Erlang distribution | Vose Software

Erlang distribution

Format: Erlang(m, b)

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(a, b) 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.


The Erlang(m,b) distribution is the same as a Gamma(a,b) distribution but with the restriction that a (i.e. m) must be a positive integer. It doesn't therefore have any advantage of the Gamma distribution in its applications. In the old days of simulation, it was easier to generate values from an Erlang distribution by simply adding together m independent Exponential distributions, and it was also possible to compute the cumulative distribution function, but this advantage is now redundant. The Erlang distribution is included in ModelRisk for greater compatibility with other simulation tools.


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 centre to calculate the number of trunks one need 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 centre or similar situation.

ModelRisk functions added to Microsoft Excel for the Erlang distribution

VoseErlang generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseErlangObject constructs a distribution object for this distribution.

VoseErlangProb returns the probability density or cumulative distribution function for this distribution.

VoseErlangProb10 returns the log10 of the probability density or cumulative distribution function.

VoseErlangFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseErlangFitObject constructs a distribution object of this distribution fitted to data.

VoseErlangFitP returns the parameters of this distribution fitted to data.


Erlang distribution equations