This work introduces a framework for analyzing the Age of Incorrect Information (AoII) in a real-time monitoring system with a generic discrete-time Markov source. We study a noisy communication system employing a hybrid automatic repeat request (HARQ) protocol, subject to a transmission rate constraint. The optimization problem is formulated as a constrained Markov decision process (CMDP), and it is shown that there exists an optimal policy that is a randomized mixture of two stationary policies. To overcome the intractability of computing the optimal stationary policies, we develop a multiple-threshold policy class where thresholds depend on the source, the receiver, and the packet count. By establishing a Markov renewal structure induced by threshold policies, we derive closed-form expressions for the long-term average AoII and transmission rate. The proposed policy is constructed via a relative value iteration algorithm that leverages the threshold structure to skip computations, combined with a bisection search to satisfy the rate constraint. To accommodate scenarios requiring lower computational complexity, we adapt the same technique to produce a simpler single-threshold policy that trades optimality for efficiency. Numerical experiments exhibit that both thresholdbased policies outperform periodic scheduling, with the multiplethreshold approach matching the performance of the globally optimal policy.

翻译：本文提出了一个框架，用于分析具有通用离散时间马尔可夫源的实时监控系统中的不正确信息年龄（AoII）。我们研究了一个采用混合自动重传请求（HARQ）协议的噪声通信系统，该协议受传输速率约束。优化问题被建模为约束马尔可夫决策过程（CMDP），并证明存在一个最优策略，该策略是两个平稳策略的随机混合。为克服计算最优平稳策略的棘突性，我们开发了一个多阈值策略类，其中阈值取决于源、接收器和数据包计数。通过建立由阈值策略诱导的马尔可夫更新结构，我们推导出长期平均AoII和传输速率的闭式表达式。所提出的策略通过相对值迭代算法构建，该算法利用阈值结构跳过计算，并结合二分搜索以满足速率约束。为适应需要较低计算复杂度的场景，我们将相同技术应用于生成更简单的单阈值策略，该策略以最优性换取效率。数值实验表明，两种基于阈值的策略均优于周期性调度，其中多阈值方法的性能与全局最优策略相匹配。