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

Abstract: 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 IID delay. The goal of the estimator is to reconstruct the real-time signal by minimizing a long-term average cost that includes both the mean squared estimation error (MSE) and the costs associated with sampling and transmission from causally received samples. Rather than pursuing the conventional MMSE estimate, our objective is to derive a policy that optimally balances estimation accuracy and resource expenditure, yielding an MSE-optimal solution under explicit cost constraints. We look for optimal online strategies for both sampling and transmission. By employing Lagrange relaxation and iterative backward induction, we derive an optimal policy that balances the trade-offs between estimation accuracy and costs. We validate our approach through comprehensive simulations, evaluating various scenarios including balanced costs, high sampling costs, high transmission costs, and different transmission delay statistics. Our results demonstrate the effectiveness and robustness of the proposed joint sampling and transmission policy in maintaining lower MSE compared to conventional periodic sampling methods. The differences are particularly striking under high delay variability. We also analyze the convergence behavior of the cost function. We believe our formulation and results provide insights into the design and implementation of efficient remote estimation systems in stochastic networks.