Optimizing Sampling for Data Freshness: Unreliable Transmissions with Random Two-way Delay

In this paper, we study a sampling problem in which fresh samples of a signal (source) are sent through an unreliable channel to a remote estimator, and acknowledgments are sent back over a feedback channel. Both the forward and feedback channels are subject to random transmission times. Motivated by distributed sensing, the estimator can estimate the real-time value of the source signal by combining the signal samples received through the channel and noisy signal observations collected from a local sensor. We prove that the estimation error is a non-decreasing function of the Age of Information (AoI) for received signal samples and design an optimal sampling strategy that minimizes the long-term average estimation error. The optimal sampler design follows a threshold strategy: If the last transmission was successful, the source waits until the expected estimation error upon delivery exceeds a threshold and then sends out a new sample. If the last transmission fails, the source immediately sends out a new sample without waiting. The threshold is the unique root of a fixed-point equation and can be solved with low complexity (e.g., by bisection search). In addition, the proposed sampling strategy is also optimal for minimizing the long-term average of general non-decreasing functions of the AoI. Its optimality holds for general transmission time distributions of the forward and feedback channels.