DOI: https://doi.org/10.15588/1607-3274-2019-2-8

THE ANALYSIS OF TWO QUEUING SYSTEMS HE2/M/1 WITH ORDINARY AND SHIFTED INPUT DISTRIBUTIONS

V. N. Tarasov

Abstract


Context. In queuing theory, G/M/1 and G/G/1 systems research is relevant in that there is still no solution in the final form for
the general case for arbitrary laws of distributions of the input flow and service time. The problem of finding a solution for the average
waiting time in queue in a closed form for two systems with ordinary and shifted hypererlangian and exponential 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/M/1 and G/G/1 with ordinary and shifted hypererlangian and exponential 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.
These expressions expand and supplement the known queuing theory formulas for the average waiting time for G/M/1 and 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.

Keywords


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

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