General capacity region for the fully-connected 3-node packet erasure network

This work considers the fully-connected 3-node packet erasure network: For each time slot, with some probabilities a packet sent by any node i may be received by both of the other nodes j and k; received only by node j (or node k); or received by neither nodes. Interference is avoided by enforcing that at most one node can transmit in each time slot. We assume that node i can always reach node j, possibly with the help of the third node k, for any i ≠ j pairs (thus the term fully-connected). One example of this model is any Wi-Fi network with 3 nodes within the hearing range of each other. We consider the most general traffic demands. Namely, there are six private-information flows with rates (R_1→2,R_1→3,R_2→1, R_2→3,R_3→1,R_3→2), respectively, and three common-information flows with rates (R_1→23,R_2→31,R_3→12), respectively. We characterize the 9-dimensional Shannon capacity region within a gap that is inversely proportional to the packet size (bits). The gap can be attributed to exchanging reception status (ACK) and can be further reduced to 0 if we allow ACK to be transmitted via a separate control channel. For normal-sized packets, say 12000 bits, our results have thus effectively characterized the capacity region. Technical contributions of this work include a new converse for many-to-many network communications and a new capacity-approaching simple linear network coding scheme.