phase-type distribution: Information and Much More from Answers.com
- ️Wed Jul 01 2015
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Probability density function (pdf) | ![]() See article for details |
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Median | no simple closed form |
Mode | no simple closed form |
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A phase-type distribution is a probability distribution that 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 phase-type distribution is dense in the field of all positive-valued distributions, that is, it can be used to approximate any positive valued distribution.
Definition
There exists a continuous-time Markov process with m + 1 states, where . The states 1,...,m are transient states and state m + 1 is an absorbing state. The process has an initial probability of starting in any of the
m + 1 phases given by the probability vector
.
The continuous phase-type distibution is the distribution of time from the processes starting until absorption in the absorbing state.
This process can be written in the form of a transition rate matrix,
where S is a m×m matrix and . Here
represents an m×1 vector with every element being 1.
Characterization
The distribution of time X until the process reaches the absorbing state is said to be
phase-type distributed and is denoted .
The distribution function of X is given by,
and the density function,
for all x > 0, where is the matrix exponential. It is usually assumed the probability of process starting in the absorbing state
is zero. The moments of the distribution function are given by,
Special cases
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.)
As the phase-type distribution is dense in the field of all positive-valued distributions, we can represent any positive valued distributions. 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 appproximation, even if the precision of the approximation can be as good as we want.
Examples
In all the following examples it is assummed that there is no probability mass at zero, that is αm + 1 = 0.
Exponential distribution
The simplest non-trivial example of phase-type distribution is the exponential distribution of parameter λ. The parameter of the phase-type distribution are : and
Hyper-exponential or mixture of exponential distribution
The mixture of exponential or hypoexponential distribution with parameter (α1,α2,α3,α4,α5) (such that and
) and (λ1,λ2,λ3,λ4,λ5) can be representend as a phase type
distribution with
and
The mixture of exponential can be characterized through its density
or its distribution function
This can be generalized to a mixture of n exponential distributions.
Erlang distribution
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),
and
The hypoexponential distribution is a generlisation of the Erlang distribution by having different rates for each transition (the non-homogeneous case).
Mixture of Erlang distribution
The mixture of two erlang distribution with parameter E(3,β1), E(3,β2) and (α1,α2) (such that α1 + α2 = 1 and ) can be representend as a
phase type distribution with
and
Coxian distribution
The Coxian distribution is a generalisation of 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,
and
where , in the case where all pi = 1 we have
the hypoexponential distribution.
The generalised Coxian distribution relaxes the condition that requires starting in the first phase.
See also
- Discrete phase-type distribution
- Continuous-time Markov process
- Exponential distribution
- Hyper-exponential distribution
- Queueing model
- Queuing theory
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.
- C. A. O'Cinneide (1990). Characterization of phase-type distributions. Communications in Statistics: Stocahstic 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.
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