On a positive recurrence criterion for multidimensional Markov chains with application to the stability of slotted-Aloha with a finite number of queues

In this paper, 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 invoking stochastic dominance arguments in conjunction with recurrence criterion due to Rosberg. 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.