The Age of Incorrect Information (AoII) is a recently proposed metric for real-time remote monitoring systems. In particular, AoII measures the time the information at the monitor is incorrect, weighted by the magnitude of this incorrectness, thereby combining the notions of freshness and distortion. This paper addresses the definition of an AoII-optimal transmission policy in a discrete-time communication scheme with a resource constraint and a hybrid automatic repeat request (HARQ) protocol. Considering an $N$-ary symmetric Markov source, the problem is formulated as an infinite-horizon average-cost constrained Markov decision process (CMDP). The source model is characterized by the cardinality of the state space and the probability of staying at the same state. Interestingly, it is proved that under some conditions, the optimal transmission policy is to never transmit. This reveals that there exists a region of the source dynamics where communication is inadequate in reducing the AoII. Elsewhere, there exists an optimal transmission policy, which is a randomized mixture of two discrete threshold-based policies that randomize at one state. The optimal threshold and the randomization component are derived analytically. Numerical results illustrate the impact of source dynamics, channel conditions, and the resource constraint on the average AoII.

翻译：不正确信息年龄（AoII）是近期提出的一种用于实时远程监控系统的指标。具体而言，AoII衡量监控端信息处于不正确状态的时间，并依据这种不正确性的严重程度进行加权，从而融合了信息新鲜度与失真的概念。本文针对具有资源约束和混合自动重传请求（HARQ）协议的离散时间通信方案，定义了AoII最优传输策略。考虑一个N元对称马尔可夫源，该问题被建模为无限时域平均代价约束马尔可夫决策过程（CMDP）。源模型由状态空间的基数及保持同一状态的概率刻画。有趣的是，我们证明在某些条件下，最优传输策略是永不传输。这表明存在一个源动态区域，在该区域内通信对于降低AoII无效。而在其他区域，存在一个最优传输策略，该策略是两种基于离散阈值的策略的随机混合，并在一个状态上进行随机化。我们解析推导了最优阈值及随机化分量。数值结果展示了源动态、信道条件及资源约束对平均AoII的影响。