  In this paper, we study a remote monitoring system where a receiver observes
a remote binary Markov source and decides whether to sample and transmit the
state through a randomly delayed channel. We adopt uncertainty of information
(UoI), defined as the entropy conditional on past observations at the receiver,
as a metric of value of information, in contrast to the traditional
state-agnostic nonlinear age of information (AoI) penalty functions. To address
the limitations of prior UoI research that assumes one-time-slot delays, we
extend our analysis to scenarios with random delays. We model the problem as a
partially observable Markov decision process (POMDP) problem and simplify it to
a semi-Markov decision process (SMDP) by introducing the belief state. We
propose two algorithms: A globally optimal bisection relative value iteration
(bisec-RVI) algorithm and a computationally efficient sub-optimal index-based
threshold algorithm to solve the long-term average UoI minimization problem.
Numerical simulations demonstrate that our sampling policies surpass
traditional zero wait and AoI-optimal policies, particularly under conditions
of large delay, with the sub-optimal policy nearly matching the performance of
the optimal one.
