Erlang distribution

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Template:Probability distribution{(k-1)!\,}</math>|

 cdf        =<math>1-\frac{\gamma(k, \lambda x)}{(k-1)!}</math>|
 mean       =<math>k/\lambda\,</math>|
 median     =|
 mode       =<math>(k-1)/\lambda\,</math> for <math>k \geq 1\,</math> |
 variance   =<math>k /\lambda^2\,</math>|
 skewness   =<math>\frac{2}{\sqrt{k}}</math>|
 kurtosis   =<math>\frac{6}{k}</math>|
 entropy    =<math>k/\lambda+(k-1)\ln(\lambda)+\ln((k-1)!)\,</math>
<math>+(1-k)\psi(k)\,</math>|
 mgf        =<math>(1 - t/\lambda)^{-k}\,</math> for <math>t < \lambda\,</math>|
 char       =<math>(1 - it/\lambda)^{-k}\,</math>|

}} The Erlang distribution is a continuous probability distribution with wide applicability primarily due to its relation to the exponential and Gamma distributions. The Erlang distribution was developed by A. K. Erlang to examine the number of telephone calls which might be made at the same time to the operators of the switching stations. This work on telephone traffic engineering has been expanded to consider waiting times in queueing systems in general. The distribution is now used in the field of stochastic processes.

1 Overview
2 Specification of the Erlang distribution
- 2.1 Probability density function
- 2.2 Cumulative distribution function
3 Occurrence
4 See also
5 External links

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Overview

The Erlang distribution is a continuous distribution, which has a positive value for all real numbers greater than zero, and is given by two parameters: the shape <math>k</math>, which is an integer, and the rate <math>\lambda</math>, which is a real. The distribution is sometimes defined using the inverse of the rate parameter, the scale <math>\theta</math>.

When the shape parameter <math>k</math> equals 1, the distribution simplifies to the exponential distribution.

The Erlang distribution is a special case of the Gamma distribution where the shape parameter <math>k</math> is an integer. In the Gamma distribution, this parameter is a real.

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Specification of the Erlang distribution

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Probability density function

The probability density function of the Erlang distribution is

<math>f(x; k,\lambda)={\lambda^k x^{k-1} e^{-\lambda x} \over (k-1)!}\quad\mbox{for }x>0.</math>

where e is the base of the natural logarithm and <math>!</math> is the factorial function. The parameter <math>k</math> is called the shape parameter and the parameter <math>\lambda</math> is called the rate parameter. An alternative, but equivalent, parametrization uses the scale parameter <math>\theta</math> which is simply the inverse of the rate parameter (i.e. <math>\theta = 1/\lambda</math>):

<math>f(x; k,\theta)=\frac{ x^{k-1} e^{-\frac{x}{\theta}} }{\theta^k(k-1)!}\quad\mbox{for }x>0.</math>

Because of the factorial function in the denominator, the Erlang distribution only defined when the parameter k is a positive integer. The Gamma distribution generalizes the Erlang by allowing its first parameter to be a real, using the gamma function instead of the factorial function.

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Cumulative distribution function

The cumulative distribution function of the Erlang distribution is

<math>F(x; k,\lambda) = 1- \frac{\gamma(k, \lambda x)}{(k-1)!}</math>

where <math>\gamma()</math> is the incomplete gamma function.

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Occurrence

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Waiting times

There are two commonly used versions of the Erlang distribution, depending on the traffic assumptions modelled:

Erlang B distribution - Does not allow queuing of blocked calls
Erlang C distribution - Allows Unlimited queuing of blocked calls until they are served

The Erlang B and C distributions are still in everyday use for traffic modelling for applications such as the design of call centres.

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Compartment models

The Erlang distribution also occurs as a description of the rate of transition of elements through a system of compartments. Such systems are widely used in biology and ecology.

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Stochastic processes

The Erlang distribution is the distribution of the sum of k independent identically distributed random variables each having an exponential distribution.

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