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：本文研究面向多源远程估计的采样与传输调度问题，其中调度器需决定何时从多个连续时间高斯-马尔可夫过程中采集样本，并通过多个信道将样本传输至远程估计器。样本传输时间在各样本和信道间独立同分布。调度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。该问题可建模为具有连续状态空间的连续时间“不宁多臂赌博机”（RMAB）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。该结果具有两种值得关注的退化情形：(i) 在单源情形中，该策略复现了针对Wiener和Ornstein-Uhlenbeck过程远程估计的早期阈值采样策略。当高斯-马尔可夫过程的瞬时估计误差跨越最优阈值时，Whittle指数恰好等于0。此外，本文还获得了针对不稳定Ornstein-Uhlenbeck过程远程估计的新型最优采样策略。(ii) 在信号不可知情形中，我们推导出基于信息年龄（AoI）的远程估计Whittle指数精确表达式，该结果补充了以往仅适用于确定性传输时间的相关结论。数值结果表明，本文所提策略优于信号不可知的AoI型Whittle指数策略以及"最大年龄优先-零等待"（MAF-ZW）策略。当部分高斯-马尔可夫过程高度不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。