Teletraffic mathematical model based on erlang and hypererlang distributions

This article presents the results of 22 / /1HE E queuing system studies with Hyper-Erlang and Erlang second-order input distributions. By Kendall’s defi nition, these systems belong to G/G/1 class with arbitrary distribution laws of input fl ow intervals and service time. In the queueing theory, the study of such systems is particularly relevant due to the fact that it is still not impossible to fi nd a solution for the average waiting time in a queue in the fi nal form. Higher-order Erlang and hyper-Erlang distributions are diffi cult to consider since they cause an increase in computational complexity when deriving a solution for the average waiting time. For the considered second-order distributions, the classical spectral decomposition method of solving the Lindley integral equation for G/G/1 systems allows obtaining a solution in closed form. The article presents the obtained spectral decomposition of the Lindley integral equation solution for the system in question and the formula for the average waiting time in the queue. The adequacy of the obtained results is confi rmed by the correctness of using the classical spectral decomposition method and the results of numerical simulation. The 22 / /1HE E system is applicable when the arrival intervals variation coeffi cient is greater than or equal to 1/ 2 and the service time variation coeffi cient is equal to 1/ 2. For practical application of the results obtained, the method of moments from probability theory is used. The results of Mathcad numerical simulation unequivocally confi rm that the average waiting time is related to the arrival intervals variation coeffi cient and the service time variation coeffi cient by a quadratic dependence.