Phase-type distributions and representations: Some results and open problems for system theory

Commault, Christian; Mocanu, Stéphane

doi:10.1080/0020717031000114986

Cited by 51 publications

(49 citation statements)

References 29 publications

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“…The representation of DPH distributions is closely related to the positive realization of the positive systems, which have been researched wide in control and system theory communities. The representation problem finding a Markov chain associated with phase-type distribution has a lot of links with positive realization problem in the control theory [3]. The relationship between the probability generating functions of the probability mass function of a DPH-distribution and a corresponding representation is very similar to the relationship between a transfer function and a corresponding state space realization.…”

Section: Introductionmentioning

confidence: 94%

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The Dimension Reduction Algorithm for the Positive Realization of Discrete Phase-Type Distributions

Kim¹

2012

Journal of the Korea Society for Industrial and Applied Mathema

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ABSTRACT. This paper provides an efficient dimension reduction algorithm of the positive realization of discrete phase type(DPH) distributions. The relationship between the representation of DPH distributions and the positive realization of the positive system is explained. The dimension of the positive realization of a discrete phase-type realization may be larger than its McMillan degree of probability generating functions. The positive realization with sufficient large dimension bound can be obtained easily but generally, the minimal positive realization problem is not solved yet. We propose an efficient dimension reduction algorithm to make the positive realization with tighter upper bound from a given probability generating functions in terms of convex cone problem and linear programming.

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Section: Introductionmentioning

confidence: 94%

“…The parameter estimation algorithm for a DPH distribution has been introduced and formalized in [3]. It is well known that general PH distributions are considerably over-parameterized and have high order representation in fitting or approximation problems [4] [5].…”

Section: Introductionmentioning

confidence: 99%

The Dimension Reduction Algorithm for the Positive Realization of Discrete Phase-Type Distributions

Kim¹

2012

Journal of the Korea Society for Industrial and Applied Mathema

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“…The phase-type representation (so called, positive representation) is strongly connected with the positive realization in system theory [5]. The relation between the Laplace-Stieltjes transform (LST) of a probability distribution and a corresponding phase-type representation is similar to that between a transfer function and a corresponding state space realization.…”

Section: Introductionmentioning

confidence: 99%

“…There are some important approaches for minimal and specially structured positive realization such as the triangular, bi-diagonal, mono-cyclic, and uni-cyclic representations [5,9]. Every phase-type distribution has a mono-cyclic representation with larger order [9].…”

Section: Introductionmentioning

confidence: 99%

An Optimization Approach for Computing a Sparse Mono-Cyclic Positive Representation

Kim¹

2016

Journal of the Korea Society for Industrial and Applied Mathema

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ABSTRACT. The phase-type representation is strongly connected with the positive realization in positive system. We attempt to transform phase-type representation into sparse mono-cyclic positive representation with as low order as possible. Because equivalent positive representations of a given phase-type distribution are non-unique, it is important to find a simple sparse positive representation with lower order that leads to more effective use in applications. A Hypo-Feedback-Coxian Block (HFCB) representation is a good candidate for a simple sparse representation. Our objective is to find an HFCB representation with possibly lower order, including all the eigenvalues of the original generator. We introduce an efficient nonlinear optimization method for computing an HFCB representation from a given phase-type representation. We discuss numerical problems encountered when finding efficiently a stable solution of the nonlinear constrained optimization problem. Numerical simulations are performed to show the effectiveness of the proposed algorithm.

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“…In particular, we make use of the fact that any matrix-exponential function with positive density on (0, ∞) and with a unique dominant eigenvalue (whose multiplicity can be greater than one) has a PH representation [16]. Furthermore, our approach is built on the monocyclic representation of PH distributions [15,6]. The monocyclic representation of PH distribution bridges several Markov-chain-related results with the analysis of matrix-exponential functions.…”

Section: Introductionmentioning

confidence: 99%

Does a given vector–matrix pair correspond to a PH distribution?

Reinecke¹,

Telek

2014

Performance Evaluation

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The analysis of practical queueing problems benefits if realistic distributions can be used as parameters. Phase type (PH) distributions can approximate many distributions arising in practice, but their practical applicability has always been limited when they are described by a non-Markovian vector-matrix pair. In this case it is hard to check whether the non-Markovian vector-matrix pair defines a non-negative matrix-exponential function or not. In this paper we propose a numerical procedure for checking if the matrix-exponential function defined by a non-Markovian vector-matrix pair can be represented by a Markovian vector-matrix pair with potentially larger size. If so, then the matrix-exponential function is non-negative.The proposed procedure is based on O'Cinneide's characterization result, which says that a nonMarkovian vector-matrix pair with strictly positive density on (0, ∞) and with a real dominant eigenvalue has a Markovian representation. Our method checks the existence of a potential Markovian representation in a computationally efficient way utilizing the structural properties of the applied representation transformation procedure.

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Phase-type distributions and representations: Some results and open problems for system theory

Abstract: International audienc

Cited by 51 publications

References 29 publications

The Dimension Reduction Algorithm for the Positive Realization of Discrete Phase-Type Distributions

The Dimension Reduction Algorithm for the Positive Realization of Discrete Phase-Type Distributions

An Optimization Approach for Computing a Sparse Mono-Cyclic Positive Representation

Does a given vector–matrix pair correspond to a PH distribution?

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