The access delay of Aloha with exponential back-off strategies

This paper provides definite results about the access delay of slotted Aloha with exponential back-off, an infinite number of stages, and queues in saturation. The analysis is performed leveraging a method recently introduced, which indicates a maximum throughput equal to 0.370 with binary back-off, and 0.4303 when the exponential back-off base is optimized, and using a delay offset equal to 2. Here we provide the exact cumulative distribution of the access delay, showing the heavy-tail behavior already conjectured by using approximate models. We also prove that the existence of k-th order delay moments depends on explicit conditions on the number of stations involved and other system parameters as well.