Mathematical delay model based on QS with hyperexponential and Erlang distributions

This article is devoted to the study and obtaining a closed-form solution for the average delay of claims in the queue for a queuing system formed by two flows with hyperexponential and Erlang distributions of intervals. The combination of these distribution laws provides the coefficient of variation of the input flow intervals large units, and for the service time - less than unity. Considering the coefficients of variation as numerical characteristics in the queuing theory is important, because the main characteristic of the queuing system is that the average delay is related to these coefficients of variation by a quadratic dependence. In queuing theory, studies of G/G/1 systems are relevant due to the fact that they can be used in modeling data transmission systems for various purposes. To solve the problem posed, the method of spectral decomposition of the solution of the integral Lindley equation was used. This method made it possible to obtain a spectral decomposition, and through it a solution for the average delay of requests in the queue for the system under consideration in a closed form. For the practical application of the results obtained, the method of moments of the theory of probability was used.