Age-Minimal Transmission for Energy Harvesting Sensors with Finite\n Batteries: Online Policies

An energy-harvesting sensor node that is sending status updates to a\ndestination is considered. The sensor is equipped with a battery of finite size\nto save its incoming energy, and consumes one unit of energy per status update\ntransmission, which is delivered to the destination instantly over an\nerror-free channel. The setting is online in which the harvested energy is\nrevealed to the sensor causally over time, and the goal is to design status\nupdate transmission policy such that the long term average age of information\n(AoI) is minimized. AoI is defined as the time elapsed since the latest update\nhas reached at the destination. Two energy arrival models are considered: a\nrandom battery recharge (RBR) model, and an incremental battery recharge (IBR)\nmodel. In both models, energy arrives according to a Poisson process with unit\nrate, with values that completely fill up the battery in the RBR model, and\nwith values that fill up the battery incrementally, unit-by-unit, in the IBR\nmodel. The key approach to characterizing the optimal status update policy for\nboth models is showing the optimality of renewal policies, in which the\ninter-update times follow a specific renewal process that depends on the energy\narrival model and the battery size. It is then shown that the optimal renewal\npolicy has an energy-dependent threshold structure, in which the sensor sends a\nstatus update only if the AoI grows above a certain threshold that depends on\nthe energy available. For both the RBR and the IBR models, the optimal\nenergy-dependent thresholds are characterized explicitly, i.e., in closed-form,\nin terms of the optimal long term average AoI. It is also shown that the\noptimal thresholds are monotonically decreasing in the energy available in the\nbattery, and that the smallest threshold, which comes in effect when the\nbattery is full, is equal to the optimal long term average AoI.\n