Effect of exponential backoff scheme and retransmission cutoff on the stability of frequency-hopping slotted ALOHA systems

The combinatorial effect of an exponential backoff scheme and retransmission cutoff on the stability of frequency-hopping slotted ALOHA systems with finite population is investigated in terms of the catastrophe theory. In the systems, the packet retransmission probabilities are geometrically distributed with respect to the number of experienced unsuccessful transmissions and a packet will be discarded after a certain number of unsuccessful transmissions. Expressions which should be satisfied at equilibrium points are first derived. Then, the cusp point and the bifurcation sets are numerically evaluated. Finally, we visualize how the exponential backoff scheme and retransmission cutoff effect depends on the bistable region. Numerical results show that the exponential backoff scheme can mitigate bistable behavior of the system with finite population. However, it is also revealed that there is asymptotically no effect of the exponential backoff scheme on the stability of the system with infinite population.