Spectral solutions for QS with distribution laws in the form of probabilistic mixtures

Background. QS are the main mathematical tool for modeling data transmission systems, which are not without reason called queuing networks. The need to regulate such characteristics of mass service systems as waiting time in a queue or queue length is due to the improvement of the quality of operation of data transmission systems. The ability to regulate these characteristics allows minimizing the waiting time in the queue in the buffers of transmitting devices, as well as the volumes of buffer memory itself. To demonstrate this possibility, the paper examines queuing systems formed by both conventional distribution laws in the form of probability mixtures and time-shifted distribution laws. Aim. In this work, the hyperexponential and hyper-Erlangian distributions of the second order were chosen as components of the QS. Based on these distribution laws, numerical-analytical models were constructed for two queuing systems with normal and shifted distribution laws, with the derivation of a solution for the main characteristic of the queuing system – the average waiting time in the queue. As is known, the remaining characteristics of the QS are derivatives of the average waiting time. Methods. The paper uses a shift of the distribution laws to the right from the zero point. To derive a solution for the average waiting time in a queue, the classical method of spectral solution of the Lindley integral equation is used based on the Laplace transform of the distribution laws that form the considered QS. The obtained calculation formulas for the average waiting time in a queue allow us to calculate the characteristics of such systems for a wide range of changes in teletraffic parameters. Results. The obtained results can be used in modern teletraffic theory in the design and modeling of various promising data transmission systems, including the volumes of buffer memory of transmitting devices. Conclusion. The shift of the distribution laws in time leads to a decrease in their variation coefficients. Due to the quadratic dependence of the average waiting time on the variation coefficients of the arrival and service time intervals, a noticeable decrease in the average waiting time follows in systems with time shifts.