Queues with Delayed Information: Analyzing the Impact of the Choice Model Function

The paper proves that the N-queue DDE with routing probabilities proportional to Ḡ(qi(t−Δ)) has symmetric equilibrium q* = λ/(Nμ) and loses stability at Δcr = arccos(μ/C)/√(C²−μ²), where C = −(λ/N) g(q*)/Ḡ(q*) = −(λ/N) h(q*), valid for C < −|μ|. It linearizes to u̇ = A u(t−Δ) − μu with A = C(I − (1/N)11^T), decomposing into a synchronous mode and N−1 identical asynchronous modes with characteristic equation s + μ − C e^{−sΔ} = 0. These are exactly the steps and formulas used by the candidate solution, which also supplies a standard transversality calculation showing the imaginary pair crosses from stable to unstable as Δ increases past Δcr. Hence, both arguments are correct and essentially the same, with the model adding a minor but standard crossing-direction detail.