Sampling of the Wiener Process for Remote Estimation Over a Channel With Random Delay

Abstract

In this paper, we consider a problem of sampling a Wiener process, with samples forwarded to a remote estimator over a channel that is modeled as a queue. The estimator reconstructs an estimate of the real-time signal value from causally received samples. We study the optimal online sampling strategy that minimizes the mean square estimation error subject to a sampling rate constraint. We prove that the optimal sampling strategy is a threshold policy, and find the optimal threshold. This threshold is determined by how much the Wiener process varies during the random service time and the maximum allowed sampling rate. 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. This reveals an interesting connection between age of information and remote estimation. Our comparisons show that the estimation error of the optimal sampling policy can be much smaller than those of age-optimal sampling, zero-wait sampling, and classic periodic sampling.

Minimizing Age of Information via Scheduling over Heterogeneous Channels

“…These results were extended to the multi-source multi-server regime in [37]. Next, there has been a significant effort in age-optimal sampling [3,24,35,36,38]. The optimal sampling policy was provided for minimizing a monotonic age function in [24,35,38].…”

“…Note that the structure of Lemma 2 is motivated by Lemma 2 in [24] and Lemma 2 in [36]. In [24] and [36], since the channel is error free, the age state at the end of each transmission is independent with history information. Thus, Lemma 2 in [24] and Lemma 2 in [36] are related with a per-sample (single transmission) control.…”

“…In [24] and [36], since the channel is error free, the age state at the end of each transmission is independent with history information. Thus, Lemma 2 in [24] and Lemma 2 in [36] are related with a per-sample (single transmission) control. However, our study does not have such a property and thus Lemma 2 arises from solving the long term average cost of the threshold type policy.…”

“…The studies in [24,35,36] also derive an exact solution to their inner-layer problem. However, their technique is using optimal stopping rules [24,36] or stochastic convex optimization [35], which is different with our study.…”

Optimizing Age Penalty in Time-Varying Networks with Markovian and Error-Prone Channel State

“…Notice that in many general scenarios, like remote estimation [ 20 , 21 ], the proper evaluation of data freshness is a function of AoI instead of AoI itself. Therefore, the metric of Cost of Update Delay (CoUD) in [ 7 ] and age penalty function in [ 9 ] have been proposed to measure data freshness in a general setting.…”