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 resource constraints on the average AoII.

翻译：错误信息年龄( AoII )是近期提出的一种用于实时远程监控系统的度量标准。具体而言，AoII衡量监控端信息处于错误状态的时间，并依据错误程度进行加权，从而融合了新鲜度与失真两个概念。本文针对离散时间通信方案中具有资源约束和混合自动重传请求( HARQ )协议的AoII最优传输策略进行了定义。考虑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 )。源模型通过状态空间基数和保持同一状态的概率来刻画。有趣的是，研究表明在某些条件下最优传输策略即为永不传输。这表明存在一个源动态区域，其中通信对降低AoII无效。在其他情况下，存在一种最优传输策略，它是两种基于离散阈值的策略的随机混合，并在一个状态下进行随机化。最优阈值和随机化分量通过解析方法推导得出。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。