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Erlang distribution

The idea in one sentence

The Erlang distribution describes the waiting time until the kth event in a Poisson process with constant rate. At k=1 it is exponential; for integer k, it is a sum of independent exponential waits.

Support and interpretation

It is positive and is a Gamma distribution with integer shape. k controls how many stages must be completed and beta the scale of each stage.

Its place in the family

Erlang is integer-shape Gamma and a sum of exponentials. It also sits behind the Poisson process: counting events in an interval and waiting for event k are two views of the same process.

An applied reading

It suits a total time made of several approximately exponential phases. If phases are dependent or have different rates, generalized Gamma or hypoexponential models may be more faithful.

Do not read k as the number of simultaneous calls: it counts accumulated stages or events in the waiting time.

From telephone calls to chains of stages

Agner Krarup Erlang developed his theory while studying telephone traffic in Copenhagen. If calls arrive through a Poisson process, the time until call number k is Erlang. The same sum describes a task that must complete k independent exponential stages with a common rate.

Erlang therefore occupies a special place between Gamma and queueing models. Its shape must be an integer, and that restriction can represent actual phases rather than numerical convenience. As k grows, total time concentrates around its mean. Stages with different rates produce a hypoexponential law instead. If the stage count is only a metaphor, a Gamma model with free shape may be more honest.

Decision guide

A good candidate when: total waiting time is the sum of an integer number of independent exponential stages sharing one rate.

Compare it with: Gamma when the effective stage count need not be integer, and Hypoexponential when stages have different rates. That structure matters more than a small fit-score difference.

References

  • SciPy reference: scipy.stats.erlang — definition and parameterization
  • Johnson, N. L., Kotz, S. & Balakrishnan, N. (1994). Continuous Univariate Distributions, 2nd ed., Vol. 1. Wiley.
  • Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. Wiley.
  • Erlang, A. K. (1909). The theory of probabilities and telephone conversations. Nyt Tidsskrift for Matematik B, 20, 33–39.

Erlang Distribution: equations and calculator

Distribution defintion

X∼Erlang(k,β)X\sim\mathrm{Erlang}\left(k,\beta\right)

Distribution domain

x∈[0,∞)x\in [0,\infty)

Parameters domain and parameters constraints

k∈N+,β∈R+k\in\mathbb{N}^{+},\beta\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=P(k,xβ)=γ(k,xβ)(k−1)!F_{X}\left(x\right)=\text{P}(k,\frac{x}{\beta})=\frac{\gamma(k,\frac{x}{\beta})}{(k-1)!}

Probability density function

fX(x)=xk−1e−xββk(k−1)!f_{X}\left(x\right)=\frac{x^{k-1} e^{-\frac{x}{\beta}}}{\beta^k(k-1)!}

Percent point function/Sample

FX−1(u)=βP−1(k,u)F^{-1}_{X}\left(u\right)=\beta \text{P}^{-1}\left(k,u\right)

Non-central parametric moments

μn′=E[Xn]=∫0∞xnfX(x)dx=βnΓ(n+k)Γ(k)\mu'_{n}=E[X^n]=\int_{0}^{\infty}x^{n}f_{X}\left(x\right)dx=\beta^{n}\frac{\Gamma\left(n+k\right)}{\Gamma(k)}

Parametric mean

Mean(X)=μ1′\mathrm{Mean}(X)=\mu'_{1}

Parametric variance

Variance(X)=μ2′−μ1′2\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}

Parametric median

Median(X)=P(k,12β)\mathrm{Median}(X)=\text{P}(k,\frac{1}{2\beta})

Parametric mode

Mode(X)=β(k−1)\mathrm{Mode}(X)=\beta\left(k-1\right)

Additional information and definitions

β:Scale parameter\beta:\text{Scale parameter}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
P(a,x)=γ(a,x)Γ(a):Regularized lower incomplete gamma function\text{P}\left(a,x\right)=\frac{\gamma(a,x)}{\Gamma(a)}:\text{Regularized lower incomplete gamma function}
P−1(a,u):Inverse of regularized lower incomplete gamma function\text{P}^{-1}\left(a,u\right):\text{Inverse of regularized lower incomplete gamma function}
γ(a,x):Lower incomplete gamma function\gamma\left(a,x\right):\text{Lower incomplete gamma function}
Γ(x):Gamma function\Gamma\left(x\right):\text{Gamma function}