  We consider a controlled double-ended queue consisting of two classes of
customers, labeled sellers and buyers. The sellers and buyers arrive in a
trading market according to two independent renewal processes. Whenever there
is a seller and buyer pair, they are matched and leave the system
instantaneously. The matching follows first-come-first-match service
discipline. Those customers who cannot be matched immediately need to wait in
the designated queue, and they are assumed to be impatient with generally
distributed patience times. The control problem is concerned with the tradeoff
between blocking and abandonment, and its objective is to choose optimal
queue-capacities (buffer lengths) for sellers and buyers to minimize an
infinite horizon discounted linear cost functional which consists of holding
costs, and penalty costs for blocking and abandonment. When the arrival
intensities of both customer classes tend to infinity in concert, we use a
heavy traffic approximation to formulate an approximate diffusion control
problem (DCP), and derive an optimal threshold policy for the DCP. Finally, we
employ the DCP solution to establish an easy-to-implement, simple
asymptotically optimal threshold policy for the original queueing control
problem.
