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). Interestingly, it is proved that, under some conditions, the optimal transmission policy is to never transmit. This reveals 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 on at most one state. The optimal threshold and the randomization component are derived analytically. Numerical results illustrate the impact of the source dynamics, channel conditions, and resource constraints on the average AoII.

翻译：错误信息年龄（AoII）是近期为实时远程监控系统提出的一项度量指标。该指标通过信息错误程度的幅度加权，量化监控端信息处于错误状态的时间，从而融合了信息新鲜度与失真度的概念。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，研究AoII最优传输策略的定义问题。考虑N元对称马尔可夫信源，该问题被建模为无限时域平均成本约束马尔可夫决策过程（CMDP）。有趣的是，研究证明在特定条件下，最优传输策略为永不传输。这揭示了信源动态特性中存在通信无法降低AoII的区间。在其他情况下，存在一种最优传输策略，该策略是两种基于离散阈值的策略在最多一个状态上进行随机化的混合策略。本文通过解析方法推导了最优阈值与随机化分量。数值结果阐明了信源动态特性、信道条件及资源约束对平均AoII的影响。