QUEUEING SYSTEMS WITH TIME LAG
Keywords:delayed system, shifted distributions, Laplace transform, Lindley integral equation, spectral decomposition method.
Context. In the queuing theory of a research of the G/G/1 systems are relevant because it is impossible to receive decisions for the average waiting time in queue in a final form in case of arbitrary laws of distributions of an input flow and service time. Therefore, the study of such systems for particular cases of input distributions is important. The problem of deriving solutions for the average waiting time in a queue in closed form for systems with distributions shifted to the right from the zero point is considered.
Objective. Getting solutions for the main characteristics of the systems – the average waiting time of requirements in the queue for queuing systems (QS) of type G/G/1 with shifted input distributions.
Methods. 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 the 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. For the first time, spectral expansions are obtained for the solution of the Lindley integral equation for systems with delay, which are used to derive formulas for the average waiting time in a queue in closed form. The paper presents the final studies for the remaining eight delay systems.
Conclusions. It is shown that in systems with delay, the average waiting time is less than in in the usual systems. The obtained formula for the average waiting time expands and complements the well-known queuing theory incomplete formula for the average waiting time for G/G/1 systems. This approach allows us to calculate the average latency for these systems in mathematical packages for a wide range of traffic parameters. 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.
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