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

TL;DR

The paper studies optimal remote estimation of a Wiener process with sampling and transmission costs over a channel with IID delay, formulating the objective as the long-term average MSE subject to cost constraints: $\\text{mse}_{\\text{opt}} = \\min_{\\pi \\in \\Pi} \\frac{ \\sum_{n=1}^{\\infty} \\\mathbb{E}[\\int_{D_n}^{D_{n+1}} (W_t - W_0)^2 dt + c_s k_n + c_\\tau] }{ \\sum_{n=1}^{\\infty} \\\mathbb{E}[D_{n+1}-D_n]}$. The authors apply Lagrange relaxation with a multiplier $\\lambda$ and use backward induction over regenerative epochs to compute optimal waiting times $Z_{n,j}$ and stopping indices $k_n$; the per-epoch optimization reduces to a DP with $E_m$ error state and transition $E_{m+1} = \\mathcal{N}(\\sqrt{E_m}, Z)^2$. They prove convergence to a unique fixed-point $g_\\infty$ and show that the optimal $\\lambda^*$ satisfies $J(\\lambda^*)=0$, ensuring the policy minimizes long-run cost. Numerical results demonstrate meaningful MSE reductions compared with periodic sampling, with robustness to delay variability and practical guidance for remote-sensing systems. $

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.

Figures (6)

• Figure 1: System Model
• Figure 2: Process Timeline
• Figure 3: MSE vs. the delay variance $\sigma^2$ for $c_s = 2$, $c_t = 5$, and $\lambda = 10$
• Figure 4: Convergence of cost over iterations for $\sigma^2 = 0.1$ and $c_s = 2$, $c_t = 5$, $\lambda = 10$
• Figure 5: Optimal waiting times vs. error for $\sigma^2 = 0$ and $c_s = 1$ as constant

Theorems & Definitions (6)

• Lemma 1
• proof
• Lemma 2
• proof
• Corollary 1
• Theorem 1: Convergence of the Optimal Policy Algorithm