DOI: https://doi.org/10.15588/1607-3274-2018-4-6

### ANALYSIS AND COMPARISON OF TWO QUEUEING SYSTEMS WITH HYPERERLANGIAN INPUT DISTRIBUTIONS

V. N. Tarasov

#### Abstract

Context. The problem of finding the solution for the mean waiting time in a closed form for a conventional system with hyperarlangian
input distributions of second order and for system with shifted hypererlangian input distributions is considered.
Objective is obtaining a solution for the main characteristic of the system-the average waiting time for queuing requirements for
a queuing system of the type G/G/1 with normal and with shifted hypererlangian input distributions of the second order.
Method. To solve this problem, we used the classical method of spectral decomposition of the solution of the Lindley integral
equation, which allows one to obtain a solution for the mean waiting time for of the systems under consideration in closed form. The
method of spectral decomposition of the solution of the Lindley integral equation occupies an important part of 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 decompositions of the solution of the Lindley integral equation for both systems were obtained, with the help of which the calculated expressions for the average waiting time in the queue for the above-mentioned systems
in closed form were derived. 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 the derived from the average waiting time.
Conclusions. It is shown that the hypererlangian distribution law of the second order, like the hyperexponential, which is threeparameter, can be determined by both the first two moments and the first three moments. The choice of such a law of probability distribution is due to the fact that its coefficient of variation covers a wider range than the hyperexponential distribution. For the shifted hypererlangian distribution law, the coefficient of variation covers an even wider range. The introduction of shifted distributions extends the scope of the QS with considering the well-known fact from queuing theory, that the average waiting time is related
to the coefficients of variations in the intervals of receipts and the time of service by a quadratic dependence. The method of spectral decomposition of the solution of the Lindley integral equation for a queuing system with hypererlangian input distributions of the
second order makes it possible to obtain a solution in closed form and this solution is published for the first time. The solution obtained supplements and extends the well-known queuing theory formula for the average waiting time in the queue for a queuing system
of type G/G/1.

#### Keywords

hypererlangian distribution law; Lindley integral equation; spectral decomposition method; Laplace transform.

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Copyright (c) 2019 V. N. Tarasov