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| {{Probability distribution |
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| name =Phase-type|
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| type =density|
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| pdf_image =|
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| cdf_image =|
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| parameters =<math>S,\; m\times m</math> subgenerator [[Matrix (mathematics)|matrix]]<br /><math>\boldsymbol{\alpha}</math>, [[probability]] [[row vector]]|
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| support =<math>x \in [0; \infty)\!</math>|
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| pdf =<math>\boldsymbol{\alpha}e^{xS}\boldsymbol{S}^{0}</math><br /> See article for details|
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| cdf =<math>1-\boldsymbol{\alpha}e^{xS}\boldsymbol{1}</math>|
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| mean =<math>-\boldsymbol{\alpha}{S}^{-1}\mathbf{1}</math>|
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| mode =no simple closed form|
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| variance =<math>2\boldsymbol{\alpha}{S}^{-2}\mathbf{1}-(\boldsymbol{\alpha}{S}^{-1}\mathbf{1})^{2}</math>|
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| median =no simple closed form|
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| skewness = <!-- <math>-6\boldsymbol{\alpha}{S}^{-3}\mathbf{1}/\sigma^{3}</math> -->|
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| kurtosis = <!-- <math>24\boldsymbol{\alpha}{S}^{-4}\mathbf{1}/\sigma^{4}-3</math> -->|
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| entropy =|
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| mgf =<math>-\boldsymbol{\alpha}(tI+S)^{-1}\boldsymbol{S}^{0}+\alpha_{0}</math>|
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| char =<math>-\boldsymbol{\alpha}(itI+S)^{-1}\boldsymbol{S}^{0}+\alpha_{0}</math>|
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| }}
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| A '''phase-type distribution''' is a [[probability distribution]] constructed by a convolution of [[exponential distribution]]s.<ref>{{cite doi|10.1017/CBO9781139226424.026}}</ref> It results from a system of one or more inter-related [[Poisson process]]es occurring in [[sequence]], or phases. The sequence in which each of the phases occur may itself be a [[stochastic process]]. The distribution can be represented by a [[random variable]] describing the time until absorption of a [[Markov process]] with one absorbing state. Each of the [[Markov process|state]]s of the Markov process represents one of the phases.
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| It has a [[discrete time]] equivalent the '''[[discrete phase-type distribution]]'''.
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| The set of phase-type distributions is dense in the field of all positive-valued distributions, that is, it can be used to approximate any positive-valued distribution.
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| ==Definition==
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| Consider a [[continuous-time Markov process]] with ''m''+1 states, where ''m'' ≥ 1, such that the states 1,...,''m'' are transient states and state 0 is an absorbing state. Further, let the process have an initial probability of starting in any of the ''m''+1 phases given by the probability vector (α<sub>0</sub>,'''α''') where α<sub>0</sub> is a scalar and '''α''' is a 1×''m'' vector.
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| The '''continuous phase-type distribution''' is the distribution of time from the above process's starting until absorption in the absorbing state.
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| This process can be written in the form of a transition rate matrix,
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| :<math>
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| {Q}=\left[\begin{matrix}0&\mathbf{0}\\\mathbf{S}^0&{S}\\\end{matrix}\right],
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| </math>
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| where ''S'' is an ''m''×''m'' matrix and '''''S'''''<sup>0</sup> = -S'''1'''. Here '''1''' represents an ''m''×1 vector with every element being 1.
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| ==Characterization==
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| The distribution of time ''X'' until the process reaches the absorbing state is said to be phase-type distributed and is denoted PH('''α''',''S'').
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| The distribution function of ''X'' is given by,
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| :<math>
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| F(x)=1-\boldsymbol{\alpha}\exp({S}x)\mathbf{1},
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| </math>
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| and the density function,
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| :<math>
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| f(x)=\boldsymbol{\alpha}\exp({S}x)\mathbf{S^{0}},
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| </math>
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| for all ''x'' > 0, where exp( · ) is the [[matrix exponential]]. It is usually assumed the probability of process starting in the absorbing state is zero (i.e. α<sub>0</sub>= 0). The moments of the distribution function are given by
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| :<math>
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| E[X^{n}]=(-1)^{n}n!\boldsymbol{\alpha}{S}^{-n}\mathbf{1}.
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| </math>
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| ==Special cases==
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| The following probability distributions are all considered special cases of a continuous phase-type distribution:
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| * [[Degenerate distribution]], point mass at zero or the '''empty phase-type distribution''' - 0 phases.
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| * [[Exponential distribution]] - 1 phase.
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| * [[Erlang distribution]] - 2 or more identical phases in sequence.
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| * Deterministic distribution (or constant) - The limiting case of an Erlang distribution, as the number of phases become infinite, while the time in each state becomes zero.
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| * Coxian distribution - 2 or more (not necessarily identical) phases in sequence, with a probability of transitioning to the terminating/absorbing state after each phase.
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| * [[Hyper-exponential distribution]] (also called a mixture of exponential) - 2 or more non-identical phases, that each have a probability of occurring in a mutually exclusive, or parallel, manner. (Note: The exponential distribution is the degenerate situation when all the parallel phases are identical.)
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| * [[Hypoexponential distribution]] - 2 or more phases in sequence, can be non-identical or a mixture of identical and non-identical phases, generalises the Erlang.
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| As the phase-type distribution is dense in the field of all positive-valued distributions, we can represent any positive valued distribution. However, the phase-type is a light-tailed or platikurtic distribution. So the representation of heavy-tailed or leptokurtic distribution by phase type is an approximation, even if the precision of the approximation can be as good as we want.
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| ==Examples==
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| In all the following examples it is assumed that there is no probability mass at zero, that is α<sub>0</sub> = 0.
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| ===Exponential distribution===
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| The simplest non-trivial example of a phase-type distribution is the exponential distribution of parameter λ. The parameter of the phase-type distribution are : '''''S''''' = -λ and '''α''' = 1.
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| ===Hyper-exponential or mixture of exponential distribution===
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| The mixture of exponential or [[hyper-exponential distribution]] with λ<sub>1</sub>,λ<sub>2</sub>,...,λ<sub>n</sub>>0 can be represented as a phase type distribution with
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| :<math>
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| \boldsymbol{\alpha}=(\alpha_1,\alpha_2,\alpha_3,\alpha_4,...,\alpha_n)
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| </math>
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| with <math>\sum_{i=1}^n \alpha_i =1</math> and
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| :<math>
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| {S}=\left[\begin{matrix}-\lambda_1&0&0&0&0\\0&-\lambda_2&0&0&0\\0&0&-\lambda_3&0&0\\0&0&0&-\lambda_4&0\\0&0&0&0&-\lambda_5\\\end{matrix}\right].
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| </math>
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| This mixture of densities of exponential distributed random variables can be characterized through
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| :<math>
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| f(x)=\sum_{i=1}^n \alpha_i \lambda_i e^{-\lambda_i x} =\sum_{i=1}^n\alpha_i f_{X_i}(x),
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| </math>
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| or its cumulative distribution function
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| :<math> F(x)=1-\sum_{i=1}^n \alpha_i e^{-\lambda_i x}=\sum_{i=1}^n\alpha_iF_{X_i}(x). </math>
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| with <math> X_i \sim Exp( \lambda_i ) </math>
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| ===Erlang distribution===
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| The Erlang distribution has two parameters, the shape an integer ''k'' > 0 and the rate λ > 0. This is sometimes denoted ''E''(''k'',λ). The Erlang distribution can be written in the form of a phase-type distribution by making ''S'' a ''k''×''k'' matrix with diagonal elements -λ and super-diagonal elements λ, with the probability of starting in state 1 equal to 1. For example ''E''(5,λ),
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| :<math>
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| \boldsymbol{\alpha}=(1,0,0,0,0),
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| </math>
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| and
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| :<math>
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| {S}=\left[\begin{matrix}-\lambda&\lambda&0&0&0\\0&-\lambda&\lambda&0&0\\0&0&-\lambda&\lambda&0\\0&0&0&-\lambda&\lambda\\0&0&0&0&-\lambda\\\end{matrix}\right].
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| </math>
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| For a given number of phases, the Erlang distribution is the phase type distribution with smallest coefficient of variation.<ref name="aldous">{{cite doi|10.1080/15326348708807067}}</ref>
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| The [[hypoexponential distribution]] is a generalisation of the Erlang distribution by having different rates for each transition (the non-homogeneous case). | |
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| ===Mixture of Erlang distribution===
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| The mixture of two Erlang distribution with parameter ''E''(3,β<sub>1</sub>), ''E''(3,β<sub>2</sub>) and (α<sub>1</sub>,α<sub>2</sub>) (such that α<sub>1</sub> + α<sub>2</sub> = 1 and for each ''i'', α<sub>''i''</sub> ≥ 0) can be represented as a phase type distribution with
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| :<math>
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| \boldsymbol{\alpha}=(\alpha_1,0,0,\alpha_2,0,0),
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| </math>
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| and | |
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| :<math>
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| {S}=\left[\begin{matrix}
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| -\beta_1&\beta_1&0&0&0&0\\
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| 0&-\beta_1&\beta_1&0&0&0\\
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| 0&0&-\beta_1&0&0&0\\
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| 0&0&0&-\beta_2&\beta_2&0\\
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| 0&0&0&0&-\beta_2&\beta_2\\
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| 0&0&0&0&0&-\beta_2\\
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| \end{matrix}\right].
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| </math>
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| ===Coxian distribution===
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| The '''Coxian distribution''' is a generalisation of the hypoexponential. Instead of only being able to enter the absorbing state from state ''k'' it can be reached from any phase. The phase-type representation is given by,
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| :<math>
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| S=\left[\begin{matrix}-\lambda_{1}&p_{1}\lambda_{1}&0&\dots&0&0\\
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| 0&-\lambda_{2}&p_{2}\lambda_{2}&\ddots&0&0\\
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| \vdots&\ddots&\ddots&\ddots&\ddots&\vdots\\
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| 0&0&\ddots&-\lambda_{k-2}&p_{k-2}\lambda_{k-2}&0\\
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| 0&0&\dots&0&-\lambda_{k-1}&p_{k-1}\lambda_{k-1}\\
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| 0&0&\dots&0&0&-\lambda_{k}
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| \end{matrix}\right]</math>
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| and
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| :<math>\boldsymbol{\alpha}=(1,0,\dots,0),</math>
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| where 0 < ''p''<sub>1</sub>,...,''p''<sub>''k''-1</sub> ≤ 1. In the case where all ''p''<sub>''i''</sub> = 1 we have the hypoexponential distribution. The Coxian distribution is extremely important as any acyclic phase-type distribution has an equivalent Coxian representation.
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| The '''generalised Coxian distribution''' relaxes the condition that requires starting in the first phase.
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| ==Generating samples from phase-type distributed random variables==
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| ''[http://webspn.hit.bme.hu/~telek/tools/butools/butools.html BuTools]'' includes methods for generating samples from phase-type distributed random variables.<ref>{{cite doi|10.1007/978-3-642-30782-9_19}}</ref>
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| ==Approximating other distributions==
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| Any distribution can be arbitrarily well approximated by a phase type distribution.<ref>{{cite doi|10.1002/0471200581.ch3}}</ref><ref>{{cite doi|10.1017/S0305004100030231}}</ref> In practice, however, approximations can be poor when the size of the approximating process is fixed. Approximating a deterministic distribution of time 1 with 10 phases, each of average length 0.1 will have variance 0.1 (because the Erlang distribution has smallest variance<ref name="aldous" />).
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| *''[http://webspn.hit.bme.hu/~telek/tools/butools/butools.html BuTools]'' a [[MATLAB]] and [[Mathematica]] script for fitting phase-type distributions to 3 specified moments
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| *''[http://www.cs.cmu.edu/~osogami/code/momentmatching/index.html momentmatching]'' a [[MATLAB]] script to fit a minimal phase-type distribution to 3 specified moments<ref>{{cite doi|10.1016/j.peva.2005.06.002}}</ref>
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| ==Fitting a phase type distribution to data==
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| Methods to fit a phase type distribution to data can be classified as maximum likelihood methods or moment matching methods.<ref>{{cite book | first1 = Andreas | last1= Lang | first2= Jeffrey L. | last2 = Arthur | chapter = Parameter approximation for Phase-Type distributions | title = Matrix Analytic methods in Stochastic Models | editor1-first = S. | editor1-last = Chakravarthy| editor2-first = Attahiru S. | editor2-last = Alfa | publisher = CRC Press | year = 1996 | isbn = 0824797663}}</ref> Fitting a phase type distribution to [[heavy-tailed distribution]]s has been shown to be practical in some situations.<ref>{{cite doi|10.1287/inte.1050.0155}}</ref>
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| *''[http://webspn.hit.bme.hu/~telek/tools.htm PhFit]'' a C script for fitting discrete and continuous phase type distributions to data<ref>{{cite doi|10.1007/3-540-46029-2_5}}</ref>
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| *''[http://home.imf.au.dk/asmus/pspapers.html EMpht]'' is a C script for fitting phase-type distributions to data or parametric distributions using an [[expectation–maximization algorithm]].<ref>{{cite jstor|4616418}}</ref>
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| *''[http://www.mi.fu-berlin.de/inf/groups/ag-tech/projects/HyperStar/index.html HyperStar]'' was developed around the core idea of making phase-type fitting simple and user-friendly, in order to advance the use of phase-type distributions in a wide range of areas. It provides a graphical user interface and yields good fitting results with only little user interaction.<ref>{{cite doi |10.1016/j.camwa.2012.03.016}}</ref>
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| *''[http://copa.uniandes.edu.co/?p=141 jPhase]'' is a Java library which can also compute metrics for queues using the fitted phase type distribution<ref>{{cite doi|10.1145/1190366.1190370}}</ref>
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| ==See also==
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| * [[Discrete phase-type distribution]]
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| * [[Continuous-time Markov process]]
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| * [[Exponential distribution]]
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| * [[Hyper-exponential distribution]]
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| * [[Queueing theory]]
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| ==References==
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| {{Reflist}}
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| * M. F. Neuts. Matrix-Geometric Solutions in Stochastic Models: an Algorithmic Approach, Chapter 2: Probability Distributions of Phase Type; Dover Publications Inc., 1981.
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| * G. Latouche, V. Ramaswami. Introduction to Matrix Analytic Methods in Stochastic Modelling, 1st edition. Chapter 2: PH Distributions; ASA SIAM, 1999.
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| * C. A. O'Cinneide (1990). ''Characterization of phase-type distributions''. Communications in Statistics: Stochastic Models, '''6'''(1), 1-57.
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| * C. A. O'Cinneide (1999). ''Phase-type distribution: open problems and a few properties'', Communication in Statistic: Stochastic Models, '''15'''(4), 731-757.
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| {{ProbDistributions|continuous-semi-infinite}}
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| {{DEFAULTSORT:Phase-Type Distribution}}
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| [[Category:Continuous distributions]]
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| [[Category:Types of probability distributions]]
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| [[Category:Probability distributions]]
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