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FA-67841 / Traffic signal timing plans / Open access

Webster cycle length: critical ratio sums every lane group of a phase · case 01

Webster cycle length returns a wrong result when critical ratio sums every lane group of a phase.

Verified by executionVariant 1 · 8 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

Each phase contributes the sum of its lane-group flow ratios instead of only its critical (largest) ratio, so multi-lane-group phases inflate Y.

THE FAILURE

Each phase contributes the sum of its lane-group flow ratios instead of only its critical (largest) ratio, so multi-lane-group phases inflate Y.

Unsuccessful approach: Averaging the lane groups still dilutes the critical movement; the heaviest lane group alone governs the phase.

Case contract

Input {phases: per-phase lists of lane-group flows veh/h, sat: saturation flow per lane group, lost: lost seconds per phase, allred: extra all-red seconds per cycle, cmin, cmax}. Y is the sum over phases of the critical (largest) flow ratio; L = lost*phases + allred. If Y >= 0.9 return 'oversaturated'. Otherwise C0 = (1.5L + 5)/(1 - Y), rounded UP to a multiple of 5 s and clamped to [cmin, cmax]. Return {cycle, lost: L, y_milli: floor(1000Y)}.

Why this case matters

Signal timing arithmetic is exact and integer or rational; a wrong rule silently produces unsafe or inefficient timing plans.

1 / The failure

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
from fractions import Fraction
import math
N = 1
observations = []
def solve(x):
    ph = x['phases']
    s = x['sat']
    Y = Fraction(0)
    for group in ph:
        Y += Fraction(sum(group), s)
    L = x['lost'] * len(ph) + x['allred']
    if Y >= Fraction(9, 10):
        return 'oversaturated'
    c0 = (Fraction(3, 2) * L + 5) / (1 - Y)
    c = -(-c0 // 5) * 5
    c = max(x['cmin'], min(x['cmax'], int(c)))
    return {'cycle': c, 'lost': L, 'y_milli': int(Y * 1000)}
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[({'phases': [[810], [810]], 'sat': 1800, 'lost': 4, 'allred': 0, 'cmin': 40, 'cmax': 180}, 'oversaturated'), ({'phases': [[600, 300, 660], [900, 300], [720], [780, 900]], 'sat': 1800, 'lost': 4, 'allred': 4, 'cmin': 40, 'cmax': 150}, 'oversaturated'), ({'phases': [[750, 870, 510], [570]], 'sat': 1900, 'lost': 4, 'allred': 6, 'cmin': 60, 'cmax': 180}, {'cycle': 110, 'lost': 14, 'y_milli': 757}), ({'phases': [[90], [90], [90]], 'sat': 1800, 'lost': 3, 'allred': 0, 'cmin': 60, 'cmax': 120}, {'cycle': 60, 'lost': 9, 'y_milli': 150}), ({'phases': [[720, 360], [540, 900]], 'sat': 1800, 'lost': 4, 'allred': 4, 'cmin': 60, 'cmax': 150}, 'oversaturated'), ({'phases': [[90, 90], [870, 840], [570], [660, 750, 780]], 'sat': 1800, 'lost': 4, 'allred': 5, 'cmin': 40, 'cmax': 90}, 'oversaturated'), ({'phases': [[60], [270, 210, 810], [600, 240]], 'sat': 1900, 'lost': 2, 'allred': 3, 'cmin': 40, 'cmax': 120}, {'cycle': 85, 'lost': 9, 'y_milli': 773}), ({'phases': [[240, 210], [780, 270]], 'sat': 1800, 'lost': 4, 'allred': 2, 'cmin': 40, 'cmax': 150}, {'cycle': 50, 'lost': 10, 'y_milli': 566})], [({'phases': [[810], [810]], 'sat': 1800, 'lost': 4, 'allred': 0, 'cmin': 40, 'cmax': 180}, 'oversaturated'), ({'phases': [[780, 510, 480], [630, 660, 330], [60, 900, 390], [480, 900]], 'sat': 1800, 'lost': 5, 'allred': 1, 'cmin': 60, 'cmax': 90}, 'oversaturated'), ({'phases': [[720, 360], [540, 900]], 'sat': 1800, 'lost': 4, 'allred': 4, 'cmin': 60, 'cmax': 150}, 'oversaturated'), ({'phases': [[720, 750, 810], [780, 900], [900, 90, 540], [150, 810, 750]], 'sat': 1900, 'lost': 2, 'allred': 1, 'cmin': 50, 'cmax': 90}, 'oversaturated'), ({'phases': [[300, 600], [450]], 'sat': 1800, 'lost': 4, 'allred': 2, 'cmin': 40, 'cmax': 120}, {'cycle': 50, 'lost': 10, 'y_milli': 583}), ({'phases': [[690], [480, 780, 120]], 'sat': 1800, 'lost': 4, 'allred': 6, 'cmin': 50, 'cmax': 150}, {'cycle': 145, 'lost': 14, 'y_milli': 816}), ({'phases': [[900, 630, 450], [810], [660], [840, 450]], 'sat': 1800, 'lost': 5, 'allred': 5, 'cmin': 40, 'cmax': 180}, 'oversaturated'), ({'phases': [[420, 510, 870], [150, 720]], 'sat': 2000, 'lost': 3, 'allred': 6, 'cmin': 50, 'cmax': 120}, {'cycle': 115, 'lost': 12, 'y_milli': 795})], [({'phases': [[150, 300, 630], [600, 60, 660]], 'sat': 1800, 'lost': 3, 'allred': 0, 'cmin': 50, 'cmax': 180}, {'cycle': 50, 'lost': 6, 'y_milli': 716}), ({'phases': [[300, 600], [450]], 'sat': 1800, 'lost': 4, 'allred': 2, 'cmin': 40, 'cmax': 120}, {'cycle': 50, 'lost': 10, 'y_milli': 583}), ({'phases': [[630], [210], [780]], 'sat': 1800, 'lost': 4, 'allred': 6, 'cmin': 50, 'cmax': 180}, 'oversaturated'), ({'phases': [[570, 780], [540, 720, 300], [150, 150], [810, 210]], 'sat': 1900, 'lost': 2, 'allred': 0, 'cmin': 60, 'cmax': 180}, 'oversaturated'), ({'phases': [[510, 720, 60], [270, 90, 840]], 'sat': 1800, 'lost': 5, 'allred': 2, 'cmin': 50, 'cmax': 120}, {'cycle': 120, 'lost': 12, 'y_milli': 866}), ({'phases': [[810], [810]], 'sat': 1800, 'lost': 4, 'allred': 0, 'cmin': 40, 'cmax': 180}, 'oversaturated'), ({'phases': [[780], [780]], 'sat': 1800, 'lost': 3, 'allred': 6, 'cmin': 40, 'cmax': 90}, {'cycle': 90, 'lost': 12, 'y_milli': 866}), ({'phases': [[90], [90], [90]], 'sat': 1800, 'lost': 3, 'allred': 0, 'cmin': 60, 'cmax': 120}, {'cycle': 60, 'lost': 9, 'y_milli': 150})], [({'phases': [[540, 210], [780, 210, 810]], 'sat': 1800, 'lost': 2, 'allred': 5, 'cmin': 40, 'cmax': 150}, {'cycle': 75, 'lost': 9, 'y_milli': 750}), ({'phases': [[720, 840], [720], [300, 630, 60]], 'sat': 2000, 'lost': 5, 'allred': 6, 'cmin': 40, 'cmax': 90}, 'oversaturated'), ({'phases': [[810], [810]], 'sat': 1800, 'lost': 4, 'allred': 0, 'cmin': 40, 'cmax': 180}, 'oversaturated'), ({'phases': [[450, 420], [540, 540, 870], [720, 420]], 'sat': 1800, 'lost': 3, 'allred': 5, 'cmin': 50, 'cmax': 120}, 'oversaturated'), ({'phases': [[120], [450, 300, 510], [330, 750, 300]], 'sat': 2000, 'lost': 4, 'allred': 6, 'cmin': 40, 'cmax': 180}, {'cycle': 105, 'lost': 18, 'y_milli': 690}), ({'phases': [[660, 510, 330], [270, 270, 270], [870]], 'sat': 1800, 'lost': 2, 'allred': 0, 'cmin': 40, 'cmax': 120}, 'oversaturated'), ({'phases': [[720, 360], [540, 900]], 'sat': 1800, 'lost': 4, 'allred': 4, 'cmin': 60, 'cmax': 150}, 'oversaturated'), ({'phases': [[300, 600], [450]], 'sat': 1800, 'lost': 4, 'allred': 2, 'cmin': 40, 'cmax': 120}, {'cycle': 50, 'lost': 10, 'y_milli': 583})], [({'phases': [[300, 600], [450]], 'sat': 1800, 'lost': 4, 'allred': 2, 'cmin': 40, 'cmax': 120}, {'cycle': 50, 'lost': 10, 'y_milli': 583}), ({'phases': [[720, 360], [540, 900]], 'sat': 1800, 'lost': 4, 'allred': 4, 'cmin': 60, 'cmax': 150}, 'oversaturated'), ({'phases': [[510, 450, 60], [180, 390], [540]], 'sat': 1800, 'lost': 5, 'allred': 2, 'cmin': 50, 'cmax': 180}, {'cycle': 155, 'lost': 17, 'y_milli': 800}), ({'phases': [[60], [600, 780], [870]], 'sat': 2000, 'lost': 2, 'allred': 1, 'cmin': 60, 'cmax': 180}, {'cycle': 110, 'lost': 7, 'y_milli': 855}), ({'phases': [[660, 330], [660], [150, 180], [690]], 'sat': 1900, 'lost': 4, 'allred': 2, 'cmin': 50, 'cmax': 150}, 'oversaturated'), ({'phases': [[660], [510]], 'sat': 1800, 'lost': 4, 'allred': 1, 'cmin': 60, 'cmax': 150}, {'cycle': 60, 'lost': 9, 'y_milli': 650}), ({'phases': [[810, 810, 240], [150, 660], [300, 240, 810], [270]], 'sat': 1800, 'lost': 5, 'allred': 0, 'cmin': 50, 'cmax': 150}, 'oversaturated'), ({'phases': [[810], [810]], 'sat': 1800, 'lost': 4, 'allred': 0, 'cmin': 40, 'cmax': 180}, 'oversaturated')]]
for i, (args, expected) in enumerate(fixtures[N-1]):
    check('timing oracle' + ' %d' % i, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
Boundary fixtureActualExpectedOutcome
timing oracle 0oversaturatedoversaturatedPassed
timing oracle 1oversaturatedoversaturatedPassed
timing oracle 2oversaturated{'cycle': 110, 'lost': 14, 'y_milli': 757}Failed
timing oracle 3{'cycle': 60, 'lost': 9, 'y_milli': 150}{'cycle': 60, 'lost': 9, 'y_milli': 150}Passed
timing oracle 4oversaturatedoversaturatedPassed
timing oracle 5oversaturatedoversaturatedPassed
timing oracle 6oversaturated{'cycle': 85, 'lost': 9, 'y_milli': 773}Failed
timing oracle 7{'cycle': 120, 'lost': 10, 'y_milli': 833}{'cycle': 50, 'lost': 10, 'y_milli': 566}Failed

SHA-256 / 4c5af8a35ae3491da95c0a6ae3d52888851689cac4e72312cdc706a1446e35fb

2 / The unsuccessful fix

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
from fractions import Fraction
import math
N = 1
observations = []
def solve(x):
    ph = x['phases']
    s = x['sat']
    Y = Fraction(0)
    for group in ph:
        Y += Fraction(sum(group), s * len(group))
    L = x['lost'] * len(ph) + x['allred']
    if Y >= Fraction(9, 10):
        return 'oversaturated'
    c0 = (Fraction(3, 2) * L + 5) / (1 - Y)
    c = -(-c0 // 5) * 5
    c = max(x['cmin'], min(x['cmax'], int(c)))
    return {'cycle': c, 'lost': L, 'y_milli': int(Y * 1000)}
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[({'phases': [[810], [810]], 'sat': 1800, 'lost': 4, 'allred': 0, 'cmin': 40, 'cmax': 180}, 'oversaturated'), ({'phases': [[600, 300, 660], [900, 300], [720], [780, 900]], 'sat': 1800, 'lost': 4, 'allred': 4, 'cmin': 40, 'cmax': 150}, 'oversaturated'), ({'phases': [[750, 870, 510], [570]], 'sat': 1900, 'lost': 4, 'allred': 6, 'cmin': 60, 'cmax': 180}, {'cycle': 110, 'lost': 14, 'y_milli': 757}), ({'phases': [[90], [90], [90]], 'sat': 1800, 'lost': 3, 'allred': 0, 'cmin': 60, 'cmax': 120}, {'cycle': 60, 'lost': 9, 'y_milli': 150}), ({'phases': [[720, 360], [540, 900]], 'sat': 1800, 'lost': 4, 'allred': 4, 'cmin': 60, 'cmax': 150}, 'oversaturated'), ({'phases': [[90, 90], [870, 840], [570], [660, 750, 780]], 'sat': 1800, 'lost': 4, 'allred': 5, 'cmin': 40, 'cmax': 90}, 'oversaturated'), ({'phases': [[60], [270, 210, 810], [600, 240]], 'sat': 1900, 'lost': 2, 'allred': 3, 'cmin': 40, 'cmax': 120}, {'cycle': 85, 'lost': 9, 'y_milli': 773}), ({'phases': [[240, 210], [780, 270]], 'sat': 1800, 'lost': 4, 'allred': 2, 'cmin': 40, 'cmax': 150}, {'cycle': 50, 'lost': 10, 'y_milli': 566})], [({'phases': [[810], [810]], 'sat': 1800, 'lost': 4, 'allred': 0, 'cmin': 40, 'cmax': 180}, 'oversaturated'), ({'phases': [[780, 510, 480], [630, 660, 330], [60, 900, 390], [480, 900]], 'sat': 1800, 'lost': 5, 'allred': 1, 'cmin': 60, 'cmax': 90}, 'oversaturated'), ({'phases': [[720, 360], [540, 900]], 'sat': 1800, 'lost': 4, 'allred': 4, 'cmin': 60, 'cmax': 150}, 'oversaturated'), ({'phases': [[720, 750, 810], [780, 900], [900, 90, 540], [150, 810, 750]], 'sat': 1900, 'lost': 2, 'allred': 1, 'cmin': 50, 'cmax': 90}, 'oversaturated'), ({'phases': [[300, 600], [450]], 'sat': 1800, 'lost': 4, 'allred': 2, 'cmin': 40, 'cmax': 120}, {'cycle': 50, 'lost': 10, 'y_milli': 583}), ({'phases': [[690], [480, 780, 120]], 'sat': 1800, 'lost': 4, 'allred': 6, 'cmin': 50, 'cmax': 150}, {'cycle': 145, 'lost': 14, 'y_milli': 816}), ({'phases': [[900, 630, 450], [810], [660], [840, 450]], 'sat': 1800, 'lost': 5, 'allred': 5, 'cmin': 40, 'cmax': 180}, 'oversaturated'), ({'phases': [[420, 510, 870], [150, 720]], 'sat': 2000, 'lost': 3, 'allred': 6, 'cmin': 50, 'cmax': 120}, {'cycle': 115, 'lost': 12, 'y_milli': 795})], [({'phases': [[150, 300, 630], [600, 60, 660]], 'sat': 1800, 'lost': 3, 'allred': 0, 'cmin': 50, 'cmax': 180}, {'cycle': 50, 'lost': 6, 'y_milli': 716}), ({'phases': [[300, 600], [450]], 'sat': 1800, 'lost': 4, 'allred': 2, 'cmin': 40, 'cmax': 120}, {'cycle': 50, 'lost': 10, 'y_milli': 583}), ({'phases': [[630], [210], [780]], 'sat': 1800, 'lost': 4, 'allred': 6, 'cmin': 50, 'cmax': 180}, 'oversaturated'), ({'phases': [[570, 780], [540, 720, 300], [150, 150], [810, 210]], 'sat': 1900, 'lost': 2, 'allred': 0, 'cmin': 60, 'cmax': 180}, 'oversaturated'), ({'phases': [[510, 720, 60], [270, 90, 840]], 'sat': 1800, 'lost': 5, 'allred': 2, 'cmin': 50, 'cmax': 120}, {'cycle': 120, 'lost': 12, 'y_milli': 866}), ({'phases': [[810], [810]], 'sat': 1800, 'lost': 4, 'allred': 0, 'cmin': 40, 'cmax': 180}, 'oversaturated'), ({'phases': [[780], [780]], 'sat': 1800, 'lost': 3, 'allred': 6, 'cmin': 40, 'cmax': 90}, {'cycle': 90, 'lost': 12, 'y_milli': 866}), ({'phases': [[90], [90], [90]], 'sat': 1800, 'lost': 3, 'allred': 0, 'cmin': 60, 'cmax': 120}, {'cycle': 60, 'lost': 9, 'y_milli': 150})], [({'phases': [[540, 210], [780, 210, 810]], 'sat': 1800, 'lost': 2, 'allred': 5, 'cmin': 40, 'cmax': 150}, {'cycle': 75, 'lost': 9, 'y_milli': 750}), ({'phases': [[720, 840], [720], [300, 630, 60]], 'sat': 2000, 'lost': 5, 'allred': 6, 'cmin': 40, 'cmax': 90}, 'oversaturated'), ({'phases': [[810], [810]], 'sat': 1800, 'lost': 4, 'allred': 0, 'cmin': 40, 'cmax': 180}, 'oversaturated'), ({'phases': [[450, 420], [540, 540, 870], [720, 420]], 'sat': 1800, 'lost': 3, 'allred': 5, 'cmin': 50, 'cmax': 120}, 'oversaturated'), ({'phases': [[120], [450, 300, 510], [330, 750, 300]], 'sat': 2000, 'lost': 4, 'allred': 6, 'cmin': 40, 'cmax': 180}, {'cycle': 105, 'lost': 18, 'y_milli': 690}), ({'phases': [[660, 510, 330], [270, 270, 270], [870]], 'sat': 1800, 'lost': 2, 'allred': 0, 'cmin': 40, 'cmax': 120}, 'oversaturated'), ({'phases': [[720, 360], [540, 900]], 'sat': 1800, 'lost': 4, 'allred': 4, 'cmin': 60, 'cmax': 150}, 'oversaturated'), ({'phases': [[300, 600], [450]], 'sat': 1800, 'lost': 4, 'allred': 2, 'cmin': 40, 'cmax': 120}, {'cycle': 50, 'lost': 10, 'y_milli': 583})], [({'phases': [[300, 600], [450]], 'sat': 1800, 'lost': 4, 'allred': 2, 'cmin': 40, 'cmax': 120}, {'cycle': 50, 'lost': 10, 'y_milli': 583}), ({'phases': [[720, 360], [540, 900]], 'sat': 1800, 'lost': 4, 'allred': 4, 'cmin': 60, 'cmax': 150}, 'oversaturated'), ({'phases': [[510, 450, 60], [180, 390], [540]], 'sat': 1800, 'lost': 5, 'allred': 2, 'cmin': 50, 'cmax': 180}, {'cycle': 155, 'lost': 17, 'y_milli': 800}), ({'phases': [[60], [600, 780], [870]], 'sat': 2000, 'lost': 2, 'allred': 1, 'cmin': 60, 'cmax': 180}, {'cycle': 110, 'lost': 7, 'y_milli': 855}), ({'phases': [[660, 330], [660], [150, 180], [690]], 'sat': 1900, 'lost': 4, 'allred': 2, 'cmin': 50, 'cmax': 150}, 'oversaturated'), ({'phases': [[660], [510]], 'sat': 1800, 'lost': 4, 'allred': 1, 'cmin': 60, 'cmax': 150}, {'cycle': 60, 'lost': 9, 'y_milli': 650}), ({'phases': [[810, 810, 240], [150, 660], [300, 240, 810], [270]], 'sat': 1800, 'lost': 5, 'allred': 0, 'cmin': 50, 'cmax': 150}, 'oversaturated'), ({'phases': [[810], [810]], 'sat': 1800, 'lost': 4, 'allred': 0, 'cmin': 40, 'cmax': 180}, 'oversaturated')]]
for i, (args, expected) in enumerate(fixtures[N-1]):
    check('timing oracle' + ' %d' % i, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
Boundary fixtureActualExpectedOutcome
timing oracle 0oversaturatedoversaturatedPassed
timing oracle 1oversaturatedoversaturatedPassed
timing oracle 2{'cycle': 80, 'lost': 14, 'y_milli': 673}{'cycle': 110, 'lost': 14, 'y_milli': 757}Failed
timing oracle 3{'cycle': 60, 'lost': 9, 'y_milli': 150}{'cycle': 60, 'lost': 9, 'y_milli': 150}Passed
timing oracle 4{'cycle': 80, 'lost': 12, 'y_milli': 700}oversaturatedFailed
timing oracle 5oversaturatedoversaturatedPassed
timing oracle 6{'cycle': 40, 'lost': 9, 'y_milli': 478}{'cycle': 85, 'lost': 9, 'y_milli': 773}Failed
timing oracle 7{'cycle': 40, 'lost': 10, 'y_milli': 416}{'cycle': 50, 'lost': 10, 'y_milli': 566}Failed

SHA-256 / 8297f9f0eae51278dc4a062989694a3f89be6e7f3c58fc25377a8295ff45c492

HELD IN THE MEMBER ARCHIVE

The verified repair and its recorded checks are member-only.

This mechanism has 8 recorded checks per implementation. The open-access tier publishes the failure and the unsuccessful fix; the repaired source that passes every check, and the observations that prove it, are available to members.

Every case sharing this mechanism uses the same contract and the same repair, so this one record is held back for all of them.

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Verification & scope

A deterministic, bounded toy model with a stipulated contract; it makes no claim of conformance to any agency manual or standard. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.

Observations recorded using Python 3.12.14 at 2026-09-29T14:47:56.576616+00:00.

Case digest / 9e4f2d738006d7b4825238574c2d0c438d88ea8491c34523c7ba411135d9a035