  Age of incorrect information (AoII) has recently been proposed as an
alternative to existing information freshness metrics for real-time sampling
and estimation problems involving information sources that are tracked by
remote monitors. Different from existing metrics, AoII penalizes the incorrect
information by increasing linearly with time as long as the source and the
monitor are de-synchronized, and is reset when they are synchronized back.
While AoII has generally been investigated for discrete time information
sources, we develop a novel analytical model in this paper for push- and
pull-based sampling and transmission of a continuous time Markov chain (CTMC)
process. In the pull-based model, the sensor starts transmitting information on
the observed CTMC only when a pull request from the monitor is received. On the
other hand, in the push-based scenario, the sensor, being aware of the AoII
process, samples and transmits when the AoII process exceeds a random
threshold. The proposed analytical model for both scenarios is based on the
construction of a discrete time MC (DTMC) making state transitions at the
embedded epochs of synchronization points, using the theory of absorbing CTMCs,
and in particular phase-type distributions. For a given sampling policy,
analytical models to obtain the mean AoII and the average sampling rate are
developed. Numerical results are presented to validate the analytical model as
well as to provide insight on optimal sampling policies under sampling rate
constraints.
