  We address a single server queue control problem (QCP) in heavy traffic,
originating from Lee and Weerasinghe (2011) (see \cite{lee2011convergence}). We
employ their single server queueing model, where the state process is the
offered waiting time process, the customer arrival process has a
state-dependent intensity, and service times and customer's patience times are
i.i.d with general distributions. We introduce an infinite-horizon discounted
cost functional generated by two types of costs associated with the queueing
model: a control cost related to the use of control and a penalty for idleness
cost. Our major contribution lies in solving the QCP with a non-trivial cost
function for the control cost. Under mild assumptions, the heavy traffic limit
of the QCP yields a stochastic control problem described by a stochastic
differential equation, which we call a diffusion control problem (DCP). We
solve the corresponding DCP with the help of the Legendre-Fenchel
transformation and a formal Hamilton-Jacobi-Bellman (HJB) equation to find an
optimal control. We provide the existence and uniqueness of the corresponding
admissible control system by a comparison result. Then we ``convert" the
optimal strategy of the DCP to the QCP of which we obtain a nearly optimal
solution. As a consequence, we deduce an asymptotically optimal solution to the
unsolved QCP in \cite{lee2011convergence}.
