Hyper-Erlang distribution

Diagram showing queueing system equivalent of a hyper-Erlang distribution

In probability theory, a hyper-Erlang distribution is a continuous probability distribution which takes a particular Erlang distribution E_i with probability p_i. A hyper-Erlang distributed random variable X has probability density function given by

A(x) = \sum_{i=1}^n p_i E_{l_i}(x)

where each p_i > 0 with the p_i summing to 1 and each of the E_{l_i} being an Erlang distribution with l_i stages each of which has parameter λ_i.^[1]^[2]^[3]

References

↑ Bocharov, P. P.; D'Apice, C.; Pechinkin, A. V. (2003). "2. Defining parameters of queueing systems". Queueing Theory. doi:10.1515/9783110936025.61. ISBN 9783110936025.
↑ Yuguang Fang; Chlamtac, I. (1999). "Teletraffic analysis and mobility modeling of PCS networks". IEEE Transactions on Communications 47 (7): 1062. doi:10.1109/26.774856.
↑ Fang, Y. (2001). "Hyper-Erlang Distribution Model and its Application in Wireless Mobile Networks". Wireless Networks (Kluwer Academic Publishers) 7 (3): 211–219. doi:10.1023/A:1016617904269.

Probability distributions

Discrete univariate with finite support

Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support

beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine Reciprocal Triangular U-quadratic uniform Wigner semicircle

[[List of probability distributions#Supported_on_semi-infinite_intervals.2C_usually_.5B0.2C.E2.88.9E.29|Continuous univariate supported on a semi-infinite interval, usually [0,∞)]]

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson S_U Landau Laplace Asymmetric Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled inverse gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional

Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular

Degenerate discrete degenerate Dirac delta function Singular Cantor

Families

Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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Hyper-Erlang distribution

See also

References