We study push-based sampling and transmission policies for a status update system consisting of a general finite-state continuous-time Markov chain (CTMC) information source with known dynamics, with the goal of minimizing the average age of incorrect information (AoII). The problem setting we investigate involves an exponentially distributed delay channel for transmissions and a constraint on the average sampling rate. We first show that the optimum sampling and transmission policy is a 'multi-threshold policy', where the thresholds depend on both the estimation value and the state of the original process, and sampling and transmission need to be initiated when the instantaneous AoII exceeds the corresponding threshold, called the estimation- and state-aware transmission (ESAT) policy. Subsequently, we formulate the problem of finding the thresholds as a constrained semi-Markov decision process (CSMDP) and the Lagrangian approach. Additionally, we propose two lower complexity sub-optimum policies, namely the estimation-aware transmission (EAT) policy, and the single-threshold (ST) policy, for which it is possible to obtain these thresholds for CTMCs with relatively larger number of states. The underlying CSMDP formulation relies on the 'multi-regime phase-type' (MRPH) distribution which is a generalization of the well-known phase-type distribution, which allows us to obtain the distribution of time until absorption in a CTMC whose transition rates change with respect to time in a piece-wise manner. The effectiveness of the proposed ESAT, EAT and ST sampling and transmission policies are shown through numerical examples, along with comparisons with a baseline scheme that transmits packets according to a Poisson process in out-of-sync periods.

翻译：我们研究由已知动态的通用有限状态连续时间马尔可夫链（CTMC）信息源构成的态势更新系统中基于推送的采样与传输策略，旨在最小化错误信息平均年龄（AoII）。我们所研究的问题场景涉及具有指数分布延迟的传输信道以及对平均采样率的约束。我们首先证明最优采样与传输策略是一种“多阈值策略”，其中阈值同时取决于估计值和原始过程的状态，当瞬时AoII超过相应阈值时需要启动采样与传输，该策略称为估计与状态感知传输（ESAT）策略。随后，我们将寻找阈值的问题表述为一个约束半马尔可夫决策过程（CSMDP），并采用拉格朗日方法求解。此外，我们提出了两种较低复杂度的次优策略，即估计感知传输（EAT）策略和单阈值（ST）策略，对于状态数相对较多的CTMC，这些策略的阈值是可求得的。基础的CSMDP公式依赖于“多体制位相型”（MRPH）分布，这是对著名位相型分布的推广，使我们能够获得在转移速率随时间分段变化的CTMC中直至吸收的时间分布。通过数值算例展示了所提出的ESAT、EAT和ST采样与传输策略的有效性，并与在失步期间按泊松过程传输数据包的基线方案进行了比较。