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-armed Bandit (RMAB) problem with a continuous state space. We prove that the bandits 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 of Gauss-Markov processes. Our results unite two theoretical frameworks that were used for remote estimation and AoI minimization: threshold-based sampling and Whittle index-based scheduling. In the single-source, single-channel scenario, we demonstrate that the optimal solution to the sampling and scheduling problem can be equivalently expressed as both a threshold-based sampling strategy and a Whittle index-based scheduling policy. Notably, the Whittle index is equal to zero if and only if two conditions are satisfied: (i) the channel is idle, and (ii) the estimation error is precisely equal to the threshold in the threshold-based sampling strategy. Moreover, the methodology employed to derive threshold-based sampling strategies in the single-source, single-channel scenario plays a crucial role in establishing indexability and evaluating the Whittle index in the more intricate multi-source, multi-channel scenario. Our numerical results show that the proposed policy achieves high-performance gain over the existing policies when some of the Gauss-Markov processes are highly unstable.

翻译：我们研究了一种面向多源远程估计的采样与传输调度问题：调度器需决定何时对多个连续时间高斯-马尔可夫过程进行采样，并通过多个信道将样本传输至远程估计器。样本传输时间在样本间及信道间独立同分布。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。该问题是一个具有连续状态空间的连续时间歇性多臂赌博机（RMAB）问题。我们证明了该赌博机具有可索引性，并推导出Whittle索引的精确表达式。据我们所知，这是首个面向多源信号感知型高斯-马尔可夫过程远程估计的Whittle索引策略。我们的研究成果统一了此前用于远程估计与AoI最小化的两种理论框架：基于阈值的采样策略与基于Whittle索引的调度策略。在单源单信道场景中，我们证明采样与调度问题的最优解可等价表示为基于阈值的采样策略与基于Whittle索引的调度策略两种形式。值得注意的是，Whittle索引为零当且仅当满足两个条件：(i) 信道空闲，(ii) 估计误差恰好等于阈值采样策略中的阈值。此外，在单源单信道场景中推导基于阈值采样策略的方法论，为更复杂的多源多信道场景中建立可索引性并计算Whittle索引起到了关键作用。数值结果表明，当部分高斯-马尔可夫过程高度不稳定时，所提策略相较于现有策略具有显著性能优势。