Optimal Sampling under Cost for Remote Estimation of the Wiener Process over a Channel with Delay

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.