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

In this paper, we consider a problem of samplinga Wiener process, with samples forwarded to a remote esti-mator via a channel with queueing and random delay. Theestimator reconstructs a real-time estimate of the signal fromcausally received samples. Motivated by recent research onAge-of-Information, we study the optimal sampling strategyfor minimizing the mean square estimation error subject to asampling-rate constraint. We prove that the optimal samplingstrategy is a threshold policy, and find the optimal threshold.This threshold is determined by the sampling-rate constraintand how much the Wiener process varies during the channeldelay. An unexpected consequence is that even in the absence ofthe sampling-rate constraint, the optimal strategy is not zero-waitsampling in which a new sample is taken once the previous sampleis delivered; rather, there is a non-zero waiting time before takingthe next sample. Further, if the sampling times are independentof the observed Wiener process, the optimal sampling problemreduces to an age-of-information optimization problem that hasbeen recently solved. Our comparisons show that the estimationerror of the optimal sampling policy is much smaller than thoseof age-optimal sampling, zero-wait sampling, and classic uniformsampling.