# RESEARCH OF QUEUEING SYSTEMS WITH SHIFTED ERLANGIAN AND EXPONENTIAL INPUT DISTRIBUTIONS

## Authors

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

## Keywords:

Erlangian and exponential distribution laws, Lindley integral equation, spectral decomposition method, Laplace transform.

## Abstract

Context. In queuing theory, the study of G/G/1 systems is particularly relevant due to the fact that until now there is no solution in the final form
in the general case. The problem of the derivation in closed form of the solution for the average waiting time in the queue for ordinary systems with
erlangian and exponential input distributions and for the same systems with shifted distributions is considered.
Objective. Obtaining a solution for the main system characteristic – the average waiting time for queue requirements for three types of queuing
systems of type G/G/1 with conventional and shifted erlangian and exponential input distributions.
Method. To solve this problem, we used the classical method of spectral decomposition of the solution of Lindley integral equation, which allows
one to obtain a solution for average the waiting time for systems under consideration in a closed form. The method of spectral decomposition of
the solution of 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 the three kinds of systems were first obtained with the
help of which the calculated expressions for the average waiting time in the queue for the above systems in a closed form were derived.
Conclusions. The introduction of the time shift parameter in the laws of input flow distribution and service time for the systems under consideration
turns them into systems with a delay with a shorter waiting time. This is due to the fact that the time shift operation reduces the coefficient of
variation in the intervals between the receipts of the requirements and their service time, and as is known from queuing theory, the average wait time
of requirements is related to these coefficients of variation by a quadratic dependence. The system with erlangian input distributions of the second
order is applicable only at a certain point value of the coefficients of variation of the intervals between the receipts of the requirements and their service
time. The same system with shifted distributions allows us to operate with interval values of coefficients of variations, which expands the scope
of these systems. Similarly the situation and with the shifted exponential distributions is. In addition, the shifted exponential distribution contains two
parameters and allows one to approximate arbitrary distribution laws using the first two moments. 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. The method of spectral decomposition of the solution of the Lindley integral equation for the systems under consideration makes it
possible to obtain a solution in a closed form and these solutions are published for the first time.

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