Erlang distribution
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| Erlang distribution | |
|---|---|
| Name | Erlang distribution |
| Type | Continuous probability distribution |
| Parameters | shape k (integer), rate λ > 0 |
| Support | x ≥ 0 |
| (λ^k x^{k-1} e^{-λ x})/(k-1)! | |
| Cdf | 1 - e^{-λ x} Σ_{n=0}^{k-1} (λ x)^n / n! |
| Mean | k/λ |
| Variance | k/λ^2 |
Erlang distribution The Erlang distribution is a two-parameter continuous probability distribution widely used in Queueing theory, Teletraffic engineering, Stochastic processes, Reliability engineering, and Survival analysis. It models the waiting time until the k-th event in a Poisson process with rate λ, where k is a positive integer; historically it was developed by Agner Krarup Erlang for telephone traffic analysis at the A/S Øresund Telephone Company and influenced later work at institutions such as Bell Labs, AT&T, and Brown University.
Definition and basic properties
The Erlang distribution is defined for nonnegative real x with integer shape parameter k ∈ {1,2,3,...} and rate parameter λ > 0. It arises as the sum of k independent, identically distributed exponential random variables with rate λ, a connection leveraged in studies at Massachusetts Institute of Technology, Princeton University, and University of Cambridge. Key properties include nonnegativity of support, unimodality for k ≥ 2, and memorylessness reduction to the exponential case when k = 1; these features were exploited in modeling by researchers at Columbia University, Harvard University, and Stanford University.
Probability density and cumulative distribution
The probability density function (pdf) is f(x; k, λ) = (λ^k x^{k-1} e^{-λ x})/(k-1)! for x ≥ 0. The cumulative distribution function (cdf) is F(x; k, λ) = 1 − e^{-λ x} Σ_{n=0}^{k-1} (λ x)^n / n!. These closed forms were central to analytical work by teams at Bell Labs Research and implementations in numerical libraries at National Institute of Standards and Technology, European Organisation for Nuclear Research, and IBM Research. The survival function and hazard function are directly obtained from the cdf and pdf and are used in practice by analysts at World Health Organization, Centers for Disease Control and Prevention, and actuarial groups at Lloyd's of London.
Moments and characteristic functions
The n-th raw moment equals E[X^n] = (k+n-1)!/(λ^n (k-1)!). In particular, the mean is μ = k/λ and the variance is σ^2 = k/λ^2. The moment-generating function is M_X(t) = (λ/(λ − t))^k for t < λ, and the characteristic function is φ_X(t) = (λ/(λ − i t))^k. These expressions have been instrumental in theoretical developments at Institute for Advanced Study, National Academy of Sciences, and applications in signal processing at MIT Lincoln Laboratory and Los Alamos National Laboratory.
Relationships to other distributions
The Erlang distribution is a special case of the Gamma distribution when the shape parameter is integer-valued, and it reduces to the exponential distribution for k = 1. It relates to the Chi-squared distribution when scaled appropriately and to the Hypoexponential distribution and Phase-type distribution in Markovian constructions used by scholars at Imperial College London and École Polytechnique Fédérale de Lausanne. Convolutions of Erlang variables appear in analyses at Oxford University, University of California, Berkeley, and University of Michigan for compound processes, and the distribution features in limit theorems referenced by the Royal Statistical Society.
Parameter estimation and inference
Common estimation techniques include maximum likelihood estimation (MLE), method of moments, and Bayesian inference with conjugate priors; for integer k one may perform profile likelihood or information-criteria model selection used by statisticians at Johns Hopkins University, University of Washington, and Carnegie Mellon University. The MLE for λ given k has closed form λ̂ = k / x̄, while estimation of integer k often involves discrete optimization or likelihood-ratio tests as practiced in studies supported by National Institutes of Health, European Commission, and statistical groups at World Bank. Bayesian approaches use priors such as the Gamma distribution for λ and reversible-jump Markov chain Monte Carlo for k, methods advanced at University College London and École Normale Supérieure.
Applications and examples
Practical applications include modeling interarrival and service times in M/M/k queue systems, telephone and packet traffic modeling pioneered in studies at A/S Øresund Telephone Company and Bell Labs, component life modeling in reliability assessments at General Electric and Siemens, and clinical trial time-to-event analyses implemented by teams at Mayo Clinic and Johns Hopkins Hospital. Examples in operations research and telecommunications appear in textbooks from Prentice Hall and research from IEEE conferences, while implementations are available in software from R Project, Python Software Foundation packages, and statistical packages by SAS Institute and StataCorp.