  We study the problem of optimizing the decisions of a preemptively capable
transmitter to minimize the Age of Incorrect Information (AoII) when the
communication channel has a random delay. We consider a slotted-time system
where a transmitter observes a Markovian source and makes decisions based on
the system status. In each time slot, the transmitter decides whether to
preempt or skip when the channel is busy. When the channel is idle, the
transmitter decides whether to send a new update. A remote receiver estimates
the state of the Markovian source based on the update it receives. We consider
a generic transmission delay and assume that the transmission delay is
independent and identically distributed for each update. This paper aims to
optimize the transmitter's decision in each time slot to minimize the AoII with
generic time penalty functions. To this end, we first use the Markov decision
process to formulate the optimization problem and derive the analytical
expressions of the expected AoIIs achieved by two canonical preemptive
policies. Then, we prove the existence of the optimal policy and provide a
feasible value iteration algorithm to approximate the optimal policy. However,
the value iteration algorithm will be computationally expensive if we want
considerable confidence in the approximation. Therefore, we analyze the system
characteristics under two canonical delay distributions and theoretically
obtain the corresponding optimal policies using the policy improvement theorem.
Finally, numerical results are presented to illustrate the performance
improvements brought about by the preemption capability.
