Phase-type distribution

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Phase-type
Parameters S,\; m\times m subgenerator matrix
\boldsymbol{\alpha}, probability row vector
Support x \in [0; \infty)\!
pdf \boldsymbol{\alpha}e^{xS}\boldsymbol{S}^{0}
See article for details
CDF 1-\boldsymbol{\alpha}e^{xS}\boldsymbol{1}
Mean -\boldsymbol{\alpha}{S}^{-1}\mathbf{1}
Median no simple closed form
Mode no simple closed form
Variance 2\boldsymbol{\alpha}{S}^{-2}\mathbf{1}-(\boldsymbol{\alpha}{S}^{-1}\mathbf{1})^{2}
MGF -\boldsymbol{\alpha}(tI+S)^{-1}\boldsymbol{S}^{0}+\alpha_{0}
CF -\boldsymbol{\alpha}(itI+S)^{-1}\boldsymbol{S}^{0}+\alpha_{0}

A phase-type distribution is a probability distribution constructed by a convolution of exponential distributions.[1] It results from a system of one or more inter-related Poisson processes 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 states of the Markov process represents one of the phases.

It has a discrete time equivalent the discrete phase-type distribution.

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.

Definition[edit]

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 (α0,α) where α0 is a scalar and α is a 1 × m vector.

The continuous phase-type distribution is the distribution of time from the above process's starting until absorption in the absorbing state.

This process can be written in the form of a transition rate matrix,


{Q}=\left[\begin{matrix}0&\mathbf{0}\\\mathbf{S}^0&{S}\\\end{matrix}\right],

where S is an m × m matrix and S0 = –S1. Here 1 represents an m × 1 vector with every element being 1.

Characterization[edit]

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).

The distribution function of X is given by,


F(x)=1-\boldsymbol{\alpha}\exp({S}x)\mathbf{1},

and the density function,


f(x)=\boldsymbol{\alpha}\exp({S}x)\mathbf{S^{0}},

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. α0= 0). The moments of the distribution function are given by


E[X^{n}]=(-1)^{n}n!\boldsymbol{\alpha}{S}^{-n}\mathbf{1}.

Special cases[edit]

The following probability distributions are all considered special cases of a continuous phase-type distribution:

  • Degenerate distribution, point mass at zero or the empty phase-type distribution - 0 phases.
  • Exponential distribution - 1 phase.
  • Erlang distribution - 2 or more identical phases in sequence.
  • 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.
  • Coxian distribution - 2 or more (not necessarily identical) phases in sequence, with a probability of transitioning to the terminating/absorbing state after each phase.
  • 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.)
  • Hypoexponential distribution - 2 or more phases in sequence, can be non-identical or a mixture of identical and non-identical phases, generalises the Erlang.

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.

Examples[edit]

In all the following examples it is assumed that there is no probability mass at zero, that is α0 = 0.

Exponential distribution[edit]

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.

Hyper-exponential or mixture of exponential distribution[edit]

The mixture of exponential or hyper-exponential distribution with λ12,...,λn>0 can be represented as a phase type distribution with


\boldsymbol{\alpha}=(\alpha_1,\alpha_2,\alpha_3,\alpha_4,...,\alpha_n)

with \sum_{i=1}^n \alpha_i =1 and


{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].

This mixture of densities of exponential distributed random variables can be characterized through


f(x)=\sum_{i=1}^n \alpha_i \lambda_i e^{-\lambda_i x} =\sum_{i=1}^n\alpha_i f_{X_i}(x),

or its cumulative distribution function

 F(x)=1-\sum_{i=1}^n \alpha_i e^{-\lambda_i x}=\sum_{i=1}^n\alpha_iF_{X_i}(x).

with  X_i \sim Exp( \lambda_i )

Erlang distribution[edit]

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),


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

and


{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].

For a given number of phases, the Erlang distribution is the phase type distribution with smallest coefficient of variation.[2]

The hypoexponential distribution is a generalisation of the Erlang distribution by having different rates for each transition (the non-homogeneous case).

Mixture of Erlang distribution[edit]

The mixture of two Erlang distribution with parameter E(3,β1), E(3,β2) and (α12) (such that α1 + α2 = 1 and for each i, αi ≥ 0) can be represented as a phase type distribution with


\boldsymbol{\alpha}=(\alpha_1,0,0,\alpha_2,0,0),

and


{S}=\left[\begin{matrix}
-\beta_1&\beta_1&0&0&0&0\\
0&-\beta_1&\beta_1&0&0&0\\
0&0&-\beta_1&0&0&0\\
0&0&0&-\beta_2&\beta_2&0\\
0&0&0&0&-\beta_2&\beta_2\\
0&0&0&0&0&-\beta_2\\
\end{matrix}\right].

Coxian distribution[edit]

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,


S=\left[\begin{matrix}-\lambda_{1}&p_{1}\lambda_{1}&0&\dots&0&0\\
                    0&-\lambda_{2}&p_{2}\lambda_{2}&\ddots&0&0\\
                    \vdots&\ddots&\ddots&\ddots&\ddots&\vdots\\
                    0&0&\ddots&-\lambda_{k-2}&p_{k-2}\lambda_{k-2}&0\\
                    0&0&\dots&0&-\lambda_{k-1}&p_{k-1}\lambda_{k-1}\\
                    0&0&\dots&0&0&-\lambda_{k}
\end{matrix}\right]

and

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

where 0 < p1,...,pk-1 ≤ 1. In the case where all pi = 1 we have the hypoexponential distribution. The Coxian distribution is extremely important as any acyclic phase-type distribution has an equivalent Coxian representation.

The generalised Coxian distribution relaxes the condition that requires starting in the first phase.

Generating samples from phase-type distributed random variables[edit]

BuTools includes methods for generating samples from phase-type distributed random variables.[3]

Approximating other distributions[edit]

Any distribution can be arbitrarily well approximated by a phase type distribution.[4][5] 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[2]).

Fitting a phase type distribution to data[edit]

Methods to fit a phase type distribution to data can be classified as maximum likelihood methods or moment matching methods.[7] Fitting a phase type distribution to heavy-tailed distributions has been shown to be practical in some situations.[8]

  • PhFit a C script for fitting discrete and continuous phase type distributions to data[9]
  • EMpht is a C script for fitting phase-type distributions to data or parametric distributions using an expectation–maximization algorithm.[10]
  • 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.[11]
  • jPhase is a Java library which can also compute metrics for queues using the fitted phase type distribution[12]

See also[edit]

References[edit]

  1. ^ Harchol-Balter, M. (2012). "Real-World Workloads: High Variability and Heavy Tails". "Performance Modeling and Design of Computer Systems". p. 347. doi:10.1017/CBO9781139226424.026. ISBN 9781139226424.  edit
  2. ^ a b Aldous, David; Shepp, Larry (1987). "The least variable phase type distribution is erlang". Stochastic Models 3 (3): 467. doi:10.1080/15326348708807067.  edit
  3. ^ Horváth, G. B.; Reinecke, P.; Telek, M. S.; Wolter, K. (2012). "Efficient Generation of PH-Distributed Random Variates". "Analytical and Stochastic Modeling Techniques and Applications". Lecture Notes in Computer Science 7314. p. 271. doi:10.1007/978-3-642-30782-9_19. ISBN 978-3-642-30781-2.  edit
  4. ^ Bolch, Gunter; Greiner, Stefan; de Meer, Hermann; Trivedi, Kishor S. (1998). "Steady-State Solutions of Markov Chains". "Queueing Networks and Markov Chains". pp. 103–151. doi:10.1002/0471200581.ch3. ISBN 0471193666.  edit
  5. ^ Cox, D. R. (2008). "A use of complex probabilities in the theory of stochastic processes". Mathematical Proceedings of the Cambridge Philosophical Society 51 (2): 313. doi:10.1017/S0305004100030231.  edit
  6. ^ Osogami, T.; Harchol-Balter, M. (2006). "Closed form solutions for mapping general distributions to quasi-minimal PH distributions". Performance Evaluation 63 (6): 524. doi:10.1016/j.peva.2005.06.002.  edit
  7. ^ Lang, Andreas; Arthur, Jeffrey L. (1996). "Parameter approximation for Phase-Type distributions". In Chakravarthy, S.; Alfa, Attahiru S. Matrix Analytic methods in Stochastic Models. CRC Press. ISBN 0824797663. 
  8. ^ Ramaswami, V.; Poole, D.; Ahn, S.; Byers, S.; Kaplan, A. (2005). "Ensuring Access to Emergency Services in the Presence of Long Internet Dial-Up Calls". Interfaces 35 (5): 411. doi:10.1287/inte.1050.0155.  edit
  9. ^ Horváth, András S.; Telek, Miklós S. (2002). "PhFit: A General Phase-Type Fitting Tool". "Computer Performance Evaluation: Modelling Techniques and Tools". Lecture Notes in Computer Science 2324. p. 82. doi:10.1007/3-540-46029-2_5. ISBN 978-3-540-43539-6.  edit
  10. ^ Asmussen, Søren; Nerman, Olle; Olsson, Marita (1996). "Fitting Phase-Type Distributions via the EM Algorithm". Scandinavian Journal of Statistics 23 (4): 419–441. JSTOR 4616418.  edit
  11. ^ Reinecke, P.; Krauß, T.; Wolter, K. (2012). "Cluster-based fitting of phase-type distributions to empirical data". Computers & Mathematics with Applications 64 (12): 3840. doi:10.1016/j.camwa.2012.03.016.  edit
  12. ^ Pérez, J. F.; Riaño, G. N. (2006). "jPhase: an object-oriented tool for modeling phase-type distributions". "Proceeding from the 2006 workshop on Tools for solving structured Markov chains (SMCtools '06)". doi:10.1145/1190366.1190370. ISBN 1595935061.  edit
  • M. F. Neuts. Matrix-Geometric Solutions in Stochastic Models: an Algorithmic 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.
  • C. A. O'Cinneide (1990). Characterization of phase-type distributions. Communications in Statistics: Stochastic Models, 6(1), 1-57.
  • C. A. O'Cinneide (1999). Phase-type distribution: open problems and a few properties, Communication in Statistic: Stochastic Models, 15(4), 731-757.