We study how well a machine can solve a general problem in queueing theory, using a neural net to predict the stationary queue-length distribution of an M/G/1 queue. This problem is, arguably, the most general queuing problem for which an analytical 'ground truth' solution exists. We overcome two key challenges: (1) generating training data that provide 'diverse' service time distributions, and (2) providing continuous service distributions as input to the neural net. To overcome (1), we develop an algorithm to sample phase-type service time distributions that cover a broad space of non-negative distributions; exact solutions of M / PH /1 (with phase-type service) are used for the training data. For (2) we find that using only the first n moments of the service times as inputs is sufficient to train the neural net. Our empirical results indicate that neural nets can estimate the stationary behavior of the M/G/1 extremely accurately.