Instability and geometric transience of the aloha protocol

In this paper, we give a simple probabilistic proof to show that the discrete time Markov chain underlying the slotted uncontrolled Aloha protocol is geometrically transient. Let P be the (?? ?? ??) transition matrix of this Markov chain. Let Pn denote the northwest (n ?? n) corner truncation of P and ??n. its largest eigenvalue. We establish that, as a consequence of geometric transience, ?? = lim ??n as n ?? ??, exists and that 0 < ?? < 1. Note that the largest eigenvalue of P equals 1. We propose 1/(1-??) as a performance measure which we show to be the limit of certain expected exit times.