<p>Many empirical studies have confirmed that service-time and patience-time distributions in service systems (e.g., call centers and health care) are far from exponentially distributed. Because non-Markovian queues are rarely amenable to analytic solutions, performance analysis often resorts to approximating methods such as heavy-traffic fluid limits or computer simulations. In this paper, we contribute to the literature on transient performance analysis of non-Markovian queues by developing a new <i>neural networks method</i>, dubbed <i>Neural network in non-Markovian Queue</i> (NeuraliNQ); we specifically focus on queues with customer abandonment. NeuraliNQ is an offline supervised learning method that uses synthetic training data to learn the system’s intrinsic characteristics. In real-time applications, NeuraliNQ can recurrently estimate the transient system waiting time performance in a finite time window. Our results confirm that NeuraliNQ is able to achieve the proper balance between efficiency and accuracy: on the one hand, it is four orders of magnitude computationally more efficient than Monte-Carlo simulations; on the other hand, it yields higher solution accuracy than standard approximation methods such as the heavy-traffic fluid model, especially when the system scale is not too large.</p>