  A real-time status updating system is considered, in which an energy
harvesting sensor is acquiring measurements regarding some physical phenomenon
and sending them to a destination through an erasure channel. The setting is
online, in which energy arrives in units according to a Poisson process with
unit rate, with arrival times being revealed causally over time. Energy is
saved in a unit-sized battery. The sensor is notified by the destination of
whether updates were erased via feedback. Updates need to reach the destination
successfully in a timely fashion, namely, such that the long term average age
of information, defined as the time elapsed since the latest successful update
has reached the destination, is minimized. First, it is shown that the optimal
status update policy has a renewal structure: successful update times should
constitute a renewal process. Then, threshold-greedy policies are investigated:
a new update is transmitted, following a successful one, only if the age of
information grows above a certain threshold; and if it is erased, then all
subsequent update attempts are greedily scheduled whenever energy is available.
The optimal threshold-greedy policy is then analytically derived.
