  In this paper, we consider a problem of sampling a Wiener process, with
samples forwarded to a remote estimator via a channel that consists of a queue
with random delay. The estimator reconstructs a real-time estimate of the
signal from causally received samples. Motivated by recent research on
age-of-information, we study the optimal sampling strategy that minimizes the
mean square estimation error subject to a sampling frequency constraint. We
prove that the optimal sampling strategy is a threshold policy, and find the
optimal threshold. This threshold is determined by the sampling frequency
constraint and how much the Wiener process varies during the channel delay. An
interesting consequence is that even in the absence of the sampling frequency
constraint, the optimal strategy is not zero-wait sampling in which a new
sample is taken once the previous sample is delivered; rather, it is optimal to
wait for a non-zero amount of time after the previous sample is delivered, and
then take the next sample. Further, if the sampling times are independent of
the observed Wiener process, the optimal sampling problem reduces to an
age-of-information optimization problem that has been recently solved. Our
comparisons show that the estimation error of the optimal sampling policy is
much smaller than those of age-optimal sampling, zero-wait sampling, and
classic uniform sampling.
