  In small-cell wireless networks where users are connected to multiple base
stations (BSs), it is often advantageous to switch off dynamically a subset of
BSs to minimize energy costs. We consider two types of energy cost: (i) the
cost of maintaining a BS in the active state, and (ii) the cost of switching a
BS from the active state to inactive state. The problem is to operate the
network at the lowest possible energy cost (sum of activation and switching
costs) subject to queue stability. In this setting, the traditional approach --
a Max-Weight algorithm along with a Lyapunov-based stability argument -- does
not suffice to show queue stability, essentially due to the temporal
co-evolution between channel scheduling and the BS activation decisions induced
by the switching cost. Instead, we develop a learning and BS activation
algorithm with slow temporal dynamics, and a Max-Weight based channel scheduler
that has fast temporal dynamics. We show using convergence of
time-inhomogeneous Markov chains, that the co-evolving dynamics of learning, BS
activation and queue lengths lead to near optimal average energy costs along
with queue stability.
