Phase Type and Matrix Exponential Distributions in Stochastic Modeling

In this paper we introduce certain Hankel matrices that can be used to study ME (matrix exponential) distributions, in particular to compute their ME orders. The Hankel matrices for a given ME probability distribution can be constructed if one of the following five types of information about the distribution is available: (i) an ME representation, (ii) its moments, (iii) the derivatives of its distribution function, (iv) its Laplace-Stieltjes transform, or (v) its distribution function. Using the Hankel matrices, a necessary and sufficient condition for a probability distribution to be an ME distribution is found and a method of computing the ME order of the ME distribution developed. Implications for the PH (phase-type) order of PH distributions are examined. The relationship between the ME order, the PH order, and a lower bound on the PH order given by Aldous and Shepp (1987) is discussed in numerical examples.

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Concentrated matrix exponential distributions with real eigenvalues

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964821000309 ◽

2021 ◽

pp. 1-17

Author(s):

András Mészáros ◽

Miklós Telek

Keyword(s):

Stochastic Models ◽

Laplace Transformation ◽

Matrix Exponential ◽

Phase Type Distribution ◽

Exponential Distributions ◽

Inverse Laplace Transformation ◽

Complex Eigenvalues ◽

Phase Type ◽

The Matrix ◽

The Laplace Transform

Abstract Concentrated random variables are frequently used in representing deterministic delays in stochastic models. The squared coefficient of variation ( $\mathrm {SCV}$ ) of the most concentrated phase-type distribution of order $N$ is $1/N$ . To further reduce the $\mathrm {SCV}$ , concentrated matrix exponential (CME) distributions with complex eigenvalues were investigated recently. It was obtained that the $\mathrm {SCV}$ of an order $N$ CME distribution can be less than $n^{-2.1}$ for odd $N=2n+1$ orders, and the matrix exponential distribution, which exhibits such a low $\mathrm {SCV}$ has complex eigenvalues. In this paper, we consider CME distributions with real eigenvalues (CME-R). We present efficient numerical methods for identifying a CME-R distribution with smallest SCV for a given order $n$ . Our investigations show that the $\mathrm {SCV}$ of the most concentrated CME-R of order $N=2n+1$ is less than $n^{-1.85}$ . We also discuss how CME-R can be used for numerical inverse Laplace transformation, which is beneficial when the Laplace transform function is impossible to evaluate at complex points.

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On matrix exponential distributions

Advances in Applied Probability ◽

10.1017/s0001867800001695 ◽

2007 ◽

Vol 39 (01) ◽

pp. 271-292 ◽

Cited By ~ 2

Author(s):

Qi-Ming He ◽

Hanqin Zhang

Keyword(s):

Distribution Function ◽

Probability Distribution ◽

Matrix Exponential ◽

Exponential Distributions ◽

Hankel Matrices ◽

Phase Type ◽

Types Of Information ◽

Necessary And Sufficient ◽

The Relationship ◽

Derivatives Of

In this paper we introduce certain Hankel matrices that can be used to study ME (matrix exponential) distributions, in particular to compute their ME orders. The Hankel matrices for a given ME probability distribution can be constructed if one of the following five types of information about the distribution is available: (i) an ME representation, (ii) its moments, (iii) the derivatives of its distribution function, (iv) its Laplace-Stieltjes transform, or (v) its distribution function. Using the Hankel matrices, a necessary and sufficient condition for a probability distribution to be an ME distribution is found and a method of computing the ME order of the ME distribution developed. Implications for the PH (phase-type) order of PH distributions are examined. The relationship between the ME order, the PH order, and a lower bound on the PH order given by Aldous and Shepp (1987) is discussed in numerical examples.

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An alternative characterization for matrix exponential distributions

Advances in Applied Probability ◽

10.1017/s0001867800003712 ◽

2009 ◽

Vol 41 (04) ◽

pp. 1005-1022

Author(s):

Mark Fackrell

Keyword(s):

Continuous Function ◽

Exponential Distribution ◽

Matrix Exponential ◽

Necessary Condition ◽

Stieltjes Transform ◽

Real Numbers ◽

Exponential Distributions ◽

Alternative Characterization

A necessary condition for a rational Laplace–Stieltjes transform to correspond to a matrix exponential distribution is that the pole of maximal real part is real and negative. Given a rational Laplace–Stieltjes transform with such a pole, we present a method to determine whether or not the numerator polynomial admits a transform that corresponds to a matrix exponential distribution. The method relies on the minimization of a continuous function of one variable over the nonnegative real numbers. Using this approach, we give an alternative characterization for all matrix exponential distributions of order three.

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Fitting with Matrix-Exponential Distributions

Stochastic Models ◽

10.1081/stm-200056227 ◽

2005 ◽

Vol 21 (2-3) ◽

pp. 377-400 ◽

Cited By ~ 24

Author(s):

Mark Fackrell

Keyword(s):

Matrix Exponential ◽

Exponential Distributions

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A closure characterisation of phase-type distributions

Journal of Applied Probability ◽

10.1017/s0021900200106655 ◽

1992 ◽

Vol 29 (01) ◽

pp. 92-103 ◽

Cited By ~ 1

Author(s):

Robert S. Maier ◽

Colm Art O'Cinneide

Keyword(s):

Analogous Result ◽

Finite Automata ◽

Automata Theory ◽

Phase Type Distribution ◽

Exponential Distributions ◽

Phase Type ◽

Phase Type Distributions ◽

Discrete Phase ◽

Rational Sequence ◽

Family Of Distributions

We characterise the classes of continuous and discrete phase-type distributions in the following way. They are known to be closed under convolutions, mixtures, and the unary ‘geometric mixture' operation. We show that the continuous class is the smallest family of distributions that is closed under these operations and contains all exponential distributions and the point mass at zero. An analogous result holds for the discrete class. We also show that discrete phase-type distributions can be regarded as ℝ+-rational sequences, in the sense of automata theory. This allows us to view our characterisation of them as a corollary of the Kleene–Schützenberger theorem on the behavior of finite automata. We prove moreover that any summable ℝ+-rational sequence is proportional to a discrete phase-type distribution.

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