Stability and Instability Conditions for Slotted Aloha with Exponential Backoff

This paper provides stability and instability conditions for slotted Aloha under the exponential backoff (EB) model with geometric law i 7→ b0 , when transmission buffers are in saturation, i.e., always full. In particular, we prove that for any number of users and for b > 1 the system is: (i) ergodic for i0 > 1, (ii) null recurrent for 0 < i0 ≤ 1, and (iii) transient for i0 = 0. Furthermore, when referring to a system with queues and Poisson arrivals, the system is shown to be stable whenever EB in saturation is stable with throughput λ0 and the system input rate is upper-bounded as λ < λ0. Index Terms Slotted Aloha, Exponential Backoff, Stability, Markov chain, Ergodicity