Preempting to Minimize Age of Incorrect Information under Random Delay

We consider the problem of optimizing the decision of a preemptive transmitter to minimize the Age of Incorrect Information (AoII) when the channel has a random delay. In the system, a transmitter observes a dynamic source and makes decisions based on the system status. When the channel is busy, the transmitter can choose whether to preempt to transmit a new update. When the channel is idle, the transmitter can choose whether or not to transmit a new update. We assume that the channel has a random delay and that this delay is independent and identically distributed for any update. At the other end of the channel is a receiver that estimates the state of the dynamic source based on the updates it receives. We adopt AoII to measure the performance of the system. Therefore, this paper aims to optimize the transmitter's action in each time slot to minimize AoII. We first use the Markov decision process to formulate the optimization problem and give the corresponding value iterative algorithm to obtain the optimal policy. However, the value iteration algorithm is computationally demanding, and some approximations are made to realize the algorithm. Hence, we theoretically analyze some canonical delay distributions and obtain the corresponding optimal policies by leveraging the policy improvement theorem.