Hyperexponential distribution

The solid blue line shows the probability density of a hyperexponential distribution using the example p ₁ = 0.9, p ₂ = 0.1, λ ₁ = 1 and λ ₂ = 20.

The hyperexponential distribution is a continuous probability distribution . In clear terms, it is a superposition of several exponential distributions .

definition

Let be ( partly ) independent , exponentially distributed random variables with rates and let be probabilities whose sum equals 1. Then the random variable is called hyperexponentially distributed if it has the following probability density : ${\displaystyle Y_{i}}$ ${\displaystyle i=1,\dotsc ,N}$ ${\displaystyle \lambda _{i}}$ ${\displaystyle p_{i}}$ ${\displaystyle X}$

${\displaystyle f_{X}(x)=\sum _{i=1}^{N}p_{i}f_{Y_{i}}(x)={\begin{cases}\sum \limits _{i=1}^{N}p_{i}\lambda _{i}e^{-\lambda _{i}\,x}&x\geq 0,\\0&x<0.\end{cases}}}$

Classification and remarks

In the case of an exponential distribution, the coefficient of variation (standard deviation divided by the expected value) is equal to 1. The term “hyper” -exponential comes from the fact that the coefficient of variation here is greater than 1 (if different occur). In contrast to this, it is less than 1 for the hypoexponential distribution . While the exponential distribution is the continuous analogue to the geometric distribution , the hyperexponential distribution is not an analogue to the hypergeometric distribution . The hyperexponential distribution is an example of a mixed distribution . ${\displaystyle \lambda _{i}}$

The utilization of an Internet connection via which either (with probability and rate ) Internet telephony or (with probability and rate ) file transfers run can serve as an application example , whereby . The total load is then distributed hyperexponentially. ${\displaystyle p}$ ${\displaystyle \lambda _{1}}$ ${\displaystyle q}$ ${\displaystyle \lambda _{2}}$ ${\displaystyle p+q=1}$

A given probability distribution, including end-load distributions , can be approximated by a hyperexponential distribution by recursively fitting different time scales ( ) using the so-called Prony method. ${\displaystyle \lambda _{i}}$

properties

The linearity of the integral results in:

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

and

${\displaystyle \operatorname {E} \left[X^{2}\right]=\int _{-\infty }^{\infty }x^{2}f(x)\,dx=\sum _{i=1}^{N}p_{i}\int _{0}^{\infty }x^{2}\lambda _{i}e^{-\lambda _{i}x}\,dx=\sum _{i=1}^{N}{\frac {2}{\lambda _{i}^{2}}}p_{i}\;.}$

With the help of the displacement theorem, the variance results:

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

Unless they are all the same size, the standard deviation is greater than the expected value. ${\displaystyle \lambda _{i}}$

The moment generating function is

${\displaystyle \operatorname {E} \left[e^{tX}\right]=\int _{-\infty }^{\infty }e^{tx}f(x)\,dx=\sum _{i=1}^{N}p_{i}\int _{0}^{\infty }e^{tx}\lambda _{i}e^{-\lambda _{i}x}\,dx=\sum _{i=1}^{N}{\frac {\lambda _{i}}{\lambda _{i}-t}}p_{i}\;.}$

See also

• Phase distribution

Footnotes and individual references

1. ↑ LN Singh, GR Dattatreya: Estimation of the Hyperexponential Density with Applications in Sensor Networks . In: International Journal of Distributed Sensor Networks . 3, No. 3, 2007, p. 311. doi : 10.1080 / 15501320701259925 .
2. ^ A. Feldmann , W. Whitt: Fitting mixtures of exponentials to long-tail distributions to analyze network performance models . In: Performance Evaluation . 31, No. 3-4, 1998, p. 245. doi : 10.1016 / S0166-5316 (97) 00003-5 .
3. ^ HT Papadopolous, C. Heavey, J. Browne: Queuing Theory in Manufacturing Systems Analysis and Design . Springer, 1993, ISBN 9780412387203 , p. 35.