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The multiple access channel (MAC) models K independent senders transmitting to a single receiver over a shared medium. The capacity region — the set of simultaneously achievable rate tuples (R_1, ..., R_K) — is characterized by sum_{i in S} R_i <= I(X_S; Y | X_{Sc}) for all subsets S of senders, where X_S denotes the inputs from senders in S. For two users: R_1 <= I(X_1;Y|X_2), R_2 <= I(X_2;Y|X_1), R_1+R_2 <= I(X_1,X_2;Y). The capacity region is achieved by joint typicality decoding or successive interference cancellation (SIC), where the receiver decodes one user at a time, subtracting each decoded signal. The MAC was the first multi-user channel whose capacity region was fully characterized.
The multiple access channel is the canonical uplink model: think of multiple cell phones transmitting to a single base station, or multiple IoT devices sending data to a gateway. Each sender has an independent message and a power constraint. The shared medium means the receiver sees a superposition of all transmitted signals plus noise. The fundamental question is: what rates can all users simultaneously achieve?
For two Gaussian users, Y = X_1 + X_2 + Z. Decoding user 1 first (treating X_2 as additional noise of power P_2): R_1 <= (1/2) log(1 + P_1/(P_2 + N)). After subtracting the decoded X_1, decode user 2 with no interference: R_2 <= (1/2) log(1 + P_2/N). Reversing the order swaps the rates. Time-sharing between the two orders traces the dominant face of the capacity region. The sum-rate boundary R_1 + R_2 <= (1/2) log(1 + (P_1+P_2)/N) is achieved by both orderings.
The key insight is that treating all other users as noise and decoding sequentially (SIC) is optimal — you do not need more sophisticated joint decoding. This is because the MAC is "informationally friendly": each user's signal adds information, not just interference. The receiver decodes one user, removes their contribution from the received signal (since it now knows what they sent), and faces a cleaner signal for the remaining users. This is unlike the interference channel, where each receiver wants only its own message and the other signals are pure interference.
The MAC has direct engineering implications. In 4G/5G uplink, non-orthogonal multiple access (NOMA) schemes use SIC-like receivers to allow users to transmit simultaneously, approaching the MAC capacity region. Traditional orthogonal schemes (TDMA, FDMA, OFDMA) divide the channel among users, which is simple but suboptimal — the capacity region of orthogonal access is strictly inside the MAC capacity region. The information-theoretic result motivates the engineering move toward non-orthogonal schemes that let users "collide" and rely on receiver intelligence to separate them.