Hyperexponential distribution

Diagram showing queueing system equivalent of a hyperexponential distribution

In probability theory, a hyperexponential distribution is a continuous probability distribution whose probability density function of the random variable X is given by

f_{X}(x)=\sum _{{i=1}}^{n}f_{{Y_{i}}}(x)\;p_{i},

where each Y_i is an exponentially distributed random variable with rate parameter λ_i, and p_i is the probability that X will take on the form of the exponential distribution with rate λ_i.^[1] It is named the hyperexponential distribution since its coefficient of variation is greater than that of the exponential distribution, whose coefficient of variation is 1, and the hypoexponential distribution, which has a coefficient of variation smaller than one. While the exponential distribution is the continuous analogue of the geometric distribution, the hyperexponential distribution is not analogous to the hypergeometric distribution. The hyperexponential distribution is an example of a mixture density.

An example of a hyperexponential random variable can be seen in the context of telephony, where, if someone has a modem and a phone, their phone line usage could be modeled as a hyperexponential distribution where there is probability p of them talking on the phone with rate λ₁ and probability q of them using their internet connection with rate λ₂.

Properties

Since the expected value of a sum is the sum of the expected values, the expected value of a hyperexponential random variable can be shown as

E[X]=\int _{{-\infty }}^{\infty }xf(x)\,dx=\sum _{{i=1}}^{n}p_{i}\int _{0}^{\infty }x\lambda _{i}e^{{-\lambda _{i}x}}\,dx=\sum _{{i=1}}^{n}{\frac {p_{i}}{\lambda _{i}}}

and

E\!\left[X^{2}\right]=\int _{{-\infty }}^{\infty }x^{2}f(x)\,dx=\sum _{{i=1}}^{n}p_{i}\int _{0}^{\infty }x^{2}\lambda _{i}e^{{-\lambda _{i}x}}\,dx=\sum _{{i=1}}^{n}{\frac {2}{\lambda _{i}^{2}}}p_{i},

from which we can derive the variance:^[2]

\operatorname {Var}[X]=E\!\left[X^{2}\right]-E\!\left[X\right]^{2}=\sum _{{i=1}}^{n}{\frac {2}{\lambda _{i}^{2}}}p_{i}-\left[\sum _{{i=1}}^{n}{\frac {p_{i}}{\lambda _{i}}}\right]^{2}=\left[\sum _{{i=1}}^{n}{\frac {p_{i}}{\lambda _{i}}}\right]^{2}+\sum _{{i=1}}^{n}\sum _{{j=1}}^{n}p_{i}p_{j}\left({\frac {1}{\lambda _{i}}}-{\frac {1}{\lambda _{j}}}\right)^{2}.

The standard deviation exceeds the mean in general (except for the degenerate case of all the λs being equal), so the coefficient of variation is greater than 1.

The moment-generating function is given by

E\!\left[e^{{tx}}\right]=\int _{{-\infty }}^{\infty }e^{{tx}}f(x)\,dx=\sum _{{i=1}}^{n}p_{i}\int _{0}^{\infty }e^{{tx}}\lambda _{i}e^{{-\lambda _{i}x}}\,dx=\sum _{{i=1}}^{n}{\frac {\lambda _{i}}{\lambda _{i}-t}}p_{i}.

Fitting

A given probability distribution, including a heavy-tailed distribution, can be approximated by a hyperexponential distribution by fitting recursively to different time scales using Prony's method.^[3]

References

↑ Singh, L. N.; Dattatreya, G. R. (2007). "Estimation of the Hyperexponential Density with Applications in Sensor Networks". International Journal of Distributed Sensor Networks. 3 (3): 311. doi:10.1080/15501320701259925.
↑ H.T. Papadopolous; C. Heavey; J. Browne (1993). Queueing Theory in Manufacturing Systems Analysis and Design. Springer. p. 35. ISBN 9780412387203.
↑ Feldmann, A.; Whitt, W. (1998). "Fitting mixtures of exponentials to long-tail distributions to analyze network performance models" (PDF). Performance Evaluation. 31 (3–4): 245. doi:10.1016/S0166-5316(97)00003-5.

Probability distributions

List

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	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace 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	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-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 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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Hyperexponential distribution

Properties

Fitting

See also

References