  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 $f_{\max}$ and samples the processes according to a
Maximum-Age-First (MAF) schedule. The samples from all processes consume random
processing delays, and then are transmitted over an erasure channel with
probability $\epsilon$. Aided by optimal structural results, we show that the
optimal sampling policy, 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.
