Research of Dual Systems H2/M/1 and M/H2/1 with Exponential and Hyperexponential Input Distributions

In this paper, we present comparative results of publications on dual systems H₂/M/1 and M/H₂/1 with hyperexponential input distributions of the second order. By Kendall's definition, these systems belong to the classes G/M/1 and M/G/1, respectively. The distribution of H₂ includes 3 unknown parameters and can therefore be approximated at the level of the first three moments. The solution for the average waiting time for the H₂/M/1 system depends on the three moments of the H₂ distribution. The average waiting time for the system M/H₂/1 does not depend on the 3rd moment of the distribution of H₂, but depends only on the first two moments and completely coincides with the Polyaczek-Khinchin formula. This is the qualitative difference between the dual systems in question H₂/M/1 and M/H₂/1. A point for comparison of these systems is the average waiting time in the queue within the same system load. These data also vary. The conclusion for the average waiting time in the queue for both systems is based on the method of spectral decomposition in solution of the integral equation Lindley.