.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/probability/queueing/plot_01_erlang_b.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_probability_queueing_plot_01_erlang_b.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_probability_queueing_plot_01_erlang_b.py:


Erlang's loss formula: how many telephone lines?
========================================================

For Poisson call arrivals offering :math:`A` erlangs of traffic to
:math:`c` lines, a call is lost with probability :math:`B(c, A)`.
Erlang's formula answers the engineer's question: how many lines keep
losses below 1%?

.. GENERATED FROM PYTHON SOURCE LINES 12-17

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.probability import erlang_b








.. GENERATED FROM PYTHON SOURCE LINES 18-20

Blocking probability against the number of lines
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 20-34

.. code-block:: Python


    servers = np.arange(0, 41)
    fig, ax = plt.subplots()
    for load in (2.0, 5.0, 10.0, 20.0):
        blocking = [erlang_b(load, int(c)) for c in servers]
        ax.semilogy(servers, blocking, label=f"A = {load:g} erlangs")
        needed = next(int(c) for c, b in zip(servers, blocking) if b < 0.01)
        print(f"A = {load:>4g} erlangs: {needed} lines keep blocking below 1%")
    ax.axhline(0.01, color="0.5", ls="--")
    ax.set_ylim(1e-6, 1.2)
    ax.set_xlabel("lines c")
    ax.set_ylabel("blocking probability B(c, A)")
    ax.legend()
    ax.set_title("Erlang B formula (1917)")



.. image-sg:: /api/gallery/probability/queueing/images/sphx_glr_plot_01_erlang_b_001.png
   :alt: Erlang B formula (1917)
   :srcset: /api/gallery/probability/queueing/images/sphx_glr_plot_01_erlang_b_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    A =    2 erlangs: 7 lines keep blocking below 1%
    A =    5 erlangs: 11 lines keep blocking below 1%
    A =   10 erlangs: 18 lines keep blocking below 1%
    A =   20 erlangs: 30 lines keep blocking below 1%

    Text(0.5, 1.0, 'Erlang B formula (1917)')




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

   **Total running time of the script:** (0 minutes 0.058 seconds)


.. _sphx_glr_download_api_gallery_probability_queueing_plot_01_erlang_b.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_01_erlang_b.ipynb <plot_01_erlang_b.ipynb>`

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

      :download:`Download Python source code: plot_01_erlang_b.py <plot_01_erlang_b.py>`

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

      :download:`Download zipped: plot_01_erlang_b.zip <plot_01_erlang_b.zip>`


.. only:: html

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

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