# Optimal Design of Queuing Systems via Compositional Stochastic Programming

## TL;DR

This paper introduces a stochastic optimization framework for designing queuing systems with arbitrary, unknown process distributions, using a compositional gradient descent algorithm that handles complex expectation-based objectives and constraints.

## Contribution

It develops a novel constrained stochastic compositional gradient descent method for queuing system design, accommodating arbitrary process distributions and providing theoretical performance guarantees.

## Key findings

- The proposed algorithm effectively handles complex queuing design problems.
- Theoretical analysis characterizes the iteration complexity of the method.
- Numerical tests validate the algorithm's efficacy and theoretical results.

## Abstract

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.