  We study a double-ended queue where buyers and sellers arrive to conduct
trades. When there is a pair of buyer and seller in the system, they
immediately transact a trade and leave. Thus there cannot be non-zero number of
buyers and sellers simultaneously in the system. We assume that sellers and
buyers arrive at the system according to independent renewal processes, and
they would leave the system after independent exponential patience times. We
establish fluid and diffusion approximations for the queue length process under
a suitable asymptotic regime. The fluid limit is the solution of an ordinary
differential equation, and the diffusion limit is a time-inhomogeneous
asymmetric Ornstein-Uhlenbeck process (O-U process). A heavy traffic analysis
is also developed, and the diffusion limit in the stronger heavy traffic regime
is a time-homogeneous asymmetric O-U process. The limiting distributions of
both diffusion limits are obtained. We also show the interchange of the heavy
traffic and steady state limits.
