Capacity of time-slotted ALOHA systems

We consider the capacity of time-slotted ALOHA systems, where multiple users synchronously send packets, which may collide at the receiver. Specific coding for ALOHA systems had previously been proposed to avoid complete loss of packets involved in collisions, but the capacity of ALOHA systems had not been previously determined. We consider capacity in terms of reliably received rate rather than transmitted rate. We consider capacity achieving strategies under AWGN for transmission of a single packet which is long enough to achieve capacity over the duration of the packet. We combine concepts from multi-access channels and broadcast channels to determine the capacity region for a single transmission of a packet in an ALOHA system. The coding for each user takes into account the possibility of collisions with other users in order to establish a capacity region. Next, we consider the case where we transmit several packets under a channel model where users receive the right to transmit the package according to independent Bernoulli processes. We can then apply the single-packet coding strategies in order to maximize the expected reliable received rate