  We propose a unified approach to establishing diffusion approximations for
queues with impatient customers within a general framework of scaling customer
patience time. The approach consists of two steps. The first step is to show
that the diffusion-scaled abandonment process is asymptotically close to a
function of the diffusion-scaled queue length process under appropriate
conditions. The second step is to construct a continuous mapping not only to
characterize the system dynamics using the system primitives, but also to help
verify the conditions needed in the first step. The diffusion approximations
can then be obtained by applying the continuous mapping theorem. The approach
has two advantages: (i) it provides a unified procedure to establish the
diffusion approximations regardless of the structure of the queueing model or
the type of patience-time scaling; (ii) and it makes the diffusion analysis of
queues with customer abandonment essentially the same as the diffusion analysis
of queues without customer abandonment. We demonstrate the application of this
approach via the single server system with Markov-modulated service speeds in
the traditional heavy-traffic regime and the many-server system in the
Halfin-Whitt regime and the non-degenerate slowdown regime.
