Age-optimal channel coding blocklength for an M/G/1 queue with HARQ

We consider a communication system in which a source transmits information updates to a destination node through a binary erasure channel (BEC). When a packet containing an information update, which consists of a fixed number of information bits, arrives at the transmitter, it gets queued in a buffer, to be encoded and sent over the channel. Before transmitting a packet, the transmitter selects a channel coding blocklength n and then uses an automatic repeat request (ARQ) protocol, whereby packets that are decoded incorrectly are repeated. The choice of the coding blocklength thus affects the end-to-end status age. However, this dependency is nontrivial since, on the one hand, the duration of a single transmission attempt is directly proportional to n, so the smaller n the better. On the other hand, a smaller value of the blocklength n yields a higher probability of decoding error, which increases the end-toend status age. Employing recent finite-blocklength information-theoretic bounds and approximations on the rate achievable on a BEC for a given blocklength and a given error probability, we study the age-optimal design of this system. We find that for any nontrivial BEC, there exists an optimal blocklength that minimizes the average age and average peak age of information.