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A non-convex optimization approach of searching algebraic degree phase-type representations for general phase-type distributions

Part of: Markov processes Basic linear algebra Distribution theory - Probability

Published online by Cambridge University Press:  03 March 2025

Yujie Liu ,
Dacheng Yao Open the ORCID record for Dacheng Yao [Opens in a new window]  and
Hanqin Zhang
Yujie Liu*
Affiliation:
National University of Singapore
Dacheng Yao*
Affiliation:
Academy of Mathematics and Systems Science, Chinese Academy of Sciences and University of Chinese Academy of Sciences
Hanqin Zhang*
Affiliation:
National University of Singapore
*
*Postal address: Department of Industrial Systems Engineering and Management, National University of Singapore, 117576, Singapore. Email: yj-liu@nus.edu.sg
**Postal address: Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China; School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China. Email: dachengyao@amss.ac.cn
***Postal address: Department of Analytics and Operations, National University of Singapore, 119245, Singapore. Email: bizzhq@nus.edu.sg
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Abstract

For a continuous-time phase-type (PH) distribution, starting with its Laplace–Stieltjes transform, we obtain a necessary and sufficient condition for its minimal PH representation to have the same order as its algebraic degree. To facilitate finding this minimal representation, we transform this condition equivalently into a non-convex optimization problem, which can be effectively addressed using an alternating minimization algorithm. The algorithm convergence is also proved. Moreover, the method we develop for the continuous-time PH distributions can be used directly for the discrete-time PH distributions after establishing an equivalence between the minimal representation problems for continuous-time and discrete-time PH distributions.

Keywords

Markov chainnon-convex optimizationphase-type distributionrational function

MSC classification

Primary: 60E05: Distributions
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 90C26: Nonconvex programming, global optimization 15A09: Matrix inversion, generalized inverses

Information

Type
Original Article
Information
Advances in Applied Probability , First View , pp. 1 - 37
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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