  We consider a transmitter-receiver pair in a slotted-time system. The
transmitter observes a dynamic source and sends updates to a remote receiver
through an error-free communication channel that suffers a random delay. We
consider two cases. In the first case, the update is guaranteed to be delivered
within a certain number of time slots. In the second case, the update is
immediately discarded once the transmission time exceeds a predetermined value.
The receiver estimates the 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 the Markov decision process and investigate the
performance of the threshold policy, under which the transmitter transmits
updates only when the transmission is allowed and the AoII exceeds the
threshold $\tau$. By delving into the characteristics of the system evolution,
we precisely compute the expected AoII achieved by the threshold policy using
the Markov chain. Then, we prove that the optimal policy exists and provide a
computable relative value iteration algorithm to estimate the optimal policy.
Furthermore, by leveraging the policy improvement theorem, we theoretically
prove that, under an easily verifiable condition, the optimal policy is the
threshold policy with $\tau=1$. Finally, numerical results are presented to
highlight the performance of the optimal policy.
