The queue GeoX/G/1/N+1 revisited

We present analytic expressions (in terms of roots of the underlying characteristic equation) for the steady-state distributions of the number of customers for the finite-state queueing model GeoX/G/1/N+1 with partial-batch rejection policy. We obtain the system-length distributions at a service-completion epoch by applying the imbedded Markov chain technique. Using the roots of the related characteristic equation, the method leads to giving a unified approach for solving both finite- and infinite-buffer systems. We find relationships between system-length distributions at departure, random, and arrival epochs using discrete renewal theory and conditioning on the system states. Based on these relationships, we obtain various performance measures and provide numerical results for the same. We also perform computational analysis and compare our results with respect to the solution obtained by solving a linear system of equations in terms of running times.