In this paper, we study a sampling and transmission scheduling problem for
multi-source remote estimation, where a scheduler determines when to take
samples from multiple continuous-time Gauss-Markov processes and send the
samples over multiple channels to remote estimators. The sample transmission
times are i.i.d. across samples and channels. The objective of the scheduler is
to minimize the weighted sum of the time-average expected estimation errors of
these Gauss-Markov sources. This problem is a continuous-time Restless
Multi-arm Bandit (RMAB) problem with a continuous state space. We prove that
the arms are indexable and derive an exact expression of the Whittle index. To
the extent of our knowledge, this is the first Whittle index policy for
multi-source signal-aware remote estimation. This result has two degenerated
cases of interest: (i) In the single-source case, the Whittle index policy
reproduces earlier threshold-based sampling policies for the remote estimation
of Wiener and Ornstein-Uhlenbeck processes. When the instantaneous estimation
error of the Gauss-Markov process crosses the optimal threshold, the Whittle
index is precisely equal to 0. In addition, a new optimal sampling policy for
the remote estimation of the unstable Ornstein-Uhlenbeck process is obtained.
(ii) In the signal-agnostic case, we find an exact expression of the Whittle
index for Age of Information (AoI)-based remote estimation, which complements
earlier results by allowing for random transmission times. Our numerical
results show that the proposed policy performs better than the signal-agnostic
AoI-based Whittle index policy and the Maximum-Age-First, Zero-Wait (MAF-ZW)
policy. The performance gain of the proposed policy is high when some of the
Gauss-Markov processes are highly unstable or when the sample transmission
times follow a heavy-tail distribution.Comment: 21 pages, 4 figures, part of this manuscript has been submitted to
ACM MobiHoc 202