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

The Join-the-Shortest-Queue (JSQ) policy is among the most widely used load balancing algorithms and has been extensively studied. However, an exact characterization of the system behavior remains challenging. Most prior research has focused on analyzing its performance in the 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, so steady-state and heavy-traffic characterizations may be less informative over practical time horizons. 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 rather than 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.

💡 Research Summary

This paper provides a rigorous finite‑time convergence analysis for the Join‑the‑Shortest‑Queue (JSQ) load‑balancing policy in its most elementary form: a continuous‑time Markov chain (CTMC) with two symmetric servers, Poisson arrivals at rate λ and exponential service times with rate μ. The traffic intensity is defined as ρ = λ/(2μ) and is assumed to lie in (0,1) so that a stationary distribution π exists. The authors’ main contribution is an explicit, non‑asymptotic bound on the total variation (TV) distance between the distribution Pₜˣ of the queue‑length vector at time t, starting from any initial state x ∈ ℕ², and the steady‑state distribution π. Specifically, they prove

  d_TV(Pₜˣ, π) ≤ C₁(λ, μ, x) · t⁻¹,

where

  C₁(λ, μ, x) = (2 − ρ)² /