Stationary distribution convergence of the offered waiting processes in heavy traffic under general patience time scaling

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 Lee and Weerasinghe (Stochastic Process Appl 121(11):2507-2552, 2011) and Reed and Ward (Math Oper Res 33(3):606-644, 2008) that the sequence of scaled offered waiting time processes converges weakly to a reflecting diffusion process with nonlinear 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.