from math import sqrt
from random import Random
def waits(arrivals, service_s):
    finish = 0.0; out = []
    for arrival in arrivals:
        finish = max(arrival, finish); out.append(finish - arrival); finish += service_s
    return out
def miss_interval(delays, deadline_s):
    hits = sum(delay > deadline_s for delay in delays); p = hits / len(delays)
    radius = 1.96 * sqrt(p * (1 - p) / len(delays))
    return p, max(0.0, p - radius), min(1.0, p + radius)
arrivals = [i * .001 for i in range(100)]
baseline = waits(arrivals, .0008)
counterfactual = waits(arrivals, .0012)
seed = 51; rng = Random(seed)
stochastic = waits([i * .001 + rng.uniform(-.0001, .0001) for i in range(400)], .00105)
interval = miss_interval(stochastic, .002)
assert max(baseline) == 0.0 and counterfactual[-1] > .019
assert interval[1] <= interval[0] <= interval[2] and seed == 51
print(f"PASS: 51 latency evidence stable={max(baseline):.6f}s overload={counterfactual[-1]:.6f}s miss95={interval}")