Hypoexponential distribution

Probability distribution
name =Hypoexponential
type =density
pdf_

cdf_

parameters =$lambda\_\{1\},dots,lambda\_\{k\}\; >\; 0,$ rates (real)
support =$x\; in\; [0;\; infty)!$
pdf =Expressed as a phase-type distribution

Has no other simple form; see article for details
cdf =Expressed as a phase-type distribution


mean =$sum^\{k\}\_\{i=1\}1/lambda\_\{i\},$
mode =$0$
variance =$2sum^\{k\}\_\{i=1\}1/lambda\_\{i\}sum\_\{n=1\}^\{i\}1/lambda\_\{n\}$
median =$ln(2)sum^\{k\}\_\{i=1\}1/lambda\_\{i\}$
skewness =no simple closed form
kurtosis =no simple closed form
entropy =
mgf =
char =

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.

Overview

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

:

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

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 $lambda\_\{i\}$ until state "k" transitions with rate $lambda\_\{k\}$ to the absorbing state "k+1". This can be written in the form of a subgenerator matrix,

:

For simplicity denote the above matrix $ThetaequivTheta(lambda\_\{1\},dots,lambda\_\{k\})$. If the probability of starting in each of the "k" states is

:

then .

Characterization

A random variable has cumulative distribution function given by,

:

and density function,

:

where 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

:

Which can be used to find moments,

:

ee also

* Exponential distribution
* Erlang distribution
* Hyper-exponential distribution
* Phase-type distribution
* Coxian distribution

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.

External references

* [http://www.cs.wm.edu/~riska/PhD-thesis-html/node9.html Phd Thesis by Alma Riska]