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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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References

Latouche, G. and Ramaswami, V. (1999). Introduction to Matrix Analytic Methods in Stochastic Modeling, SIAM, Philadelphia, PA.CrossRefGoogle Scholar
Neuts, M. F. (1981). Matrix Geometric Solutions in Stochastic Models: An Algorithmic Approach, Johns Hopkins University Press, Baltimore, MD.Google Scholar
Bladt, M. (1996). The variance constant for the actual waiting time of the $Ph/Ph/1$ queue. Ann. Appl. Probab. 6, 766–777.CrossRefGoogle Scholar
Dai, J. G., He, S., and Tezcan, T. (2010). Many-server diffusion limits for $G/Ph/n+GI$ queues. Ann. Appl. Probab. 20, 1854–1890.CrossRefGoogle Scholar
Latouche, G. and Taylor, P. (2002). Matrix-Analytic Methods: Theory and Applications, World Scientific, Hackensack, NJ.CrossRefGoogle Scholar
Asmussen, S. and Albrecher, H. (2000). Ruin Probabilities, World Scientific, Hong Kong.CrossRefGoogle Scholar
Alfa, A. S. and Li, W. (2002). A homogeneous PCS network with Markov call arrival process and phase type cell residence time. Wirel. Netw. 8, 597–605.CrossRefGoogle Scholar
He, Q. M. (2014). Fundamentals of Matrix-Analytic Methods, Springer, New York.CrossRefGoogle Scholar
Song, J. and Zipkin, P. (1993). Inventory control in a fluctuating demand environment. Oper. Res. 41, 351–370.CrossRefGoogle Scholar
O’Cinneide, C. A. (1989). On non-uniqueness of representations of phase-type distributions. Stoch. Models 5, 247–259.CrossRefGoogle Scholar
Dmitriev, N. and Dynkin, E.B. (1945). On the characteristic numbers of a stochastic matrix. C.R. (Doklady) Acad. Sci. URSS (N.S.) 49, 159–162.Google Scholar
Dmitriev, N. and Dynkin, E.B. (1946). On the characteristic numbers of stochastic matrices. Bull. Acad. Sci. URSS. Ser. Math. [Izvestia Akad. Nauk SSSR] 10, 167–184 (in Russian with English summary).Google Scholar
O’Cinneide, C. A. (1991). Phase-type distributions and invariant polytopes. Adv. Appl. Probab. 23, 515–535.CrossRefGoogle Scholar
Commault, C. and Chemla, J. P. (1996). An invariant of representations of PH distributions and some applications. J. Appl. Probab. 33, 368–381.CrossRefGoogle Scholar
Fackrell, M., He, Q. M., Taylor, P., and Zhang, H. (2010). The algrebaic degree of phase-type distributions. J. Appl. Probab. 47, 611–629.CrossRefGoogle Scholar
Aldous, A. and Shepp, L., L. (1987). The least variable phase type distribution is Erlang. Stoch. Models 3, 467–473.Google Scholar
Marshall, A. W. and Olkin, I. (1979). Inequalities: Theory of Majorization and its Applications, Academic, New York.Google Scholar
O’Cinneide, C. A. (1991). Phase-type distributions and majorization. Ann. Appl. Probab. 1, 219–227.Google Scholar
He, Q. M. and Zhang, H. (2005). A note on unicyclic representations of phase type distributions. Stoch. Models 21, 465–483.CrossRefGoogle Scholar
He, Q. M. and Zhang, H. (2006). Spectral polynomial algorithms for computing bi-diagonal representations for phase type distributions and matrix-exponential distributions. Stoch. Models 22, 289–317.CrossRefGoogle Scholar
He, Q. M. and Zhang, H. (2006). PH-invariant polytopes and Coxian representations of phase type distributions. Stoch. Models 22, 383–409.CrossRefGoogle Scholar
He, Q. M. and Zhang, H. (2008). An algorithm for computing minimal Coxian representations. INFORMS J. Comput. 20, 179–190.CrossRefGoogle Scholar
Mészáros, A., Horváth, G., and Telek, M. (2013). Representation transformations for finding Markovian representations. In Analytical and Stochastic Modeling Techniques and Applications: 20th International Conference, LNCS 7984, 277-291, Springer Berlin Heidelberg.CrossRefGoogle Scholar
Telek, M. and Horváth, G. (2007). A minimal representation of Markov arrival processes and a moments matching method. Perform. Evaluation 64, 1153–1168.CrossRefGoogle Scholar
Commault, C. and Mocanu, S. (2003). PH distributions and representations: some results and open problems for system theory. Int. J. Control 76, 566–580.CrossRefGoogle Scholar
Benvenuti, L. (2022). Minimal positive realizations: a survey. Automatica 143, 110422.CrossRefGoogle Scholar
Liu, Y., Yao, D., and Zhang, H. (2023). Minimal positive realizations for linear systems with two-different-pole transfer functions. Eur. J. Control 69, 100720.CrossRefGoogle Scholar
O’Cinneide, C. A. (1993). Triangular order of triangular phase-type distributions. Stoch. Models 9, 507–529.CrossRefGoogle Scholar
Asmussen, S. and Bladt, M. (1997). Renewal theory and queueing algorithms for matrix-exponential distributions. In Matrix-Analytic Methods in Stochastic Models, eds Alfa, A. and Chakravarthy, S. R., Marcel Dekker, New York, pp. 313–341.Google Scholar
O’Cinneide, C. A. (1990). Characterization of phase-type distributions. Stoch. Models 6, 1–57.CrossRefGoogle Scholar
Lancaster, P. and Tismenetsky, M. (1985). The Theory of Matrices, Academic Press, New York.Google Scholar
Bertsekas, D. and Tsitsiklis, J. (2015). Parallel and Distributed Computation: Numerical Methods, Athena Scientific, Belmont, MA.Google Scholar
Lee, G. M., Tam, N. N., and Yen, N. D. (2005). Quadratic Programming and Affine Variational Inequalities: A Qualitative Study, Springer, New York.Google Scholar
Grippo, L. and Sciandrone, M. (2000). On the convergence of the block nonlinear Gauss-Seidel method under convex constraints. Oper. Res. Lett. 26, 127–136.CrossRefGoogle Scholar
Maier, R. S. (1991). The algebraic construction of phase-type distributions. Stoch. Models 7, 573–602.CrossRefGoogle Scholar
Dehon, M. and Latouche, G. (1982). A geometric interpretation of the relations between the exponential and the generalized Erlang distributions. Adv. Appl. Probab. 14, 885–897.CrossRefGoogle Scholar
O’Cinneide, C. A. (1999). Phase-type distributions: open problems and a few properties. Stoch. Models 15, 731–757.CrossRefGoogle Scholar