  In the small-cell networks (SCNs) with multiple small-cell base stations
(ScBSs), the joint design of beamforming vectors, user scheduling and ScBS
sleeping is investigated with the constraints on proportional rate. A long-term
grid-energy expenditure minimization problem is formulated for the considered
SCNs, which are powered by the smart grid and natural renewable energy. Since
the scheduled user indicators are coupled with the beamforming vectors, the
formulated problem is challenging to handle. In order to decouple the
beamforming vectors from the scheduled user indicators, the Lyapunov
optimization technique is used. As a result, a practical two-scale algorithm is
proposed to allocate the user scheduling indicators and ScBS sleeping variables
at the coarse-grained granularity (frame) as well as obtain the beamforming
vectors at the fine-grained granularity (slot). Numerical results are used to
verify the performance of the proposed two-scale algorithm.
