🔗 The 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", ${\displaystyle \mu ,}$ the reciprocal of the rate, is sometimes used instead.

The Erlang distribution with shape parameter ${\displaystyle k=1}$ 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 ${\displaystyle 1/\lambda }$ each.

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.