In this paper, we study a remote monitoring system where a receiver observes a remote binary Markov source and decides whether to sample and fetch the source's state over a randomly delayed channel. Due to transmission delay, the observation of the source is imperfect, resulting in the uncertainty of the source's state at the receiver. We thus use uncertainty of information as the metric to characterize the performance of the system. Measured by Shannon's entropy, uncertainty of information reflects how much we do not know about the latest source's state in the absence of new information. The current research for uncertainty of information idealizes the transmission delay as one time slot, but not under random delay. Moreover, uncertainty of information varies with the latest observation of the source's state, making it different from other age of information related functions. Motivated by the above reasons, we formulate a uncertainty of information minimization problem under random delay. Typically, such a problem which takes actions based on the imperfect observations can be modeled as a partially observed Markov decision process. By introducing belief state, we transform this process into a semi-Markov decision process. To solve this problem, we first provide an optimal sampling policy employing a two layered bisection relative value iteration algorithm. Furthermore, we propose a sub-optimal index policy with low complexity based on the special properties of belief state. Numerical simulations illustrate that both of the proposed sampling policies outperforms two other benchmarks. Moreover, the performance of the sub-optimal policy approaches to that of the optimal policy, particularly under large delay.

翻译：本文研究了一种远程监测系统，其中接收端观测一个远程二元马尔可夫源，并决定是否通过随机延迟信道采样并获取源状态。由于传输延迟，接收端对源状态的观测存在不完整性，导致接收端对源状态存在不确定性。因此，我们采用信息不确定性作为度量系统性能的指标。信息不确定性以香农熵衡量，反映了在缺乏新信息时，我们对最新源状态的不了解程度。当前关于信息不确定性的研究将传输延迟理想化为一个时隙，而非随机延迟。此外，信息不确定性随最新源状态观测值的变化而变化，这使其区别于其他与信息年龄相关的函数。基于上述原因，我们构建了随机延迟下的信息不确定性最小化问题。通常，这种基于不完整观测采取行动的问题可建模为部分可观测马尔可夫决策过程。通过引入信念状态，我们将该过程转化为半马尔可夫决策过程。为解决该问题，我们首先提出了一种采用两层对分相对值迭代算法的最优采样策略。此外，基于信念状态的特殊性质，我们提出了一种低复杂度的次优指标策略。数值仿真表明，两种提出的采样策略均优于两个基准方法。同时，次优策略的性能接近最优策略，尤其是在大延迟条件下。