File size: 10,108 Bytes
c383c28 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 | #!/usr/bin/env python3
"""Synthetic reference study for the Supervisory Capacity Model.
This is not evidence about people or deployed LLMs. It separates an exact
logistic-model recovery control from a queue-model mismatch test and detects
violations caused by correlated arrivals, batching, and switching costs.
"""
from __future__ import annotations
import json
from pathlib import Path
import numpy as np
from scipy.optimize import least_squares
SEED = 20260828
TEAM_SIZES = np.array([1, 2, 3, 4, 5, 6, 8, 10], dtype=float)
SESSIONS_PER_SIZE = 500
DURATION_MIN = 30
LAMBDA = 0.25
BUDGET = 1.0
TYPE_NAMES = ("perception", "comprehension", "projection")
TYPE_PROBS = np.array([0.50, 0.30, 0.20])
TYPE_COSTS = np.array([0.60, 1.00, 1.60])
TYPE_DEADLINES = np.array([2.0, 3.0, 4.0])
CONDITIONS = ("additive", "correlated", "batched", "switching")
def logistic(n: np.ndarray, floor: float, span: float, midpoint: float, tau: float) -> np.ndarray:
z = np.clip((n - midpoint) / tau, -40.0, 40.0)
return floor + span / (1.0 + np.exp(z))
def fit_curve(x: np.ndarray, y: np.ndarray) -> np.ndarray:
def residual(theta: np.ndarray) -> np.ndarray:
return logistic(x, *theta) - y
initial = np.array([0.12, 0.82, 4.0, 1.0])
lower = np.array([0.0, 0.05, 0.25, 0.10])
upper = np.array([0.60, 1.00, 15.0, 8.0])
return least_squares(residual, initial, bounds=(lower, upper)).x
def minute_arrivals(rng: np.random.Generator, n_agents: int, condition: str,
minute: int, event_rate: float = LAMBDA,
cost_scale: float = 1.0):
events: list[list[float | int]] = []
independent_rate = 0.68 * event_rate if condition == "correlated" else event_rate
for agent in range(n_agents):
for _ in range(rng.poisson(independent_rate)):
kind = int(rng.choice(len(TYPE_NAMES), p=TYPE_PROBS))
events.append([
minute + rng.random(), agent, kind,
float(TYPE_COSTS[kind] * cost_scale),
])
if condition == "correlated":
for _ in range(rng.poisson(event_rate - independent_rate)):
kind = int(rng.choice(len(TYPE_NAMES), p=TYPE_PROBS))
when = minute + rng.random()
for agent in range(n_agents):
events.append([
when, agent, kind, float(TYPE_COSTS[kind] * cost_scale),
])
if condition == "batched" and events:
by_kind: dict[int, list[int]] = {}
for index, event in enumerate(events):
by_kind.setdefault(int(event[2]), []).append(index)
for kind, indices in by_kind.items():
if len(indices) > 1:
total = (TYPE_COSTS[kind] * cost_scale
* (1.0 + 0.25 * (len(indices) - 1)))
per_event = float(total / len(indices))
for index in indices:
events[index][3] = per_event
return events
def simulate_session(rng: np.random.Generator, n_agents: int, condition: str,
event_rate: float = LAMBDA, budget: float = BUDGET,
cost_scale: float = 1.0,
deadline_scale: float = 1.0) -> float:
events: list[list[float | int]] = []
for minute in range(DURATION_MIN):
events.extend(minute_arrivals(
rng, n_agents, condition, minute, event_rate, cost_scale
))
events.sort(key=lambda event: (float(event[0]), rng.random()))
if not events:
return np.nan
available = 0.0
previous_agent = None
correct = 0
for arrival, agent, kind, cost in events:
switch = 0.35 if condition == "switching" and previous_agent not in (None, agent) else 0.0
start = max(float(arrival), available)
completion = start + (float(cost) + switch) / budget
recovered = (completion - float(arrival)
<= TYPE_DEADLINES[int(kind)] * deadline_scale)
p_correct = 0.95 if recovered else 0.15
correct += int(rng.random() < p_correct)
available = completion
previous_agent = agent
return correct / len(events)
def bootstrap_midpoint(
rng: np.random.Generator, values: dict[int, np.ndarray], repetitions: int = 400
) -> tuple[float, float]:
estimates = []
for _ in range(repetitions):
means = np.array(
[rng.choice(values[int(n)], size=len(values[int(n)]), replace=True).mean() for n in TEAM_SIZES]
)
estimates.append(float(fit_curve(TEAM_SIZES, means)[2]))
return tuple(np.quantile(estimates, [0.025, 0.975]))
def direct_logistic_control(rng: np.random.Generator) -> dict[str, object]:
"""Generate scores from the fitted model itself as a code recovery check."""
truth = np.array([0.15, 0.80, 4.35, 0.90])
values: dict[int, np.ndarray] = {}
means = []
for n in TEAM_SIZES.astype(int):
probability = float(logistic(np.array([n]), *truth)[0])
scores = rng.binomial(1, probability, size=SESSIONS_PER_SIZE).astype(float)
values[n] = scores
means.append(float(scores.mean()))
theta = fit_curve(TEAM_SIZES, np.array(means))
interval = bootstrap_midpoint(rng, values)
return {
"true_floor": float(truth[0]),
"true_span": float(truth[1]),
"true_midpoint": float(truth[2]),
"true_tau": float(truth[3]),
"fitted_floor": float(theta[0]),
"fitted_span": float(theta[1]),
"fitted_midpoint": float(theta[2]),
"fitted_tau": float(theta[3]),
"midpoint_ci": list(map(float, interval)),
"means": means,
}
def sensitivity_battery(rng: np.random.Generator) -> list[dict[str, float | str]]:
"""One-factor-at-a-time queue sensitivity; descriptive, not inferential."""
settings = [("baseline", 1.0)]
for factor in ("event_rate", "budget", "cost_scale", "deadline_scale"):
settings.extend([(factor, 0.8), (factor, 1.2)])
records = []
for factor, multiplier in settings:
values = {
"event_rate": LAMBDA,
"budget": BUDGET,
"cost_scale": 1.0,
"deadline_scale": 1.0,
}
if factor != "baseline":
values[factor] *= multiplier
means = []
for n in TEAM_SIZES.astype(int):
scores = np.array([
simulate_session(rng, n, "additive", **values)
for _ in range(200)
])
means.append(float(scores[np.isfinite(scores)].mean()))
fitted = float(fit_curve(TEAM_SIZES, np.array(means))[2])
predicted = float(
values["budget"]
/ (values["event_rate"] * values["cost_scale"]
* np.dot(TYPE_PROBS, TYPE_COSTS))
)
records.append({
"factor": factor,
"multiplier": multiplier,
"event_rate": values["event_rate"],
"budget": values["budget"],
"cost_scale": values["cost_scale"],
"deadline_scale": values["deadline_scale"],
"predicted_midpoint": predicted,
"fitted_midpoint": fitted,
"bias": fitted - predicted,
})
return records
def main() -> None:
rng = np.random.default_rng(SEED)
all_values: dict[str, dict[int, np.ndarray]] = {}
results = {"direct_logistic_control": direct_logistic_control(rng)}
for condition in CONDITIONS:
condition_values = {}
means = []
for n in TEAM_SIZES.astype(int):
scores = np.array(
[simulate_session(rng, n, condition) for _ in range(SESSIONS_PER_SIZE)]
)
scores = scores[np.isfinite(scores)]
condition_values[n] = scores
means.append(float(scores.mean()))
all_values[condition] = condition_values
theta = fit_curve(TEAM_SIZES, np.array(means))
interval = bootstrap_midpoint(rng, condition_values)
results[condition] = {
"means": means,
"floor": float(theta[0]),
"span": float(theta[1]),
"midpoint": float(theta[2]),
"tau": float(theta[3]),
"midpoint_ci": list(map(float, interval)),
}
predicted_midpoint = BUDGET / (LAMBDA * float(np.dot(TYPE_PROBS, TYPE_COSTS)))
results["design"] = {
"seed": SEED,
"sessions_per_size": SESSIONS_PER_SIZE,
"duration_minutes": DURATION_MIN,
"lambda": LAMBDA,
"mean_service_cost": float(np.dot(TYPE_PROBS, TYPE_COSTS)),
"budget": BUDGET,
"predicted_midpoint": predicted_midpoint,
}
results["sensitivity_battery"] = sensitivity_battery(rng)
root = Path(__file__).resolve().parent
(root / "simulation_summary.json").write_text(json.dumps(results, indent=2) + "\n")
rows = ["n additive correlated batched switching"]
for index, n in enumerate(TEAM_SIZES.astype(int)):
rows.append(
f"{n} "
+ " ".join(f"{results[c]['means'][index]:.4f}" for c in CONDITIONS)
)
(root / "simulation_curves.dat").write_text("\n".join(rows) + "\n")
table = [
r"\begin{tabular}{lccc}",
r"\toprule",
r"Condition & nominal $n_0$ & fitted $n_{\mathrm{mid}}$ (95\% bootstrap) & diagnostic \\",
r"\midrule",
]
diagnostics = {
"additive": "reference",
"correlated": "shared bursts",
"batched": "joint service",
"switching": "agent-switch cost",
}
for condition in CONDITIONS:
record = results[condition]
lo, hi = record["midpoint_ci"]
pred = f"{predicted_midpoint:.2f}" if condition == "additive" else "4.35 (naive)"
label = condition.capitalize()
table.append(
f"{label} & {pred} & {record['midpoint']:.2f} "
f"[{lo:.2f}, {hi:.2f}] & {diagnostics[condition]} \\\\"
)
table.extend([r"\bottomrule", r"\end{tabular}"])
(root / "simulation_table.tex").write_text("\n".join(table) + "\n")
print(json.dumps(results, indent=2))
if __name__ == "__main__":
main()
|