Analysis of a system with exponential and hyper-Erlang distributions by the method of spectral decomposition of the solution the Lindley integral equation

In this work, we obtain the spectral decomposition of the solution of the Lindley integral equation for a queuing system with a Poisson input flow of requirements and a hyper-Erlang distribution of the service time. Based on it, a calculation formula is derived for the average queue waiting time for this system in a closed form. As you know, all other characteristics of the queuing systems are derived from the average waiting time. The resulting calculation formula complements and extends the well-known Polyachek-Khinchin formula in queuing theory for M/G/1 systems. In the queueing theory, studies of private systems of the M/G/1 type are relevant due to the fact that they are still actively used in the modern theory of teletraffic.