Instability of backoff protocols with arbitrary arrival rates

In contention resolution, multiple processors are trying to coordinate to send discrete messages through a shared channel with sharply limited communication. If two processors inadvertently send at the same time, the messages collide and are not transmitted successfully. An important case is acknowledgement-based contention resolution, in which processors cannot listen to the channel at all; all they know is whether or not their own messages have got through. This situation arises frequently in both networking and cloud computing. One particularly important example of an acknowledgement-based contention resolution protocol is binary exponential backoff. Variants of binary exponential backoff are used in both Ethernet and TCP/IP, and both Google Drive and AWS instruct their users to implement it to handle busy periods. In queueing models, where each processor has a queue of messages, stable acknowledgement-based protocols are already known (H{\aa}stad et al., SICOMP 1996). In queue-free models, where each processor has a single message but processors arrive randomly, it is widely conjectured that no stable acknowledgement-based protocols exist for any positive arrival rate of processors. Despite exciting recent results for full-sensing protocols which assume greater listening capabilities of the processors (see e.g. Bender et al. STOC 2020 or Chen et al. PODC 2021), this foundational question remains open even for backoff protocols unless the arrival rate of processors is at least 0.42 (Goldberg et al. SICOMP 2004). We prove the conjecture for all backoff protocols outside of a tightly-constrained special case, and set out the remaining technical obstacles to a full proof.