Optimal Sampling for Uncertainty-of-Information Minimization in a Remote Monitoring System

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.