Erlang distribution

Erlang distribution which is a continuous probability distribution, a generalization of the exponential distribution, and a special case of the gamma - distribution. You Agner Krarup Erlang was developed by the statistical modeling of the interval lengths between phone calls.

Erlang distribution which is used in the queuing theory to detect the distribution of the amount of time between events of a Poisson process, for example the arrival of customers, as well as in quality assurance for the description of lives. In call centers, this distribution is used for workforce planning to determine the required number of agents on the basis of the expected call volume in the time interval.

The Erlang distribution density provides the distribution of the probability that after the lapse of the location or time interval the -th event occurs if one expects events per unit interval (see derivation ). It describes a chain of events seriatim. The most likely distance to the -th event ( mode ) is smaller than the average value ( expected value ) because shorter event intervals occur frequently. If one fills the sorted by magnitude intervals of the respective individual events in a histogram, so this shows, accordingly, an exponential distribution.

  • 4.1 Relationship to the exponential distribution
  • 4.2 Relationship to the Poisson distribution
  • 4.3 Relationship to the continuous uniform distribution
  • 4.4 Relationship to the gamma distribution


The Erlang distribution with parameters ( a positive real number ), and ( a natural number ) is a special gamma distribution characterized by the density function

Is determined, and is different from the general gamma distribution by limiting to natural numbers in the second parameter.

Erlangverteilte a random variable, the probability of that is within the interval of the distribution function

Been called with, or the incomplete gamma function.


It is a local or a time variable and the constant occurrence frequency of events in the unit interval, then the distribution of the size of the clearance is sought until the occurrence of the -th event. What is the size of the possible areas for events that have already occurred distributed?

This distribution results from the Poisson distribution: the probability of the occurrence of events within the interval

One does not now consider this expression for fixed range as a function of the probability of events, but at a given as a function of the probability of the field size, the result after normalization still necessary ( integral of density equal to one)

The probability density

The Erlang distribution.


As a erlangverteilte random variable is the sum of independent and identically exponentially distributed with parameter random variables results in the following properties.

Expected value

The Erlang distribution has the expected value


Analogously, the variance to


Mode, the maximum density, is attached

Characteristic function

From the characteristic function of an exponentially distributed random variables we obtain the erlangverteilten a random variable:

Moment generating function

Analogously for the moment generating function


Is the entropy of the Erlang distribution

Where ψ ( p) is the digamma - function referred.

Relations with other distributions

Relationship to the exponential distribution

  • The Erlang distribution is a generalization of the exponential distribution, since it goes on for in this.
  • There are many, all exponentially distributed with the same parameter random variables are stochastically independent given. Then the random variable is Erlang distributed with parameters and.

Relationship to the Poisson distribution

  • For a Poisson process the random number of events up to a defined point in time using Poisson distribution is determined, the random time until th event is Erlang distributed. In this case, the Erlang distribution changes into an exponential distribution with which the time to the first random event and the time can be determined between two consecutive events.
  • The Erlang distribution is the conjugate to the Poisson distribution distribution.
  • Is valid for the distribution functions of the Erlang distribution and the Poisson distribution

Relationship to the continuous uniform distribution

An Erlang distribution can be generated as a convolution of uniformly continuous distributed functions:

Relationship to a gamma distribution,

The explanation of this relationship is found at the beginning of the article in the definition.