  We study push-based sampling and transmission policies for a status update
system consisting of a general finite-state continuous-time Markov chain (CTMC)
information source with known dynamics, with the goal of minimizing the average
age of incorrect information (AoII). The problem setting we investigate
involves an exponentially distributed delay channel for transmissions and a
constraint on the average sampling rate. We first show that the optimum
sampling and transmission policy is a 'multi-threshold policy', where the
thresholds depend on both the estimation value and the state of the original
process, and sampling and transmission need to be initiated when the
instantaneous AoII exceeds the corresponding threshold, called the estimation-
and state-aware transmission (ESAT) policy. Subsequently, we formulate the
problem of finding the thresholds as a constrained semi-Markov decision process
(CSMDP) and the Lagrangian approach. Additionally, we propose two lower
complexity sub-optimum policies, namely the estimation-aware transmission (EAT)
policy, and the single-threshold (ST) policy, for which it is possible to
obtain these thresholds for CTMCs with relatively larger number of states. The
underlying CSMDP formulation relies on the 'multi-regime phase-type' (MRPH)
distribution which is a generalization of the well-known phase-type
distribution, which allows us to obtain the distribution of time until
absorption in a CTMC whose transition rates change with respect to time in a
piece-wise manner. The effectiveness of the proposed ESAT, EAT and ST sampling
and transmission policies are shown through numerical examples, along with
comparisons with a baseline scheme that transmits packets according to a
Poisson process in out-of-sync periods.
