  Well-designed queuing systems form the backbone of modern communications,
distributed computing, and content delivery architectures. Designs balancing
infrastructure costs and user experience indices require tools from teletraffic
theory and operations research. A standard approach to designing such systems
involves formulating optimization problems that strive to maximize the
pertinent utility functions while adhering to quality-of-service and other
physical constraints. In many cases, formulating such problems necessitates
making simplistic assumptions on arrival and departure processes to keep the
problem simple. This work puts forth a stochastic optimization framework for
designing queuing systems where the exogenous processes may have arbitrary and
unknown distributions. We show that many such queuing design problems can
generally be formulated as stochastic optimization problems where the objective
and constraint are non-linear functions of expectations. The compositional
structure obviates the use of classical stochastic approximation approaches
where the stochastic gradients are often required to be unbiased. To this end,
a constrained stochastic compositional gradient descent algorithm is proposed
that utilizes a tracking step for the expected value functions. The
non-asymptotic performance of the proposed algorithm is characterized via its
iteration complexity. Numerical tests allow us to validate the theoretical
results and demonstrate the efficacy of the proposed algorithm.
