Characterising the Finite Capacity GI/M/1 Queue with Renewal Output

This note proves that the only stationary GI/M/1/N queueing systems (i.e., systems having fixed waiting room capacity N(0 \leqq N \leqq \infty )) with renewal output are the M/M/1/0 and stable (traffic intensity < 1) M/M/1/\infty systems.