  In this paper, we generalize a positive recurrence criterion for
multidimensional discrete-time Markov chains over countable state spaces due to
Rosberg (JAP, Vol. 17, No. 3, 1980). We revisit the stability analysis of well
known slotted-Aloha protocol with finite number of queues. Under standard
modeling assumptions, we derive a sufficient condition for the stability by
applying our positive recurrence criterion. Our sufficiency condition for
stability is linear in arrival rates and does not require knowledge of the
stationary joint statistics of queue lengths. We believe that the technique
reported here could be useful in analyzing other stability problems in
countable space Markovian settings. Toward the end, we derive some sufficient
conditions for instability of the protocol.
