Transformation of Queueing Systems into Systems with Time Delay

The purpose of this work is to demonstrate the possibility of transforming conventional queuing systems by shifting the distribution laws by introducing the shift parameter into systems with time lag. This leads to qualitatively different systems of the G/G/1 type with completely different characteristics. The paper considers various queueing systems (QS) formed by such distributions as: exponential, hyperexponential, Erlang. In contrast to the classical theory, this article considers QS with distribution laws shifted to the right from the zero point. The article provides an overview of the author's results for the average waiting time in a queue in a closed form for systems with input distributions shifted to the right from the zero point. To obtain them, we used the method of spectral decomposition of the solution of the Lindley integral equation. The article presents the spectral decompositions of the solution of the Lindley integral equation for three systems out of sixteen possible and with their help formulas for calculating the average waiting time in the queue are derived. It is shown that in systems with delay, the average waiting time is much shorter than in conventional systems. The proposed approach makes it possible to calculate the average waiting time for these systems in mathematical packages for a wide range of changes in traffic parameters. Considering the fact that packet delay variation (jitter) in the telecommunications standard is defined as the spread of latency around its average value, then jitter can be determined through the variance of latency.