  Effective control of time-sensitive industrial applications depends on the
real-time transmission of data from underlying sensors. Quantifying the data
freshness through age of information (AoI), in this paper, we jointly design
sampling and non-slot based scheduling policies to minimize the maximum
time-average age of information (MAoI) among sensors with the constraints of
average energy cost and finite queue stability. To overcome the intractability
involving high couplings of such a complex stochastic process, we first focus
on the single-sensor time-average AoI optimization problem and convert the
constrained Markov decision process (CMDP) into an unconstrained Markov
decision process (MDP) by the Lagrangian method. With the infinite-time average
energy and AoI expression expended as the Bellman equation, the single-sensor
time-average AoI optimization problem can be approached through the
steady-state distribution probability. Further, we propose a low-complexity
sub-optimal sampling and semi-distributed scheduling scheme for the
multi-sensor scenario. The simulation results show that the proposed scheme
reduces the MAoI significantly while achieving a balance between the sampling
rate and service rate for multiple sensors.
