Research of two queuing systems M/E2/1 with ordinary and shifted input distributions

Teletraffic theory often uses queuing systems like M/G/1 and G/G/1. Studies of the latter are still relevant due to the fact that it is impossible to obtain solutions for the average waiting time in the queue in the final form in the general case. This article presents the results for two queuing systems: for a regular M/E2/1 system with exponential and Erlangian distributions, as well as this system with distributions shifted to the right from the zero point. The operation of shifting the laws of distributions in this case transforms the M/G/1 system into a G/G/1 type system due to a decrease in the coefficient of variation of the intervals of the input flow into the system. As it turned out, for the distribution laws under consideration, the spectral decomposition method used for solving the Lindley integral equation for G/G/1 systems allows us to obtain a solution for the average waiting time in the final form. It is shown that in such a system with a delay in time, the average waiting time for requests in the queue can be many times shorter than in a similar conventional system. This follows from the fact that the time-shift operation of the distribution laws reduces the coefficient of variation of the intervals between arrivals and service time. At the same time, it is known that the average waiting time for requirements in the queue for the system depends directly on the squares of these variation coefficients. The M/E2/1 system is applicable only when the coefficient of variation of the intervals of arrival equal to 1 and the coefficient of variation of the service time is equal to 1/ 2, and the system with delay is applicable when the coefficients of variation of the intervals of arrival in the range (0, 1) and the coefficients of variation of the time of service from the interval (0, 1/ 2), which dramatically expands the scope of these systems. To derive solutions by the average waiting time in the queue, the spectral decomposition method of solving the Lindley integral equation, known in queuing theory, was used.