Birnbaum, Esary and Saunders’s k-out-of-n systems (1961)

Birnbaum, Esary and Saunders (1961) put system reliability on a general footing: a structure function says which combinations of working components keep the system working. The \(k\)-out-of-\(n\) system, which works if at least \(k\) of \(n\) identical components do, spans the range from parallel (\(k = 1\)) to series (\(k = n\)), and its reliability is a binomial tail (k_of_n_reliability()). The example checks it by enumerating component states and recovers the mean life of each architecture.

import itertools

import matplotlib.pyplot as plt
import numpy as np
from scipy.integrate import quad

from engineeringkit.reliability import exponential_reliability, k_of_n_reliability

Enumerate the structure function

Four engines, aircraft safe with any \(k\) working, each with reliability 0.95 for the flight.

n, r = 4, 0.95
for k in range(1, n + 1):
    states = itertools.product([0, 1], repeat=n)
    exact = sum(np.prod([r if s else 1 - r for s in state]) for state in states if sum(state) >= k)
    print(f"{k}-out-of-{n}: binomial {k_of_n_reliability(n, k, r):.6f}, enumeration {exact:.6f}")
    assert np.isclose(k_of_n_reliability(n, k, r), exact)

1-out-of-4: binomial 0.999994, enumeration 0.999994
2-out-of-4: binomial 0.999519, enumeration 0.999519
3-out-of-4: binomial 0.985981, enumeration 0.985981
4-out-of-4: binomial 0.814506, enumeration 0.814506

Mean life of each architecture

With exponential components the \(k\)-out-of-\(n\) MTTF is \(\frac{1}{\lambda}\sum_{i=k}^{n} 1/i\). Time unit: hours.

lam = 1e-3
t = np.linspace(0, 4000, 300)
fig, ax = plt.subplots(figsize=(6.5, 3.8))
for k in range(1, n + 1):
    R = [k_of_n_reliability(n, k, float(x)) for x in exponential_reliability(t, lam)]
    mttf = quad(lambda s, k=k: k_of_n_reliability(n, k, float(exponential_reliability(s, lam))), 0, np.inf)[0]
    exact = sum(1 / i for i in range(k, n + 1)) / lam
    ax.plot(t, R, label=f"{k}-out-of-{n}, MTTF {mttf:.0f} h")
    assert np.isclose(mttf, exact)
ax.set(xlabel="time [h]", ylabel="system reliability")
ax.legend()
fig.tight_layout()

plt.show()