  An energy harvesting sensor that is sending status updates to a destination
through an erasure channel is considered, in which transmissions are prone to
being erased with some probability $q$, independently from other transmissions.
The sensor, however, is unaware of erasure events due to lack of feedback from
the destination. Energy expenditure is normalized in the sense that one
transmission consumes one unit of energy. The sensor is equipped with a
unit-sized battery to save its incoming energy, which arrives according to a
Poisson process of unit rate. The setting is online, in which energy arrival
times are only revealed causally after being harvested, and the goal is to
design transmission times such that the long term average age of information
(AoI), defined as the time elapsed since the latest update has reached the
destination successfully, is minimized. The optimal status update policy is
first shown to have a renewal structure, in which the time instants at which
the destination receives an update successfully constitute a renewal process.
Then, for $q\leq\frac{1}{2}$, the optimal renewal policy is shown to have a
threshold structure, in which a new status update is transmitted only if the
AoI grows above a certain threshold, that is shown to be a decreasing function
of $q$. While for $q>\frac{1}{2}$, the optimal renewal policy is shown to be
greedy, in which a new status update is transmitted whenever energy is
available.
