Research and comparison of dual systems E2/M/1 and M/E2/1

This article presents the comparative results of original research on dual E2/M/1 and M/E2/1 systems with second order exponential and erlangian input distributions. By Kendall’s definition, these systems belong to the classes G/M/1 and M/G/1, respectively. In queuing theory, the systems M/G/1 and G/M/1 are widely used. Investigations of G/M/1 systems are particularly relevant due to the fact that there is still no solution in the final form in the general case, with arbitrary laws of the distribution of intervals of the input flow. Using a higher order erlangian distribution is difficult to derive a solution for the average waiting time due to increasing computational complexity. For such distribution laws, the classical spectral decomposition method for solving the Lindley integral equation for G/G/1 systems allows one to obtain a solution in closed form. The E2/M/1 system is applicable when the coefficient of variation of the arrival intervals is equal 1/ 2 to the coefficient of variation of the service time equal to 1, and the system M/E2/1 is applicable when the coefficient of variation of the interval of receipt is 1 and the coefficient of variation of the service time is equal 1/ 2. To derive solutions, the classical method of spectral decomposition of the solution of the Lindley integral equation is used. The results of numerical simulation indicate a slight difference in the considered dual systems due to the relatively small coefficients of variation of the used distribution laws. In the case of other laws of distributions, dual systems G/M/1 and M/G/1 will give rather different results.