Age of Actuation and Timeliness: Semantics in a Wireless Power Transfer System

In this paper, we investigate a model relevant to semantics-aware goal-oriented communications, and we propose a new metric that incorporates the utilization of information in addition to its timelines. Specifically, we consider the transmission of observations from an external process to a battery-powered receiver through status updates. These updates inform the receiver about the process status and enable actuation if sufficient energy is available to achieve a goal. We focus on a wireless power transfer (WPT) model, where the receiver receives energy from a dedicated power transmitter and occasionally from the data transmitter when they share a common channel. We analyze the Age of Information (AoI) and propose a new metric, the $\mathit{\text{Age}}\mathit{\text{of}}\mathit{\text{Actuation}}\mathit{\text{(AoA),}}\mathit{\text{which}}\mathit{\text{is}}\mathit{\text{relevant}}\mathit{\text{when}}\mathit{\text{the}}\mathit{\text{receiver}}\mathit{\text{utilizes}}\mathit{\text{the}}\mathit{\text{status}}\mathit{\text{updates}}\mathit{\text{to}}\mathit{\text{perform}}\mathit{\text{actions}}\mathit{\text{in}}\mathit{\text{a}}\mathit{\text{timely}}\mathit{\text{manner}}$. We provide analytical characterizations of the average AoA and the violation probability of the AoA, demonstrating that AoA generalizes AoI. Moreover, we introduce and analytically characterize the $\mathit{\text{Probability}}\mathit{\text{of}}\mathit{\text{Missing}}\mathit{\text{Actuation}}\mathit{\text{(PoMA)}}$; this metric becomes relevant also $\mathit{\text{to}}\mathit{\text{quantify}}\mathit{\text{the}}\mathit{\text{incurred}}\mathit{\text{cost}}\mathit{\text{of}}\mathit{\text{a}}\mathit{\text{missed}}\mathit{\text{action}}$. We formulate unconstrained and constrained optimization problems for all the metrics and present numerical evaluations of our analytical results. This proposed set of metrics goes beyond the traditional timeliness metrics since the synergy of different flows is now considered.