The average waiting time in a $H_2/H_2/1$ queueing system with delay

In queueing 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. Here $G$ on Kendall's symbolics means arbitrary distribution law of intervals between requirements of an input flow and service time.
In this article, the task of determination of characteristics of a $H_2/H_2/1$ queueing system with delay of the $G/G/1$ type is considered using the classical method of spectral decomposition of the solution of the Lindley integral equation.
As input distributions for the considered system, probabilistic mixtures of exponential distributions shifted to the right of the zero point are chosen, that is, hyperexponential distributions $H_2$. For such distribution laws, the method of spectral decomposition allows one to obtain a solution in closed form. It is shown that in such systems with a delay, the average waiting time for calls in the queue is less than in conventional systems. This is due to the fact that the operation of time shift reduces the coefficients of variation of the intervals between the receipts and the service time, and as is known from queueing theory, the average wait time of requirements is related to these coefficients of variation by a quadratic dependence. The $H_2/H_2/1$ queueing system with a delay can quite well be used as a mathematical model of modern teletraffic.