  We consider the semantics-aware remote estimation of a discrete-state Markov
source with normal (low-priority) and alarm (high-priority) states. Erroneously
announcing a normal state at the destination when the source is actually in an
alarm state (i.e., missed alarm error) incurs a significantly higher cost than
falsely announcing an alarm state when the source is in a normal state (i.e.,
false alarm error). Moreover, successive reception of an estimation error may
cause significant lasting impact, e.g., maintenance cost and misoperations.
Motivated by this, we assign different costs to different estimation errors and
introduce two new age metrics, namely the Age of Missed Alarm (AoMA) and the
Age of False Alarm (AoFA), to account for the lasting impact incurred by
different estimation errors. Notably, the two age processes evolve dependently
and can distinguish between different types of estimation errors and different
synced states. The aim is to achieve an optimal trade-off between the cost of
estimation error, lasting impact, and communication utilization. The problem is
formulated as an average-cost, countably infinite state-space Markov decision
process (MDP). We show that the optimal policy exhibits a switching-type
structure, making it amenable to policy storage and algorithm design. Notably,
when the source is symmetric and states are equally important, the optimal
policy has identical thresholds, i.e., threshold-type. Theoretical and
numerical results underscore that our approach extends the current
understanding of the Age of Incorrect Information (AoII) and the cost of
actuation error (CAE), showing that they are specific instances within our
broader framework.
