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.

翻译：本文研究了多源远程估计中的采样与传输调度问题，其中调度器需确定何时从多个连续时间高斯-马尔可夫过程采集样本，并通过多信道将样本传输至远程估计器。不同样本与信道的传输时间独立同分布。调度器的目标是最小化这些高斯-马尔可夫源的时间平均期望估计误差的加权和。该问题属于具有连续状态空间的连续时间多臂赌博机问题。我们证明各臂具有可索引性，并推导出Whittle指数的精确表达式。据我们所知，这是首个面向多源信号感知远程估计的Whittle指数策略。该结果包含两个具有研究价值的退化情形：（i）单源情形下，Wiener过程与Ornstein-Uhlenbeck过程的远程估计中，Whittle指数策略复现了早期基于阈值的采样策略。当高斯-马尔可夫过程的瞬时估计误差跨越最优阈值时，Whittle指数精确等于0。此外，针对非稳定Ornstein-Uhlenbeck过程的远程估计，本文提出了一种新的最优采样策略。（ii）信号不可知情形下，我们推导出基于信息年龄的远程估计中Whittle指数的精确表达式，通过允许随机传输时间补充了早期研究成果。数值结果表明，所提策略优于基于信号不可知信息年龄的Whittle指数策略以及最大年龄优先-零等待策略。当部分高斯-马尔可夫过程高度非稳定或样本传输时间服从重尾分布时，所提策略的性能增益尤为显著。