  We consider a slotted-time system with a transmitter-receiver pair. In the
system, a transmitter observes a dynamic source and sends updates to a remote
receiver through a communication channel. We assume that the channel is
error-free but suffers a random delay. Moreover, when an update has been
transmitted for too long, the transmission will be terminated immediately, and
the update will be discarded. We assume the maximum transmission time is
predetermined and is not controlled by the transmitter. The receiver will
maintain estimates of the current state of the dynamic source using the
received updates. In this paper, we adopt the Age of Incorrect Information
(AoII) as the performance metric and investigate the problem of optimizing the
transmitter's action in each time slot to minimize AoII. We first characterize
the optimization problem using Markov Decision Process and evaluate the
performance of some canonical transmission policies. Then, by leveraging the
policy improvement theorem, we prove that, under a simple and easy-to-verify
condition, the optimal policy for the transmitter is the one that initiates a
transmission whenever the channel is idle and AoII is not zero. Lastly, we take
the case where the transmission time is geometrically distributed as an
example. For this example, we verify the condition numerically and provide
numerical results that highlight the performance of the optimal policy.
