  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 using signal-independent sampling policies, in
which sampling instances are chosen independently from the processes' values.
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.
