  We study a sequence of single server queues with customer abandonment
(GI/GI/1+GI) under heavy traffic. The patience time distributions vary with the
sequence, which allows for a wider scope of applications. It is known ([20,
18]) that the sequence of scaled offered waiting time processes converges
weakly to a reflecting diffusion process with non-linear drift, as the traffic
intensity approaches one. In this paper, we further show that the sequence of
stationary distributions and moments of the offered waiting times, with
diffusion scaling, converge to those of the limit diffusion process. This
justifies the stationary performance of the diffusion limit as a valid
approximation for the stationary performance of the GI/GI/1+GI queue.
Consequently, we also derive the approximation for the abandonment probability
for the GI/GI/1+GI queue in the stationary state.
