On the single-slot capacity of random access over a Gaussian MAC

Consider a Gaussian multiple access channel (MAC) with two users, which do not always have a message to transmit. Neither user knows if the other is transmitting or not. Each user has two possible achievable rates depending on whether the other user is transmitting or not. Hence, in general, any coding scheme is characterized by a 4-tuple of rates. For given power constraints, we prove inner and outer bounds to the (4-dimensional) capacity region.