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Convergence Rate Analysis of the Join-the-Shortest-Queue System

Yuanzhe Ma, Siva Theja Maguluri

TL;DR

This paper addresses the finite-time convergence of a two-server Join-the-Shortest-Queue system under JSQ routing. By directly analyzing the original continuous-time Markov chain and employing coupling-based hitting-time techniques, the authors derive a non-asymptotic total-variation bound of order $O\left(\frac{1}{(1-\rho)^3}\frac{1}{t}\right)$ for $\rho<1$, with explicit constants. A refined bound on the auxiliary constant $K(\rho)$ is provided for $\rho<1/\sqrt{2}$, and a corollary gives a mean-queue-length convergence rate of $O\left(\frac{1}{(1-\sqrt{\rho})^5}\frac{1}{\sqrt{t}}\right)$. These results deliver reliable transient performance guarantees without relying on diffusion approximations, and lay groundwork for extending transient analyses to larger or heterogeneous JSQ systems.

Abstract

The Join-the-Shortest-Queue (JSQ) policy is among the most widely used routing algorithms for load balancing systems and has been extensively studied. Despite its simplicity and optimality, exact characterization of the system remains challenging. Most prior research has focused on analyzing its performance in steady-state in certain asymptotic regimes such as the heavy-traffic regime. However, the convergence rate to the steady-state in these regimes is often slow, calling into question the reliability of analyses based solely on the steady-state and heavy-traffic approximations. To address this limitation, we provide a finite-time convergence rate analysis of a JSQ system with two symmetric servers. In sharp contrast to the existing literature, we directly study the original system as opposed to an approximate limiting system such as a diffusion approximation. Our results demonstrate that for such a system, the convergence rate to its steady-state, measured in the total variation distance, is $O \left(\frac{1}{(1-ρ)^3} \frac{1}{t} \right)$, where $ρ\in (0,1)$ is the traffic intensity.

Convergence Rate Analysis of the Join-the-Shortest-Queue System

TL;DR

This paper addresses the finite-time convergence of a two-server Join-the-Shortest-Queue system under JSQ routing. By directly analyzing the original continuous-time Markov chain and employing coupling-based hitting-time techniques, the authors derive a non-asymptotic total-variation bound of order for , with explicit constants. A refined bound on the auxiliary constant is provided for , and a corollary gives a mean-queue-length convergence rate of . These results deliver reliable transient performance guarantees without relying on diffusion approximations, and lay groundwork for extending transient analyses to larger or heterogeneous JSQ systems.

Abstract

The Join-the-Shortest-Queue (JSQ) policy is among the most widely used routing algorithms for load balancing systems and has been extensively studied. Despite its simplicity and optimality, exact characterization of the system remains challenging. Most prior research has focused on analyzing its performance in steady-state in certain asymptotic regimes such as the heavy-traffic regime. However, the convergence rate to the steady-state in these regimes is often slow, calling into question the reliability of analyses based solely on the steady-state and heavy-traffic approximations. To address this limitation, we provide a finite-time convergence rate analysis of a JSQ system with two symmetric servers. In sharp contrast to the existing literature, we directly study the original system as opposed to an approximate limiting system such as a diffusion approximation. Our results demonstrate that for such a system, the convergence rate to its steady-state, measured in the total variation distance, is , where is the traffic intensity.

Paper Structure

This paper contains 19 sections, 13 theorems, 48 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main contribution
  3. Related work
  4. Notations
  5. Model and results
  6. Model
  7. Main result
  8. Proof of Theorem \ref{['thm:main']}
  9. Coupling with arrival and service
  10. Proof of Theorem \ref{['thm:main']}
  11. An improved upper bound of $K(\rho)$ for $\rho < \frac{1}{\sqrt{2}}$
  12. Proof outline of Corollary \ref{['coro-moment-difference']}
  13. Conclusion and future work
  14. Proof of lemmas in Section \ref{['sec:proof-of-thm']}
  15. Proof of Lemma \ref{['lemma-JSQ-hitting-time-heavy-traffic']}
  16. ...and 4 more sections

Key Result

Theorem 1

Consider a JSQ system as described in Section sec:model with $\rho = \frac{\lambda}{2\mu} < 1$. Let the initial state of the queue length be $\bm{x}$ and $\mathbb{P}_{\bm{x}}^t$ be the distribution of its queue length vector $\bm{q}(t)$ at time $t$, and let $\pi$ be its steady state, then for all $t where and $K(\rho)$ is a constant defined as Note that $K(\rho) \le 1$ and $K(\rho) = 1$ for $\rh

Figures (2)

  • Figure 1: Finite-time performance of a JSQ system at different values of $\rho$, where $q_i(t)$ denotes the queue length at time $t$ for server $i$. The results are based on 1000 simulations of trajectories to estimate the sum of the expected queue length vectors. Here we consider a JSQ system with Poisson arrival process with rate $\rho$ and two homogeneous servers following $\mathsf{Exp}\left(1 \right)$ with traffic intensity $\rho = 0.5$ (the left figure) and $\rho = 0.999$ (the right figure), respectively. The pair of plots clearly show that the convergence rates of JSQ systems heavily depend on $\rho$.
  • Figure 2: Left: plot of the bound \ref{['eqn:TV-bound']} as a function of $t$ for different values of $\rho$ and $x_1= x_2=10, \lambda = 10$. Right: plot of $K(\rho)$\ref{['eqn:def-K']} as a function of $\rho$.

Theorems & Definitions (21)

  • Theorem 1
  • Corollary 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • proof : Proof of Theorem \ref{['thm:main']}
  • Lemma 6
  • Lemma 7
  • ...and 11 more