Erlang distribution

**Erlang**
Probability density function
Cumulative distribution function
Parameters	$k > 0\,$ shape (real) $\lambda > 0\,$ 1/scale (real)
Support	$x \in [0; \infty)\!$
pdf	$x^{k-1}\lambda^k\frac{\exp\left(-\lambda x\right)}{(k-1)!\,}$
cdf	$\frac{\gamma(k, \lambda x)}{(k-1)!}$
Mean	$k/\lambda\,$
Median
Mode	$(k-1)/\lambda\,$ for $k \geq 1\,$
Variance	$k /\lambda^2\,$
Skewness	$\frac{2}{\sqrt{k}}$
Kurtosis	$\frac{6}{k}$
Entropy	$k/\lambda+(k-1)\ln(\lambda)+\ln((k-1)!)\,$ $+(1-k)\psi(k)\,$
mgf	$(1 - t/\lambda)^{-k}\,$ for $t < \lambda\,$
Char. func.	$(1 - it/\lambda)^{-k}\,$

In probability theory and statistics, the Erlang distribution is a continuous probability distribution developed by A. K. Erlang to predict waiting times in queueing systems, particularly in the case of telephone traffic engineering. The Erlang distribution is the distribution of the sum of k independent identically distributed random variables each having an exponential distribution. The probability density function of the Erlang distribution is

$f(x|k,\lambda)={\lambda^k x^{k-1} e^{-\lambda x} \over (k-1)!}\quad\mbox{for }x>0.$

The Erlang distribution is thus a special case of the gamma distribution for which k is a positive integer.

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 centers.

External links