Hypoexponential distribution

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Hypoexponential
Probability density function
Cumulative distribution function
Parameters	$\lambda_{1},\dots,\lambda_{k} > 0\,$ rates (real)
Support	$x \in [0; \infty)\!$
Probability density function (pdf)	Expressed as a phase-type distribution $-\boldsymbol{\alpha}e^{x\Theta}\Theta\boldsymbol{1}$ Has no other simple form; see article for details
Cumulative distribution function (cdf)	Expressed as a phase-type distribution $1-\boldsymbol{\alpha}e^{x\Theta}\boldsymbol{1}$
Mean	$\sum^{k}_{i=1}1/\lambda_{i}\,$
Median	$\ln(2)\sum^{k}_{i=1}1/\lambda_{i}$
Mode	$0$
Variance	$2\sum^{k}_{i=1}1/\lambda_{i}\sum_{n=1}^{i}1/\lambda_{n}$
Skewness	no simple closed form
Excess kurtosis	no simple closed form
Entropy
Moment-generating function (mgf)	$\boldsymbol{\alpha}(tI-\Theta)^{-1}\Theta\mathbf{1}$
Characteristic function	$\boldsymbol{\alpha}(itI-\Theta)^{-1}\Theta\mathbf{1}$

In probability theory the hypoexponential distribution or the generalized Erlang distribution is a continuous distribution, that has found use in the same fields as the Erlang distribution, such as queueing theory, teletraffic engineering and more generally in stochastic processes. It is called the hypoexponetial distribution as it has a coefficient of variation less than one, compared to the hyper-exponential distribution which has coefficient of variation greater than one and the exponential distribution which has coefficient of variation of one.

1 Overview
- 1.1 Relation to the phase-type distribution
2 Characterization
3 See also
4 References
5 External References

[edit] Overview

The Erlang distibution is a series of k exponential distributions all with rate $λ$ . The hypoexponential is a series of k exponential distributions each with their own rate $λ i$ , the rate of the $i t h$ exponential distribution. If we have k independentally distributed exponential random variables $\boldsymbol{X}_{i}$ , then the random variable,

$\boldsymbol{X}=\sum^{k}_{i=1}\boldsymbol{X}_{i}$

is hypoexponentially distributed. The hypoexponential has a minimum coefficient of variation of $1 / k$ .

[edit] Relation to the phase-type distribution

As a result of the definition it is easier to consider this distribution as a special case of the phase-type distribution. The phase-type distribution is the time to absorption of a finite state Markov process. If we have a k+1 state process, where the first k states are transient and the state k+1 is an absorbing state, then the distribution of time from the start of the process until the absorbing state is reached is phase-type distributed. This becomes the hypoexpoexponential if we start in the first 1 and move skip-free from state i to i+1 with rate $λ i$ until state k transitions with rate $λ k$ to the absorbing state k+1. This can be written in the form of a subgenerator matrix,

$\left[\begin{matrix}-\lambda_{1}&\lambda_{1}&0&\dots&0&0\\ 0&-\lambda_{2}&\lambda_{2}&\ddots&0&0\\ \vdots&\ddots&\ddots&\ddots&\ddots&\vdots\\ 0&0&\ddots&-\lambda_{k-2}&\lambda_{k-2}&0\\ 0&0&\dots&0&-\lambda_{k-1}&\lambda_{k-1}\\ 0&0&\dots&0&0&-\lambda_{k} \end{matrix}\right]\; .$

For simplicity denote the above matrix $\Theta\equiv\Theta(\lambda_{1},\dots,\lambda_{k})$ . If the probability of starting in each of the k states is

$\boldsymbol{\alpha}=(1,0,\dots,0)$

then $Hypo(\lambda_{1},\dots,\lambda_{k})=PH(\boldsymbol{\alpha},\Theta)$ .

[edit] Characterization

A random variable $\boldsymbol{X}\sim Hypo(\lambda_{1},\dots,\lambda_{k})$ has cumulative distribution function given by,

$F(x)=1-\boldsymbol{\alpha}e^{x\Theta}\boldsymbol{1}$

and density function,

$f(x)=-\boldsymbol{\alpha}e^{x\Theta}\Theta\boldsymbol{1}\; ,$

where $\boldsymbol{1}$ is a column vector of ones of the size k and $e A$ is the matrix exponential of A.

The distribution has Laplace transform of

$\mathcal{L}\{f(x)\}=-\boldsymbol{\alpha}(sI-\Theta)^{-1}\Theta\boldsymbol{1}$

Which can be used to find moments,

$E[X^{n}]=(-1)^{n}n!\boldsymbol{\alpha}\Theta^{-n}\boldsymbol{1}\; .$

[edit] See also

[edit] References

M. F. Neuts. Matrix-Geometric Solutions in Stochastic Models: an Algorthmic Approach, Chapter 2: Probability Distributions of Phase Type; Dover Publications Inc., 1981.
G. Latouche, V. Ramaswami. Introduction to Matrix Analytic Methods in Stochastic Modelling, 1st edition. Chapter 2: PH Distributions; ASA SIAM, 1999
Colm A. O'Cinneide (1999). Phase-type distribution: open problems and a few properties, Communication in Statistic - Stochastic Models, 15(4), 731-757.

[edit] External References

Phd Thesis by Alma Riska

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Hypoexponential distribution

From Wikipedia, the free encyclopedia

Contents

[edit] Overview

[edit] Relation to the phase-type distribution

[edit] Characterization

[edit] See also

[edit] References

[edit] External References

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