Solution for queuing systems E2/M/1 with ordinary and shifted input distributions

In queuing theory, the studies of the systems G/M [g1] 1 and G/G/1 are particularly relevant because there still exist no final solution for the general case. In this article we present results on (QS) G/G[F2] /1 and G/G/1 queuing systems (QS): E2/M/1 with Erlang and exponential input distributions, and also systems with delay in time, respectively. For the system with time delay the Erlang distribution of the second order shifted to the right from the zero point is selected as the input distribution along with the shifted exponential distribution. For such distribution laws, the classical method of spectral decomposition of the solution of the Lindley’s integral equation for systems G/G/1 allows to obtain a closed-form solution. It is shown that in a system with time delay the average waiting time in queue is less than in the usual system. This is explained by the fact that the time shift reduces the coefficients of variation of the intervals between the arrival and the service time, and as known from queuing theory, the average waiting time is related to these coefficients of variation by a quadratic dependence. The E2/M/1 system works only for the arrival intervals variation factor equal to and the service time variation factor equal to 1, while the system allows us to work with the arrival intervals variation factor in the range of (0, ) and the service time variation factor in the range of (0, 1), which expands the scope of these systems. For deriving the solutions, the classical method of spectral decomposition of the solution of the Lindley’s integral equation was used.