Broadcast erasure channel with feedback: The two multicast case - Algorithms and bounds

We consider the single hop broadcast packet erasure channel (BPEC) with two multicast sessions (each of them destined to a different group of N users) and regularly available instantaneous receiver ACK/NACK feedback. Using the insight gained from recent work on BPEC with unicast and degraded messages [1], [2], we propose a virtual queue based session-mixing algorithm, which does not require knowledge of channel statistics and achieves capacity for N = 2 and iid erasures. Since the extension of this algorithm to N > 2 is not straightforward, we describe a simple algorithm which outperforms standard timesharing for arbitrary N and is actually asymptotically better than timesharing, for any finite N, as the erasure probability goes to zero. We finally provide, through an information-theoretic analysis, sufficient but not necessary asymptotic conditions between N and n (the number of transmissions) for which the achieved sum rate, under any coding scheme, is essentially identical to that of timesharing.