Erlang distribution

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

• a positive integer ${\displaystyle k,}$ the "shape", and
• a positive real number ${\displaystyle \lambda ,}$ the "rate". The "scale", ${\displaystyle \beta ,}$ the reciprocal of the rate, is sometimes used instead.

Erlang

Probability density function
Cumulative distribution function
Parameters	${\displaystyle k\in \{1,2,3,\ldots \},}$ shape ${\displaystyle \lambda \in (0,\infty ),}$ rate alt.: ${\displaystyle \beta =1/\lambda ,}$ scale
Support	${\displaystyle x\in [0,\infty )}$
PDF	${\displaystyle {\frac {\lambda ^{k}x^{k-1}e^{-\lambda x}}{(k-1)!}}}$
CDF	${\displaystyle P(k,\lambda x)={\frac {\gamma (k,\lambda x)}{(k-1)!}}=1-\sum _{n=0}^{k-1}{\frac {1}{n!}}e^{-\lambda x}(\lambda x)^{n}}$
Mean	${\displaystyle {\frac {k}{\lambda }}}$
Median	No simple closed form
Mode	${\displaystyle {\frac {1}{\lambda }}(k-1)}$
Variance	${\displaystyle {\frac {k}{\lambda ^{2}}}}$
Skewness	${\displaystyle {\frac {2}{\sqrt {k}}}}$
Excess kurtosis	${\displaystyle {\frac {6}{k}}}$
Entropy	${\displaystyle (1-k)\psi (k)+\ln \left[{\frac {\Gamma (k)}{\lambda }}\right]+k}$
MGF	${\displaystyle \left(1-{\frac {t}{\lambda }}\right)^{-k}}$ for ${\displaystyle t<\lambda }$
CF	${\displaystyle \left(1-{\frac {it}{\lambda }}\right)^{-k}}$

The Erlang distribution is the distribution of a sum of ${\displaystyle k}$ independent exponential variables with mean ${\displaystyle 1/\lambda }$ each. Equivalently, it is the distribution of the time until the kth event of a Poisson process with a rate of ${\displaystyle \lambda }$. The Erlang and Poisson distributions are complementary, in that while the Poisson distribution counts the number of events that occur in a fixed amount of time, the Erlang distribution counts the amount of time until the occurrence of a fixed number of events. When ${\displaystyle k=1}$, the distribution simplifies to the exponential distribution. The Erlang distribution is a special case of the gamma distribution wherein the shape of the distribution is discretised.

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 also used in the field of stochastic processes.