# Erlang distribution

Template:Probability distribution The Erlang distribution is a two parameter family of continuous probability distributions with support ${\displaystyle x\;\in \;(0,\,\infty )}$. The two parameters are:

The Erlang distribution with shape parameter ${\displaystyle k}$ equals to 1 simplifies to the exponential distribution. It is a special case of the Gamma distribution. It is the distribution of a sum of ${\displaystyle k}$ independent exponential variables with mean ${\displaystyle \mu }$.

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 fields of stochastic processes and of biomathematics.

## Characterization

### Probability density function

The probability density function of the Erlang distribution is

${\displaystyle f(x;k,\lambda )={\lambda ^{k}x^{k-1}e^{-\lambda x} \over (k-1)!}\quad {\mbox{for }}x,\lambda \geq 0,}$

The parameter k is called the shape parameter, and the parameter ${\displaystyle \lambda }$ is called the rate parameter.

An alternative, but equivalent, parametrization uses the scale parameter ${\displaystyle \mu }$, which is the reciprocal of the rate parameter (i.e., ${\displaystyle \mu =1/\lambda }$):

${\displaystyle f(x;k,\mu )={\frac {x^{k-1}e^{-{\frac {x}{\mu }}}}{\mu ^{k}(k-1)!}}\quad {\mbox{for }}x,\mu \geq 0.}$

When the scale parameter ${\displaystyle \mu }$ equals 2, the distribution simplifies to the chi-squared distribution with 2k degrees of freedom. It can therefore be regarded as a generalized chi-squared distribution for even numbers of degrees of freedom.

Because of the factorial function in the denominator, the Erlang distribution is only defined when the parameter k is a positive integer. In fact, this distribution is sometimes called the Erlang-k distribution (e.g., an Erlang-2 distribution is an Erlang distribution with k = 2). The gamma distribution generalizes the Erlang distribution by allowing k to be any real number, using the gamma function instead of the factorial function.

### Cumulative distribution function (CDF)

The cumulative distribution function of the Erlang distribution is

${\displaystyle F(x;k,\lambda )={\frac {\gamma (k,\lambda x)}{(k-1)!}},}$

where ${\displaystyle \gamma ()}$ is the lower incomplete gamma function. The CDF may also be expressed as

${\displaystyle F(x;k,\lambda )=1-\sum _{n=0}^{k-1}{\frac {1}{n!}}e^{-\lambda x}(\lambda x)^{n}.}$

## Notes

1. Template:Cite doi
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4. Banneheka BMSG, Ekanayake GEMUPD (2009) "A new point estimator for the median of gamma distribution". Viyodaya J Science, 14:95-103
5. http://www.xycoon.com/erlang_random.htm
6. C. Chatfield and G.J. Goodhardt: “A Consumer Purchasing Model with Erlang Interpurchase Times”; Journal of the American Statistical Association, Dec. 1973, Vol.68, pp.828-835
7. Cox, D.R. (1967) Renewal Theory, p20, Methuen.