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

Erlang equations




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!}



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


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.



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.






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