  In this paper, we consider joint antenna selection and optimal beamforming
for energy efficient delay minimization. We assume multiple-input multi-output
(MIMO) system with full duplex simultaneous wireless information and power
transfer (FD-SWIPT) where each sensor is equipped with a power splitting (PS)
system and can simultaneously receive both energy and information from the
aggregator (AGG). We show that the antenna selection and beamforming power
control policies are adaptive to the energy state information (ESI), the queue
state information (QSI) and the channel state information (CSI). We develop an
analytical framework for energy efficient delay-optimal control problem based
on the theory of infinite horizon partially observable Markov decision process
(POMDP). The infinite-horizon POMDP problem is transformed into an equivalent
value Bellman program and solved by near-optimal point-based Heuristic Search
Value Iteration (PB-HSVI) method under specific standard conditions. The
proposed solution outcome is a set of sub-optimal antenna selection and
beamforming control policies. Simulation results reveal an effective trade-off
between the contradictory objectives (i.e. delay and power consumption) and
show the enhancement in delay by using FD-SWIPT systems in comparison to Half
Duplex (HD)-SWIPT systems.
