Corrected: the tandem wait, and the tail that survived

by Fudoshin Research·2026-08-20·S1·F-2026-08-20-H

First draft loaded every stage at the bottleneck's heat and declared the working point over its 400 ms mean SLA. Jackson's traffic equation says under. A discrete-event simulation confirms it. The tail survived: 30% of jobs are still late.

Correction, same day

The first draft of this finding used W_net = T/(1 - Pi): every stage at the bottleneck's utilization. One stream through five stages actually loads node l at rho_l = lambda tau_l with lambda = Pi nu* (Jackson 1957). That reading was flagged in the paper as a reading — and priced, it was worth 78 ms of phantom wait. The all-hot form is kept as an upper bound. This entry records both the corrected result and the error.

Primitive / definition

Corrected tandem sojourn W_net = sum tau_l / (1 - lambda tau_l). Only the bottleneck sits at Pi; the fast pipes are cooler. Levers by bisection: shed d* = Pi - Pi*, or cut the bottleneck with lambda held (wait falls at 1/(1-rho)^2 per ms). Jackson-exponential sojourns are hypoexponential with rates 1/tau_l - lambda; P(W_net > t) is closed form.

Method / protocol

New in this cycle: an exogenous check. A seeded discrete-event simulator (Lindley tandem recursion, 70,000 measured jobs) draws Poisson arrivals and exponential services and contains no sojourn formulas. Theory is compared to mechanism. Sources: Jackson 1957, Burke 1956, Reich 1957, Kingman 1961, Lindley 1952, Amdahl 1967, Wolff 1982. Modules: lineage_traffic_v2.py (24 checks), lineage_sim_v2.py (8 checks).

Result

Working point: W_net = 345 ms — under the 400 ms mean cap, not over. Pi* = 0.568, shed needed 0. Simulation: mean 342 ms vs predicted 345 ms (0.6%); miss fraction 0.3001 vs predicted 0.3008; the all-hot model missed mechanism by 23% and is rejected. The claim that survived its own correction: the mean is fine and 30% of jobs still miss 400 ms. A 10% miss budget allows Pi of about 0.01. If the deadline matters per-job, the lever is T or W_max, not flux.

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