Research of system with shifted hyper-erlang and exponential input distributions

This article presents the results of studies on the HE2/M/1 queuing system with hyper-Erlang and exponential input distributions shifted to the right from the zero point. By Kendall’s definition, the HE2/M/1 system with conventional distributions is of type G/M/1, for which, in general the solution for the average queue waiting time is not known. The same system with shifted distributions is transformed into the G/G/1 system, for which, in the general case, the solution for the average waiting time is also unknown. Considering the fact that, starting with a coefficient of variation equal to four, the distribution of hyper-Erlang has a heavy tail, the system in question can have an active application in the modern theory of teletraffic. Using higher-order hyper-Erlang distributions is difficult to derive a solution for the average waiting time of requests in a queue due to increasing computational complexity. For the hyper-Erlangian distribution law, as well as the hyperexponential law, the spectral decomposition method for solving the Lindley integral equation makes it possible to obtain a final solution. The article presents the results on the spectral decomposition of the solution of the Lindley integral equation for the queuing system HE2/M/1 with shifted distributions, as well as the calculation formula for the average waiting time for claims in the queue. It is shown that the HE2/M/1 system with shifted distributions is a system with a time lag and provides a shorter waiting time compared to a conventional system. The adequacy of the results is confirmed by the correct use of the classical method of spectral decomposition and the results of numerical simulation. To derive the obtained results, as well as for numerical calculations, the well-known method of moments of probability theory is used.