One Ratio, Three Regimes: How Faster Adjustment Turns Damping Links into Amplifying Ones

Many systems pass work, goods or information along a chain of units. Each unit sees its own state with a delay T and adjusts with a time scale q. We ask what happens to a disturbance along the chain when q falls while T stays the same. For a unit that passes work on in proportion to its own delayed stock, the ratio r = T/q sets the regime of each link: for r <= 1/2 every nonzero frequency is attenuated; for 1/2 < r < pi/2 the unit is stable but low frequencies are amplified at each link; for r >= pi/2 the unit is not asymptotically stable. These thresholds are the classical ones of delayed car-following models; we apply them to flow chains. For a unit that reorders to restore its own stock (the anchoring-and-adjustment rule of the beer game) with lead time L > 0, slow fluctuations are amplified: |G|^2 = 1 + w^2 L (2q + beta L) + O(w^4); this does not hold at every frequency. At zero reading delay the unit is a known transfer function (Riddalls and Bennett, 2000); the new part is the reading delay, which does not enter the leading coefficient. Simulations agree with the gain formulas within 0.2% at the main settings, and within 2% across the regimes at small time steps; the Euler bias in the noise runs is largely explained by the exact discrete-time response of the scheme. In a chain of 60 links with noise of 10% of the mean flow, the variance stops growing after about 30 links; we report this as an observation in one setting, not as a general bound. The evidence is analysis and simulation; real data has not been used. The record contains the paper (PDF and Markdown), its four figures, and a code package with all scripts, their outputs and a reproduction check. Use of AI: AI agents (large language models) took part in drafting, coding, translation and review, together with the author. The author set the questions, reviewed the results, and takes responsibility for the paper.