Sampling for Remote Estimation of the Wiener Process over an Unreliable Channel

TL;DR

This paper develops a simple threshold-based sampling policy for remote Wiener process estimation over unreliable channels, optimizing mean square error despite transmission errors and delays.

Contribution

It proves the optimality of a threshold policy in a complex setting with unreliable transmissions and introduces a low-complexity algorithm to find the optimal threshold.

Findings

Optimal threshold policy minimizes estimation error.

The algorithm efficiently computes the threshold.

Numerical results show superiority over existing policies.

Abstract

In this paper, we study a sampling problem where a source takes samples from a Wiener process and transmits them through a wireless channel to a remote estimator. Due to channel fading, interference, and potential collisions, the packet transmissions are unreliable and could take random time durations. Our objective is to devise an optimal causal sampling policy that minimizes the long-term average mean square estimation error. This optimal sampling problem is a recursive optimal stopping problem, which is generally quite difficult to solve. However, we prove that the optimal sampling strategy is, in fact, a simple threshold policy where a new sample is taken whenever the instantaneous estimation error exceeds a threshold. This threshold remains a constant value that does not vary over time. By exploring the structure properties of the recursive optimal stopping problem, a low-complexity iterative algorithm is developed to compute the optimal threshold. This work generalizes previous research by incorporating both transmission errors and random transmission times into remote estimation. Numerical simulations are provided to compare our optimal policy with the zero-wait and age-optimal policies.