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Erlang 3P distribution

Quick view

The three-parameter Erlang distribution adds a location shift to the Erlang waiting-time model. It represents several random stages that begin after a fixed delay or known minimum time.

If you are coming from another distribution

It reduces to the two-parameter Erlang when loc=0. It is also a shifted integer-shape Gamma, but not a noncentral Gamma: the delay is added after the random waiting time.

History and terminology

The 3P label belongs to a software parameterization. The original motivation remains Erlang’s telephone-traffic work, although the model is now used in reliability and queues.

A familiar situation

In a service system, loc may represent setup, warm-up, or unavoidable transport before stochastic stages. This is a strong interpretation: without a physical delay, the extra flexibility may be unnecessary.

Fitting with care

Location is close to the sample minimum and can be hard to estimate with few observations; compare it with an unshifted Erlang.

The queue starts after a fixed delay

Erlang 3P separates a duration into two parts: a fixed delay loc and a random sum of k stages. A service process might first require registration of nearly constant duration and then pass through several phases with exponential times.

That separation should exist in the mechanism or be supported by the experimental design. If the initial phase also varies, treating it as constant pushes its variation into the other parameters. Moreover, loc does not make the Markov chain of stages more complex; it only moves its origin. Interpreting k as a number of phases still requires an integer value and comparable stage rates.

Decision guide

A good candidate when: Erlang stages begin after a fixed delay or physically identifiable minimum time.

Compare it with: unshifted Erlang and 3P Gamma. Separate deterministic delay from random stages; estimating both from a small sample can confound the two effects.

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.
  • Erlang, A. K. (1909). The theory of probabilities and telephone conversations. Nyt Tidsskrift for Matematik B, 20, 33–39.
  • Evans, M., Hastings, N. & Peacock, B. (2000). Statistical Distributions, 3rd ed. Wiley.

Erlang 3P Distribution: equations and calculator

Distribution defintion

X∼Erlang3P(k,β,Loc)X\sim\mathrm{Erlang_{3P}}\left(k,\beta,\text{Loc}\right)

Distribution domain

x∈[Loc,∞)x\in\left[\text{Loc},\infty\right)

Parameters domain and parameters constraints

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

Cumulative distribution function

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

Probability density function

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

Percent point function/Sample

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

Non-central parametric moments

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

Parametric mean

Mean(X)=Loc+μ~1′\mathrm{Mean}(X)=\text{Loc}+\tilde{\mu}'_{1}

Parametric variance

Variance(X)=μ~2′−μ~1′2\mathrm{Variance}(X)=\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1}

Parametric skewness

Skewness(X)=μ~3′−3μ~2′μ~1′+2μ~1′3(μ~2′−μ~1′2)1.5\mathrm{Skewness}(X)=\frac{\tilde{\mu}'_{3}-3\tilde{\mu}'_{2}\tilde{\mu}'_{1}+2\tilde{\mu}'^{3}_{1}}{(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})^{1.5}}

Parametric kurtosis

Kurtosis(X)=μ~4′−4μ~1′μ~3′+6μ~1′2μ~2′−3μ~1′4(μ~2′−μ~1′2)2\mathrm{Kurtosis}(X)=\frac{\tilde{\mu}'_{4}-4\tilde{\mu}'_{1}\tilde{\mu}'_{3}+6\tilde{\mu}'^{2}_{1}\tilde{\mu}'_{2}-3\tilde{\mu}'^{4}_{1}}{(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})^{2}}

Parametric median

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

Parametric mode

Mode(X)=Loc+β⋅(k−1)\mathrm{Mode}(X)=\text{Loc}+\beta\cdot\left(k-1\right)

Additional information and definitions

X~∼Erlang(k,β)\tilde{X}\sim\mathrm{Erlang}\left(k,\beta\right)
Loc:Location parameter\text{Loc}:\text{Location parameter}
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}