In this paper, we address the optimal sampling of a Wiener process under sampling and transmission costs, with the samples being forwarded to a remote estimator over a channel with random IID delay. The goal of the estimator is to reconstruct an estimate of the real-time signal value from causally received samples. Our study focuses on the optimal online strategy for both sampling and transmission, aiming to minimize the mean square estimation error. We establish that the optimal strategy involves threshold policies for both sampling and transmission, and we derive the optimal thresholds. We utilize Lagrange relaxation and backward induction as our methodology, revealing the problem of minimizing estimation error, under the assumption that sampling and transmission times are independent of the observed Wiener process. Our comparative analysis demonstrates that the estimation error achieved by the optimal joint sampling and transmission policy is significantly lower than that of age-optimal sampling, zero-wait sampling, periodic sampling, and policies that optimize only the sampling times.

翻译：本文研究在采样与传输成本约束下，对Wiener过程进行最优采样的问题，其中采样数据需通过具有独立同分布随机延迟的信道传输至远程估计器。估计器的目标是根据因果接收的采样数据重构实时信号值的估计值。本研究聚焦于采样与传输的联合在线最优策略，旨在最小化均方估计误差。我们证明最优策略采用采样与传输的双阈值机制，并推导出最优阈值表达式。通过拉格朗日松弛法与逆向归纳法，在采样和传输时间与观测Wiener过程独立的假设下，揭示了最小化估计误差问题的本质。对比分析表明，联合最优采样传输策略所实现的估计误差显著低于时效最优采样、零等待采样、周期采样以及仅优化采样时刻的策略。