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Hidden Convexity in Queueing Models

Xin Chen, Linwei Xin, Minda Zhao

TL;DR

This paper reveals a hidden convexity in the joint control of arrival and service rates for GI/GI/1 queues by a strategic change of variables, leading to convex reformulations of a non-convex long-run cost objective. It shows that, for Gamma-based interarrival and service times (including M/M/1 and M/GI/1 extensions), the reformulated problem satisfies a Polyak-Łojasiewicz-Kurdyka (PLK) condition, ensuring global convergence of first-order methods. The key technical advance is proving that the expected queue length under a square-root transformation of the traffic intensity is convex for Gamma families, enabling a convex representation via Reformulations (R1) and (R2) and, in turn, global optimality guarantees. However, the convexity does not extend to mixtures of exponentials, highlighting the limits of hidden convexity. Overall, the results provide principled guidance for online/offline optimization in queueing systems and suggest directions for extending hidden convexity to broader distributions and dynamic decision settings.

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.

Hidden Convexity in Queueing Models

TL;DR

This paper reveals a hidden convexity in the joint control of arrival and service rates for GI/GI/1 queues by a strategic change of variables, leading to convex reformulations of a non-convex long-run cost objective. It shows that, for Gamma-based interarrival and service times (including M/M/1 and M/GI/1 extensions), the reformulated problem satisfies a Polyak-Łojasiewicz-Kurdyka (PLK) condition, ensuring global convergence of first-order methods. The key technical advance is proving that the expected queue length under a square-root transformation of the traffic intensity is convex for Gamma families, enabling a convex representation via Reformulations (R1) and (R2) and, in turn, global optimality guarantees. However, the convexity does not extend to mixtures of exponentials, highlighting the limits of hidden convexity. Overall, the results provide principled guidance for online/offline optimization in queueing systems and suggest directions for extending hidden convexity to broader distributions and dynamic decision settings.

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 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.

Paper Structure

This paper contains 17 sections, 8 theorems, 74 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Problem Setting
  4. Convex Reformulation
  5. Reformulation I
  6. Reformulation II
  7. Special Case: $M/M/1$ Queue
  8. $\Gamma_k/\Gamma_m/1$ Queue
  9. Counterexample: Mixture of Exponential Distributions
  10. Polyak-Łojasiewicz-Kurdyka (PŁK) Condition
  11. Conclusion
  12. Omitted Proofs in Section \ref{['sec-reformulation']}
  13. Reformulation \ref{['opt-gg1-r1']}
  14. Reformulation \ref{['opt-gg1-r2']}: $M/M/1$ Queue
  15. Reformulation \ref{['opt-gg1-r2']}: $\Gamma_k/\Gamma_m/1$ Queue
  16. ...and 2 more sections

Key Result

Theorem 1

If Assumptions assumption-general model and assumption:lambda_hat hold, the optimization problem opt-gg1-r1 is convex.

Figures (1)

  • Figure 1: Plots of $\hat{l}_n(\tau)$ and $\sum_{n=1}^N \hat{l}_n(\tau)$ when $S_n, T_n \sim \exp(1)$.

Theorems & Definitions (17)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 1: PŁK Condition
  • Theorem 5
  • proof
  • proof
  • Lemma 1
  • ...and 7 more