Analysis of queues with hyperexponential arrival distributions

We study $\mathrm{H_2/H}_2/1$, $\mathrm{H_2/M}/1$ and $\mathrm{M/H}_2/1$ queueing systems with hyperexponential arrival distributions for the purpose of finding a solution for the mean waiting time in the queue. To this end we use the spectral decomposition method for solving the Lindley integral equation. For practical application of the obtained results, we use the method of moments. Since the hyperexponential distribution law has three unknown parameters, it allows to approximate arbitrary distributions with respect to the first three moments. The choice of this distribution law is due to its simplicity and the fact that in the class of distributions with coefficients of variation greater than 1, such as log-normal, Weibull, etc., only the hyperexponential distribution makes it possible to obtain an analytical solution.