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

Stationary Distribution Convergence of the Offered Waiting Processes in Heavy Traffic under General Patience Time Scaling