Timely Multi-Process Estimation Over Erasure Channels With and Without Feedback

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.