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In this paper, we consider a problem of sampling a Wiener process, withsamples forwarded to a remote estimator via a channel that consists of a queuewith random delay. The estimator reconstructs a real-time estimate of thesignal from causally received samples. Motivated by recent research onage-of-information, we study the optimal sampling strategy that minimizes themean square estimation error subject to a sampling frequency constraint. Weprove that the optimal sampling strategy is a threshold policy, and find theoptimal threshold. This threshold is determined by the sampling frequencyconstraint and how much the Wiener process varies during the channel delay. Aninteresting consequence is that even in the absence of the sampling frequencyconstraint, the optimal strategy is not zero-wait sampling in which a newsample is taken once the previous sample is delivered; rather, it is optimal towait for a non-zero amount of time after the previous sample is delivered, andthen take the next sample. Further, if the sampling times are independent ofthe observed Wiener process, the optimal sampling problem reduces to anage-of-information optimization problem that has been recently solved. Ourcomparisons show that the estimation error of the optimal sampling policy ismuch smaller than those of age-optimal sampling, zero-wait sampling, andclassic uniform sampling.

Remote Estimation of the Wiener Process over a Channel with Random Delay