Phase-type distribution

2014-01-08T01:33:17Z

204.11.229.52:

[[File:Hyperexponential.svg|thumb|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

:<math> f_X(x) = \sum_{i=1}^n f_{Y_i}(x)\;p_i,</math>

where each ''Y''''i'' is an [[exponential distribution|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''.<ref name=SinghDatta>{{cite doi|10.1080/15501320701259925}}</ref> 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 ''λ''1 and probability ''q'' of them using their internet connection with rate ''λ''2.

==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

:<math> E[X] = \int_{-\infty}^\infty x f(x) \, dx= \sum_{i=1}^n p_i\int_0^\infty x\lambda_i e^{-\lambda_ix} \, dx = \sum_{i=1}^n \frac{p_i}{\lambda_i}</math>

and

:<math> 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_ix} \, dx = \sum_{i=1}^n \frac{2}{\lambda_i^2}p_i,</math>

from which we can derive the variance:<ref>{{cite book|author=H.T. Papadopolous, C. Heavey, and J. Browne|title=Queueing Theory in Manufacturing Systems Analysis and Design|year=1993|publisher=Springer|isbn=9780412387203|page=35|url=http://books.google.com/books?id=9pf5MCf9VDYC&pg=PA35}}</ref>

:<math>\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.
</math>

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

The [[moment-generating function]] is given by

:<math>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.</math>

==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]].<ref>{{cite doi|10.1016/S0166-5316(97)00003-5}}</ref>

==See also==
* [[Phase-type distribution]]
* [[Hyper-Erlang distribution]]

==References==
{{Reflist}}

{{ProbDistributions|continuous-semi-infinite}}

{{DEFAULTSORT:Hyper-Exponential Distribution}}
[[Category:Continuous distributions]]
[[Category:Probability distributions]]

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Phase-type distribution