/1 systems with arbitrary laws distributions of input flow 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 Biographies

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

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

### N. F. Bakhareva, Volga State University of Telecommunications and Informatics, Samara

Dr. Sc., Professor, Head of Department of Informatics and Computer Engineering

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