View source for Erlang distribution

The probability distribution concentrated on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036180/e0361801.png" /> with density

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036180/e0361802.png" /></td> </tr></table>

where the integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036180/e0361803.png" /> and the real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036180/e0361804.png" /> are parameters. The characteristic function of the Erlang distribution has the form

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and the mathematical expectation and variance are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036180/e0361806.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036180/e0361807.png" />, respectively.

The Erlang distribution is special case of the [[Gamma-distribution|gamma-distribution]]: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036180/e0361808.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036180/e0361809.png" /> is the density of the gamma-distribution for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036180/e03618010.png" />, and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036180/e03618011.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036180/e03618012.png" /> the Erlang distribution is the same as the [[Exponential distribution|exponential distribution]] with parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036180/e03618013.png" />. The Erlang distribution with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036180/e03618014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036180/e03618015.png" /> can be found as the distribution of the sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036180/e03618016.png" /> independent random variables having the same exponential distribution with parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036180/e03618017.png" />. As <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036180/e03618018.png" />, the Erlang distribution tends to the [[Degenerate distribution|degenerate distribution]] concentrated at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036180/e03618019.png" />.

The selection of the Erlang distribution from the system of gamma-distributions is explained by its use in queueing theory. In many random queueing processes the Erlang distribution appears as the distribution of intervals among random events or as the distribution of the queueing time. Sometimes the Erlang distribution is defined as the gamma-distribution with the density

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It is named for A. Erlang, who was the first to construct mathematical models in queueing problems.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  T.L. Saaty,   "On elements of queueing theory with applications" , McGraw-Hill  (1961)</TD></TR></table>