  In this paper, we analyze how well a machine can solve a general problem in
queueing theory. To answer this question, we use a deep learning model to
predict the stationary queue-length distribution of an $M/G/1$ queue (Poisson
arrivals, general service times, one server). To the best of our knowledge,
this is the first time a machine learning model is applied to a general
queueing theory problem. We chose $M/G/1$ queue for this paper because it lies
"on the cusp" of the analytical frontier: on the one hand exact solution for
this model is available, which is both computationally and mathematically
complex. On the other hand, the problem (specifically the service time
distribution) is general. This allows us to compare the accuracy and efficiency
of the deep learning approach to the analytical solutions.
  The two key challenges in applying machine learning to this problem are (1)
generating a diverse set of training examples that provide a good
representation of a "generic" positive-valued distribution, and (2)
representations of the continuous distribution of service times as an input. We
show how we overcome these challenges.
  Our results show that our model is indeed able to predict the stationary
behavior of the $M/G/1$ queue extremely accurately: the average value of our
metric over the entire test set is $0.0009$. Moreover, our machine learning
model is very efficient, computing very accurate stationary distributions in a
fraction of a second (an approach based on simulation modeling would take much
longer to converge). We also present a case-study that mimics a real-life
setting and shows that our approach is more robust and provides more accurate
solutions compared to the existing methods. This shows the promise of extending
our approach beyond the analytically solvable systems (e.g., $G/G/1$ or
$G/G/c$).
