.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/reliability/systems/plot_03_k_out_of_n.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_reliability_systems_plot_03_k_out_of_n.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_reliability_systems_plot_03_k_out_of_n.py:


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 :math:`k`-out-of-:math:`n` system,
which works if at least :math:`k` of :math:`n` identical components do,
spans the range from parallel (:math:`k = 1`) to series (:math:`k = n`), and
its reliability is a binomial tail
(:func:`~engineeringkit.reliability.k_of_n_reliability`). The example checks
it by enumerating component states and recovers the mean life of each
architecture.

.. GENERATED FROM PYTHON SOURCE LINES 17-25

.. code-block:: Python

    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








.. GENERATED FROM PYTHON SOURCE LINES 26-31

Enumerate the structure function
--------------------------------

Four engines, aircraft safe with any :math:`k` working, each with
reliability 0.95 for the flight.

.. GENERATED FROM PYTHON SOURCE LINES 31-39

.. code-block:: Python


    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)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    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




.. GENERATED FROM PYTHON SOURCE LINES 40-45

Mean life of each architecture
------------------------------

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

.. GENERATED FROM PYTHON SOURCE LINES 45-60

.. code-block:: Python


    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()



.. image-sg:: /api/gallery/reliability/systems/images/sphx_glr_plot_03_k_out_of_n_001.png
   :alt: plot 03 k out of n
   :srcset: /api/gallery/reliability/systems/images/sphx_glr_plot_03_k_out_of_n_001.png
   :class: sphx-glr-single-img






.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.070 seconds)


.. _sphx_glr_download_api_gallery_reliability_systems_plot_03_k_out_of_n.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/reliability/systems/plot_03_k_out_of_n.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_03_k_out_of_n.ipynb <plot_03_k_out_of_n.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_03_k_out_of_n.py <plot_03_k_out_of_n.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_03_k_out_of_n.zip <plot_03_k_out_of_n.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_