For a remote estimation system, we study age of incorrect information (AoII), which is a recently proposed semantic-aware freshness metric. In particular, we assume an information source observing a discrete-time finite-state Markov chain (DTMC) and employing push-based transmissions of status update packets towards the monitor which is tasked with remote estimation of the source. The source-to-monitor channel delay is assumed to have a general discrete-time phase-type (DPH) distribution, whereas the zero-delay reverse channel ensures that the source has perfect information on AoII and the remote estimate. A multi-threshold transmission policy is employed where packet transmissions are initiated when the AoII process exceeds a threshold which may be different for each estimation value. In this general setting, our goal is to minimize the weighted sum of time average of an arbitrary function of AoII and estimation, and transmission costs, by suitable choice of the thresholds. We formulate the problem as a semi-Markov decision process (SMDP) with the same state-space as the original DTMC to obtain the optimum multi-threshold policy whereas the parameters of the SMDP are obtained by using a novel stochastic tool called dual-regime absorbing Markov chain (DR-AMC), and its corresponding absorption time distribution named as dual-regime DPH (DR-DPH).

翻译：针对远程估计系统，我们研究了错误信息年龄（AoII）这一近期提出的语义感知新鲜度度量指标。具体而言，我们假设一个信息源观测离散时间有限状态马尔可夫链（DTMC），并采用推送方式向监测器传输状态更新数据包，监测器负责对信源进行远程估计。假设信源至监测器的信道延迟服从一般离散时间相位型（DPH）分布，而零延迟反向信道确保信源能准确获取AoII及远程估计值。系统采用多阈值传输策略，即当AoII过程超过特定阈值时触发数据包传输，该阈值可随估计值不同而变化。在此通用设定下，我们的目标是通过合理选择阈值，最小化AoII与估计值的任意函数的时间平均值加权和及传输成本。我们将该问题建模为与原始DTMC状态空间相同的半马尔可夫决策过程（SMDP），以获取最优多阈值策略；同时利用一种称为双机制吸收马尔可夫链（DR-AMC）的新型随机工具及其对应的吸收时间分布——双机制DPH（DR-DPH）——推导SMDP参数。