.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/reliability/systems/plot_01_von_neumann_redundancy.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_01_von_neumann_redundancy.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_01_von_neumann_redundancy.py:


Von Neumann's reliable machines from unreliable parts (1956)
============================================================

John von Neumann's 1956 lectures asked how a computer, or a brain, can
work when every component is unreliable. Two answers came out. Active
parallel redundancy, :math:`1 - (1 - r)^n`
(:func:`~engineeringkit.reliability.parallel_reliability`), survives if any
copy survives. For logic, where a wrong output is as bad as none, he used
*majority voting*: three copies and a voter give :math:`3r^2 - 2r^3`, and
voting on voters drives reliability toward one, but only if each part is
better than a coin toss, :math:`r > 1/2`.

.. GENERATED FROM PYTHON SOURCE LINES 16-21

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.reliability import exponential_reliability, k_of_n_reliability, parallel_reliability








.. GENERATED FROM PYTHON SOURCE LINES 22-26

Parallel copies over a mission
------------------------------

Each copy fails at :math:`10^{-3}` per hour (time unit: hours).

.. GENERATED FROM PYTHON SOURCE LINES 26-35

.. code-block:: Python


    t = np.linspace(0, 3000, 200)
    r = exponential_reliability(t, 1e-3)
    fig, axes = plt.subplots(1, 2, figsize=(10, 3.8))
    for n in (1, 2, 3, 5):
        axes[0].plot(t, [parallel_reliability(*[ri] * n) for ri in r], label=f"{n} in parallel")
    axes[0].set(xlabel="time [h]", ylabel="system reliability", title="Active parallel redundancy")
    axes[0].legend()




.. image-sg:: /api/gallery/reliability/systems/images/sphx_glr_plot_01_von_neumann_redundancy_001.png
   :alt: Active parallel redundancy
   :srcset: /api/gallery/reliability/systems/images/sphx_glr_plot_01_von_neumann_redundancy_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x11120df90>



.. GENERATED FROM PYTHON SOURCE LINES 36-40

Majority voting, applied again and again
----------------------------------------

Each level of the hierarchy is a 2-out-of-3 vote on the level below.

.. GENERATED FROM PYTHON SOURCE LINES 40-61

.. code-block:: Python


    p = np.linspace(0, 1, 201)
    levels = [p]
    for _ in range(4):
        levels.append(np.array([k_of_n_reliability(3, 2, float(x)) for x in levels[-1]]))
    for i, values in enumerate(levels):
        axes[1].plot(p, values, label=f"{i} levels of voting")
    axes[1].set(xlabel="component reliability r", ylabel="system reliability", title="Von Neumann's majority voting")
    axes[1].legend()
    fig.tight_layout()
    for r0 in (0.4, 0.5, 0.9):
        chain = [r0]
        for _ in range(4):
            chain.append(k_of_n_reliability(3, 2, chain[-1]))
        print(f"r = {r0}: " + " -> ".join(f"{x:.4f}" for x in chain))

    assert np.isclose(k_of_n_reliability(3, 2, 0.5), 0.5)
    assert levels[-1][180] > 0.9999 and levels[-1][80] < levels[0][80] / 3
    assert parallel_reliability(0.5, 0.5, 0.5) == 0.875

    plt.show()




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

 .. code-block:: none

    r = 0.4: 0.4000 -> 0.3520 -> 0.2845 -> 0.1967 -> 0.1009
    r = 0.5: 0.5000 -> 0.5000 -> 0.5000 -> 0.5000 -> 0.5000
    r = 0.9: 0.9000 -> 0.9720 -> 0.9977 -> 1.0000 -> 1.0000





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

   **Total running time of the script:** (0 minutes 0.055 seconds)


.. _sphx_glr_download_api_gallery_reliability_systems_plot_01_von_neumann_redundancy.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_01_von_neumann_redundancy.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_von_neumann_redundancy.ipynb <plot_01_von_neumann_redundancy.ipynb>`

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

      :download:`Download Python source code: plot_01_von_neumann_redundancy.py <plot_01_von_neumann_redundancy.py>`

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

      :download:`Download zipped: plot_01_von_neumann_redundancy.zip <plot_01_von_neumann_redundancy.zip>`


.. only:: html

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

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