# TELETRAFFIC MODEL BASED ON HE2/H2/1 SYSTEMS WITH ORDINARY AND WITH 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:

Hypererlangian and hyperexponential distribution laws, Lindley integral equation, spectral decomposition method, Laplace transform.

## Abstract

Context. The problem of deriving a solution for the average waiting time in a closed form queue for an ordinary system with second-order hyper-Erlang and hyperexponential input distributions and a system with shifted hyper-Erlang and hyperexponential input distributions is considered.

Objective. Obtaining a solution for the main characteristic of the system – the average waiting time for requirements in the queue for a queuing system of type G/G/1 with conventional and shifted second-order hyper-Erlang and hyperexponential input distributions.

Method. To solve this problem, we used the classical method of spectral decomposition of the solution of the Lindley integral equation, which allows us to obtain a solution for the average waiting time for the systems in question in a closed form. The spectral decomposition method for solving the Lindley integral equation occupies an important part of the theory of G/G/1 systems. 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 decompositions of the solution of the Lindley integral equation for both systems were obtained, with the help of which calculation formulas for the average waiting time in the queue for the above systems in closed form are derived. This approach allows you to calculate the average waiting time for these systems in mathematical packages for a wide range of traffic parameters. All other system characteristics are derived from the average waiting time.

Conclusions. It is shown that the hypererlang second-order distribution law, as well as the hyperexponential one, which is threeparameter, can be determined by both the first two moments and the first three moments. The choice of this law of probability distribution is because its coefficient of variation covers a wider range than for hyperexponential distribution. For shifted hypererlang and hyperexponential distribution laws, the coefficients of variation decrease and cover an even wider range than for conventional distributions. The introduction of time-shifted distributions expands the scope of QS taking into account the well-known fact from the queuing theory that the average waiting time is associated with the coefficients of variation of the intervals of arrivals and the service time by a quadratic dependence. The spectral decomposition method for solving the Lindley integral equation for a queuing system with second-order hyper-Erlang and hyperexponential input distributions allows us to obtain a solution in a closed form and this solution is published for the first time. The resulting solution complements and extends the well-known queuing theory formula for the average queue waiting time for queuing systems of type G/G/1.

## 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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