We consider a sensor that samples an N-state continuous-time Markov chain
(CTMC)-based information source process, and transmits the observed state of
the source, to a remote monitor tasked with timely tracking of the source
process. The mismatch between the source and monitor processes is quantified by
age of incorrect information (AoII), which penalizes the mismatch as it stays
longer, and our objective is to minimize the average AoII under an average
sampling rate constraint. We assume a perfect reverse channel and hence the
sensor has information of the estimate while initiating a transmission or
preempting an ongoing transmission. First, by modeling the problem as an
average cost constrained semi-Markov decision process (CSMDP), we show that the
structure of the problem gives rise to an optimum threshold policy for which
the sensor initiates a transmission once the AoII exceeds a threshold depending
on the instantaneous values of both the source and monitor processes. However,
due to the high complexity of obtaining the optimum policy in this general
setting, we consider a relaxed problem where the thresholds are allowed to be
dependent only on the estimate. We show that this relaxed problem can be solved
with a novel CSMDP formulation based on the theory of absorbing MCs, with a
computational complexity of O(N4), allowing one to obtain optimum
policies for general CTMCs with over a hundred states