Spectral solution for a delay system with hyper-Erlang distributions

The article is devoted to the construction of a mathematical model for delaying claims in a queue in the form of a queuing system described by two flows with the laws of distribution of time intervals shifted to the right by hyper-Erlang distributions of the second order. In the queuing theory, the study of systems G/G/1 is relevant because there is no solution in the final form for the general case. Therefore, various partial distribution laws are used as an arbitrary distribution law G in the study of such systems. In this case, the use of the shifted hyper-Erlang distribution law provides the coefficient of variation of the input flow arrival intervals and service time over the entire interval (0, ∞). To solve the problem, we used the method of spectral solution of the Lindley integral equation, which plays an important role in the queuing theory. This method made it possible to obtain a solution for the average delay of requests in the queue for the considered system in a closed form. As is known, the remaining characteristics of the queuing system are derivatives of the average delay of requests in the queue.