Stability and irreducibility of queueing systems with finite capacity

In this paper, we obtain a readily verifiable condition of stability for GI/G/1 queueing systems with finite capacity. A necessary and sufficient condition of irreducibility of the queueing size process is involved. Under this assumption, we derive general conditions of recurrence (positive recurrence) for the general process describing the state of the system. The conditions of irreducibility and recurrence are based on restrictions over the supports of the interarrival and the service distributions, which are easy to check in practice. The positive recurrence is also connected to the first moments of both distributions.