Age-optimal IR-HARQ Design in the Presence of Non-trivial Propagation Delay

We investigate on the age of information (AoI) of a communication system in which status updates arriving at the source are transmitted to the receiver through a binary erasure channel (BEC). An incremental redundancy hybrid automatic repeat request (IR-HARQ) scheme with a given maximum number of retransmissions is adopted to combat erasures. We put special interests on the scenarios where the propagation delays are non-trivial, e.g., the space communications and satellite communications. Intuitively, there exists an upper bound on the propagation delay, only within which using retransmissions is beneficial to AoI. However, characterization of such an upper bound has received little attention. In this paper, by formulating and solving the age-optimal redundancy allocation problem for the adopted IR-HARQ scheme, explicit expressions of the optimal redundancy allocation results for each transmission round are derived, based on which the upper bound of the propagation delay for beneficial retransmissions is obtained. Numerical results are also provided. The results shed important light on the IR-HARQ design for AoI-critical communications in the presence of non-trivial propagation delay.