Erlang Distribution

The Erlang distribution, also known as the Erlang-k distribution, models the waiting time until the $n$th event occurs in a Poisson process. It is fundamental in queueing theory and telecommunications engineering.

The Erlang distribution has parameters $n$ (shape) and $\theta$ (scale). Its probability density function (PDF) is:

The cumulative distribution function (CDF) is:

Parameter constraints: $n\in {\mathbb{N}}^{+}$ and $\theta \in {\mathbb{R}}^{+}$

The Erlang distribution finds widespread use in many practical applications:

• In call centers, it models both customer waiting times and service durations in multi-server queuing systems.

• Reliability engineers use it to analyze the time until system failure when multiple components or stages are involved.

• In computer networks, it helps predict packet transmission delays across multiple network nodes.

• Insurance companies apply it to model the total processing time of claims that go through multiple stages of review.

The Erlang distribution has several important mathematical relationships with other probability distributions:

• The Erlang distribution is a special case of the Gamma distribution, where the shape parameter is constrained to be a positive integer.

• When you sum $n$ independent random variables that each follow an Exponential($\theta$) distribution, the resulting sum follows an Erlang($n,\theta$) distribution.

• In the special case where $n=1$, the Erlang distribution is identical to the exponential distribution with the same scale parameter.

The Erlang distribution is implemented by the ErlangDistribution class. It has one constructor which takes the number of occurrences and the waiting time (or the shape and scale parameters) as arguments. The first argument must be an integer. The following constructs an Erlang distribution with $n$ = 10 and waiting time 7.6:

var erlang = new ErlangDistribution(10, 7.6);

Dim erlang = New ErlangDistribution(10, 7.6)

No code example is currently available or this language may not be supported.

let erlang = ErlangDistribution(10, 7.6)

The ErlangDistribution class has two specific properties, ShapeParameter, which returns the shape parameter of the distribution, and ScaleParameter, which returns the scale parameter.

ErlangDistribution has one static ($Shared$ in Visual Basic) method, Sample, which generates a random sample using a user-supplied uniform random number generator.

var random = new Pcg32();
double sample = ErlangDistribution.Sample(random, 10, 7.6);

Dim random = New Pcg32()
Dim sample = ErlangDistribution.Sample(random, 10, 7.6)

No code example is currently available or this language may not be supported.

let random = Pcg32()
let sample = ErlangDistribution.Sample(random, 10.0, 7.6)

The above example uses the Pcg32 to generate uniform random numbers.

For more information on the Erlang distribution, refer to the following sources:

• Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). "Continuous Univariate Distributions, Volume 1", Chapter 17. Wiley Series in Probability and Statistics.

• Forbes, C., Evans, M., Hastings, N., & Peacock, B. (2011). "Statistical Distributions", Chapter 15. Wiley Series in Probability and Statistics.

Other Resources