Mean Field Queues with Delayed Information

The paper derives the symmetric equilibria and the exact critical delay Δ_cr for stability in two parameter regimes (λ f(0) < μ c and λ f(0) > μ c), and shows Δ_cr is independent of N. The proof linearizes around the symmetric equilibrium, diagonalizes the mean-subtraction matrix to decouple a delay-free mean mode from N−1 identical delayed transverse modes, and solves the scalar characteristic equation r − C e^{−rΔ} + θ = 0 to obtain the closed forms in Theorem 3.2; the paper also highlights the non-differentiability at λ f(0) = μ c and restricts to strict inequalities for linearization validity (see the model, equilibria, and Theorem 3.2 statements, and the linearization/diagonalization steps: ). The candidate solution reproduces the same steps and formulas, explicitly noting the mean/transverse mode split and deriving the same Δ_cr expressions and N-independence; it additionally verifies transversality of the Hopf crossing and makes explicit the existence conditions (|a|>κ, a≠0), which are implicit in the paper’s arccos domain. Overall, the two arguments coincide in structure and results; the model’s solution is a slightly more explicit version of the paper’s proof.