Timely Multi-Process Estimation with Erasures

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 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.