Hidden Convexity in Queueing Models

TL;DR

This paper reveals a hidden convexity in queueing control problems, enabling the use of first-order methods for globally optimal solutions by reformulating the problem through a specific change of variables.

Contribution

It provides a theoretical foundation for the observed empirical success of first-order methods in non-convex queueing control problems by identifying a convex reformulation and establishing the PLK condition.

Findings

The problem admits a convex reformulation after a change of variables.

First-order methods converge globally due to the PLK condition.

A new convexity property of expected queue length under a square-root transformation.

Abstract

We study the joint control of arrival and service rates in queueing systems with the objective of minimizing long-run expected cost minus revenue. Although the objective function is non-convex, first-order methods have been empirically observed to converge to globally optimal solutions. This paper provides a theoretical foundation for this empirical phenomenon by characterizing the optimization landscape and identifying a hidden convexity: the problem admits a convex reformulation after an appropriate change of variables. Leveraging this hidden convexity, we establish the Polyak-Lojasiewicz-Kurdyka (PLK) condition for the original control problem, which excludes spurious local minima and ensures global convergence for first-order methods. Our analysis applies to a broad class of $GI/GI/1$ queueing models, including those with Gamma-distributed interarrival and service times. As a key ingredient in the proof, we establish a new convexity property of the expected queue length under a square-root transformation of the traffic intensity.