We consider a multi-process remote estimation system observing $K$ independent Ornstein-Uhlenbeck processes. In this system, a shared sensor samples the $K$ processes in such a way that the long-term average sum mean square error (MSE) is minimized. The sensor operates under a total sampling frequency constraint $f_{\max}$. The samples from all processes consume random processing delays in a shared queue and then are transmitted over an erasure channel with probability $\epsilon$. We study two variants of the problem: first, when the samples are scheduled according to a Maximum-Age-First (MAF) policy, and the receiver provides an erasure status feedback; and second, when samples are scheduled according to a Round-Robin (RR) policy, when there is no erasure status feedback from the receiver. Aided by optimal structural results, we show that the optimal sampling policy for both settings, under some conditions, is a \emph{threshold policy}. We characterize the optimal threshold and the corresponding optimal long-term average sum MSE as a function of $K$, $f_{\max}$, $\epsilon$, and the statistical properties of the observed processes. Our results show that, with an exponentially distributed service rate, the optimal threshold $\tau^*$ increases as the number of processes $K$ increases, for both settings. Additionally, we show that the optimal threshold is an \emph{increasing} function of $\epsilon$ in the case of \emph{available} erasure status feedback, while it exhibits the \emph{opposite behavior}, i.e., $\tau^*$ is a \emph{decreasing} function of $\epsilon$, in the case of \emph{absent} erasure status feedback.

翻译：我们考虑一个观测$K$个独立Ornstein-Uhlenbeck过程的多过程远程估计系统。在该系统中，一个共享传感器以最小化长期平均和均方误差（MSE）的方式对$K$个过程进行采样。传感器在总采样频率约束$f_{\max}$下运行。来自所有过程的样本在共享队列中经历随机处理延迟，随后通过擦除概率为$\epsilon$的信道传输。我们研究了该问题的两种变体：第一种，样本按最大年龄优先（MAF）策略调度，且接收器提供擦除状态反馈；第二种，样本按轮询（RR）策略调度，且接收器不提供擦除状态反馈。借助最优结构结果，我们证明在两种设置下，在一定条件下最优采样策略为*阈值策略*。我们刻画了最优阈值及相应的最优长期平均和MSE，并将其表示为$K$、$f_{\max}$、$\epsilon$以及被观测过程统计特性的函数。结果表明，在服务速率服从指数分布的情况下，两种设置中的最优阈值$\tau^*$均随过程数量$K$的增大而增大。此外，我们证明：在*存在*擦除状态反馈的情况下，最优阈值是$\epsilon$的*增函数*；而在*不存在*擦除状态反馈的情况下，则呈现*相反行为*，即$\tau^*$是$\epsilon$的*减函数*。