  An asymptotic framework for optimal control of multiclass stochastic
processing networks, using formal diffusion approximations under suitable
temporal and spatial scaling, by Brownian control problems (BCP) and their
equivalent workload formulations (EWF), has been developed by Harrison (1988).
This framework has been implemented in many works for constructing
asymptotically optimal control policies for a broad range of stochastic network
models. To date all asymptotic optimality results for such networks correspond
to settings where the solution of the EWF is a reflected Brownian motion in the
positive orthant with normal reflections. In this work we consider a well
studied stochastic network which is perhaps the simplest example of a model
with more than one dimensional workload process. In the regime considered here,
the singular control problem corresponding to the EWF does not have a simple
form explicit solution, however by considering an associated free boundary
problem one can give a representation for an optimal controlled process as a
two dimensional reflected Brownian motion in a Lipschitz domain whose boundary
is determined by the solution of the free boundary problem. Using the form of
the optimal solution we propose a sequence of control policies, given in terms
of suitable thresholds, for the scaled stochastic network control problems and
prove that this sequence of policies is asymptotically optimal. As suggested by
the solution of the EWF, the policy we propose requires a server to idle under
certain conditions which are specified in terms of the thresholds determined
from the free boundary.
