The **Erlang distribution** is a probability distribution originally created by A.K. Erlang to model the number of telephone calls that an operator at a switching station may receive at once.

The distribution is used in telephone traffic engineering, queueing systems, mathematical biology, and other fields to model a variety of real-world phenomena.

**Properties of the Erlang Distribution**

The Erlang distribution has the following probability density function:

**f(x; k, μ) = x ^{k-1}e^{-x/μ} / μ^{k}(k-1)!**

where:

**k:**The shape parameter. This must be a positive integer.**μ:**The scale parameter. This must be a positive real number.

It turns out that the Erlang distribution is a special case of the Gamma distribution when the shape parameter *k* is restricted to only positive real integers.

Note that the scale parameter is the reciprocal of the rate parameter, λ, i.e. μ = 1/λ.

The Erlang distribution has the following properties:

**Mean:**k/λ**Mode:**(k-1)/λ**Variance:**k/λ^{2}**Skewness:**2/√k**Kurtosis:**6/k

The Erlang distribution has the following relationships with other distributions:

- When the shape parameter, k, is equal to 1 the Erlang distribution is equal to the exponential distribution.
- When the scale parameter, μ, is equal to 2 the Erlang distribution is equal to a Chi-Squared distribution with 2 degrees of freedom.

**Visualizing the Erlang Distribution**

The following plot shows the shape of the Erlang distribution when it takes on different parameters:

It’s interesting to see just how much the shape of the distribution changes depending on the values used for the shape and scale parameters.

**Note:** You can find the R code used to generate the plot of Erlang distributions here.

**Use Cases**

The Erlang distribution is used in a variety of real-world settings including:

**1. Call Centers**

The Erlang distribution is used to model the time in between incoming calls at a call center along with the expected number of calls.

This allows call centers to know what their staffing capacity should be during different times of the day so they can handle the incoming calls in a timely fashion without losing money by staffing too many employees during a given shift.

**2. Medical Settings**

The Erlang distribution is widely used to model cell cycle time distribution, which has a variety of different applications in medical settings.

**3. Retail Settings**

The Erlang distribution is used by retailers for modeling the frequency of interpurchase times by consumers.

This gives retailers and other businesses an idea of how often a given consumer is expected to purchase a product or service from them. This helps businesses with inventory control as well as staffing.

**Additional Resources**

An Introduction to the Normal Distribution

An Introduction to the Binomial Distribution

An Introduction to the Poisson Distribution