Queues with Updating Information: Finding the Amplitude of Oscillations

The paper derives, for the two-queue updating model with multinomial logit choice, the exact step map on each update interval (equation (2.7)) and then proves Theorem 2.2: in the post-Hopf periodic steady state the minimum L solves ρ − L = e^{−µΔ} L + ρ e^{−θL}/(e^{−θL}+e^{−θ(ρ−L)})(1−e^{−µΔ}), and the oscillation amplitude equals ρ/2 − L. The derivation proceeds by identifying successive extrema at update times, obtaining coupled equations for U and L, and using L+U=ρ to reduce to a scalar fixed point; the amplitude follows immediately (equations (2.14)–(2.18)) . The candidate solution reproduces the same step map and fixed-point equation, with a closely aligned proof structure. Its only substantive difference is an explicit observation that in the periodic regime the total s(t)=q1(t)+q2(t) solves ds/dt=λ−µs and hence must be the constant ρ, which is consistent with the paper’s L+U=ρ and is implied by the exact update map as well . Therefore, both are correct and essentially the same proof.