Research and comparison of pairs of dual systems with hyper-erlang and exponential distributions

This article presents the results of research on HE2/M/1 and M/HE2/1 mass service systems with hyper-Erlang and exponential input distributions. By Kendall’s definition, these systems belong to classes G/M/1 and M/G/1 respectively, and also form a dual pair. In the queueing theory, studies of such systems are relevant because they are actively used in the modern theory of teletraffic. The use of higher-order hyper-Erlang distributions to derive a solution for the average waiting time of requirements in the queue is difficult due to the increasing complexity of computation. For the hyperErlang distribution law, as well as for the hyperexponential law, the spectral decomposition method of the solution makes it possible to obtain a solution in its final form. The article presents the results of the spectral decomposition of the Lindley solution for HE2/M/1 and M/HE2/1 mass maintenance systems, as well as the computational formulas for the average waiting time of the requirements in the queue. The appropriateness of the obtained results is confirmed by the correctness of using the classical method of spectral decomposition and the results of numerical simulation. To derive the results, as well as for numerical calculations, the well-known method of moments of probability theory is used.