**Erlang(m,b) - no Crystal Ball distribution = Gamma(0,b,a) with a = m, when a is an integer**

The Erlang distribution (or m-Erlang distribution) is a probability distribution developed by A. K. Erlang. It is a special case of the Gamma distribution. A Gamma(0, b, a) distribution is equal to an Erlang(m,b) distribution with a = m, when a is an integer. Examples of the Erlang distribution are given below:

Unlike the Gamma distribution, the Erlang does have a cumulative distribution function.

F(x)=1-exp\Big(\frac{-x}{\beta}\Big)\displaystyle\sum_{i-0}^{m-1}\frac{x^{i}}{\beta ^{i}i!} |

#### Generation

The Erlang distribution is the same as a Gamma distribution, as long as a is an integer.

#### Uses

The Erlang distribution is used to predict waiting times in queuing systems, etc. where a Poisson process is in operation, in the same way as a Gamma distribution.

#### Comments

A.K. Erlang worked a lot in traffic modeling. There are thus two other Erlang distributions, both used in modeling traffic:

Erlang B distribution: this is the easier of the two, and can be used, for example, in a call center to calculate the number of trunks one needs to carry a certain amount of phone traffic with a certain "target service".

Erlang C distribution: this formula is much more difficult and is often used, for example, to calculate how long callers will have to wait before being connected to a human in a call center or similar situation.