< List of probability distributions < *Erlang distribution*

The **Erlang distribution** (sometimes called the *Erlang-k distribution*) was developed by A.K. Erlang, who worked as a telecommunications engineer for the Copenhagen Telephone Company in Denmark in 1917. . It is a continuous probability distribution that is was used historically to determine real life phenomena such as the number of phone calls which can be made simultaneously to switching station operators and other communication systems. Today, its applications have grown significantly and it is an important tool for many businesses and industries.

## What Is the Erlang Distribution?

The Erlang distribution describes the arrivals of customers or requests over a given period of time, as well as their relative service times. It is a generalization of the exponential distribution [2]: while exponential distribution models the time interval to the first event, the Erlang distribution models the time interval to the*k*th event. In other words, the Erlang distribution is a sum of *k* exponentially distributed variables [3].

The Erlang distribution is also special case of the gamma distribution, and arises when the gamma shape parameter is an integer [4].

The distribution is defined by two parameters, k and μ , where:

*k*= a shape parameter. This must be a positive integer (a whole number without a fractional part). If k =*1*, the Erlang distribution becomes the exponential distribution.- μ = a scale parameter, which must be a positive real number (a real number is any number found on the number line, including fractions). If μ = 2, the distribution becomes the chi-squared distribution.

The probability density function (PDF) is:

An equivalent PDF includes λ, a measure of rate (number of expected items or calls in a given time period), which is related to μ by μ = 1/λ.

Other properties:

## How Does It Work?

The Erlang distribution relies on two factors: arrival rates (λ) and service rates (μ). The arrival rate refers to how frequently customers arrive or requests are made over a given period of time; this value must be greater than 0 and less than or equal to μ (the service rate). The service rate refers to how quickly each customer’s request can be serviced; this value must also be greater than 0. By using these two values, you can calculate your desired outcomes—such as the average waiting time before someone’s request is serviced or how many requests will be completed within a certain amount of time—with accuracy and precision.

## Applications of the Erlang Distribution

Today, the applications of the Erlang distribution have spread far beyond its original purpose in telecommunications engineering, and it’s now used in many different industries and business settings across various sectors all around the world. For example, call centers use it to calculate call volumes so they can staff accordingly; banks utilize it when determining staffing levels for banking services such as ATMs; medical facilities use it to forecast patient wait times; airlines use it when managing their flight schedules; manufacturers employ it when scheduling production lines; and retail stores utilize it when planning sales promotions or special events such as Black Friday sales or holiday discounts.

## References

[1] Johndberglund, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0, via Wikimedia Commons

[2] Erlang Distribution. Retrieved August 11, 2023 from: https://courses.engr.illinois.edu/bioe310/sp2020/Lecture_13_Erlang_Gamma_Gaussian.pdf

[3] Matlab exercise

[4] Gamma (and Erlang) distribution. Retrieved August 11, 2023 from: https://valelab4.ucsf.edu/svn/3rdpartypublic/boost/libs/math/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/gamma_dist.html

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