Phase-type 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. It is named the hyper-exponential 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 less than one. While the exponential distribution is the continuous analogue of the geometric distribution, the hyper-exponential distribution is not analogous to the hypergeometric distribution. The hyper-exponential distribution is an example of a mixture density.

An example of a hyper-exponential 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 hyper-exponential 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 of the hyper-exponential distribution

Since the expected value of a sum is the sum of the expected values, the expected value of a hyper-exponential 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:

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

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.

See also

• Phase-type distribution
• Hyper-Erlang distribution

References

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