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.

我们研究了客户放弃的单服务器队列序列 ( \(GI/GI/1+GI\)) 在交通繁忙的情况下。耐心时间分布随序列而变化，这允许更广泛的应用范围。众所周知，Lee 和 Weerasinghe（Stochastic Process Appl 121(11):2507–2552, 2011）和 Reed and Ward（Math Oper Res 33(3):606–644, 2008）按比例提供的等待时间过程的序列收敛随着交通强度接近 1，弱到具有非线性漂移的反射扩散过程。在本文中，我们进一步证明了所提供的等待时间的平稳分布和矩的序列，随着扩散缩放，收敛到极限扩散过程的那些。这证明了扩散极限的平稳性能是\(GI/GI/1+GI\)的平稳性能的有效近似值队列。因此，我们还推导出了静止状态下\(GI/GI/1+GI\)队列的放弃概率的近似值。