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On matrix exponential distributions

Part of: Basic linear algebra Foundations of probability theory

Published online by Cambridge University Press:  01 July 2016

Qi-Ming He  and
Hanqin Zhang
Qi-Ming He*
Affiliation:
Dalhousie University
Hanqin Zhang*
Affiliation:
Chinese Academy of Sciences
*
∗ Postal address: Department of Industrial Engineering, Dalhousie University, Halifax, NS B3J 2X4, Canada. Email address: qi-ming.he@dal.ca
∗∗ Postal address: Institute of Applied Mathematics, Chinese Academy of Sciences, 100080 Beijing, P. R. China. Email address: hanqin@amt.ac.cn
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Abstract

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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.

Keywords

Hankel matrixmatrix exponential distributionphase-type distributionmatrix-analytic methods

MSC classification

Primary: 60A99: None of the above, but in this section
Secondary: 15A18: Eigenvalues, singular values, and eigenvectors
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 1 , March 2007 , pp. 271 - 292
Copyright
Copyright © Applied Probability Trust 2007 

References

Aldous, D. and Shepp, L. (1987). The least variable phase type distribution is Erlang. Stoch. Models 3, 467–473.Google Scholar
Asmussen, S. (1987). Applied Probability and Queues. John Wiley, New York.Google Scholar
Asmussen, S. and Bladt, M. (1996). Renewal theory and queueing algorithms for matrix-exponential distributions. In Proc. First Internat. Conf. Matrix Analytic Methods Stoch. Models, eds Alfa, A. S. and Chakravarthy, S., Marcel Dekker, New York, pp. 313–341.Google Scholar
Benvenuti, L. and Farina, L. (2004). A tutorial on the positive realization problem. IEEE Trans. Automatic Control 49, 651–664.CrossRefGoogle Scholar
Bladt, M. and Neuts, M. F. (2003). Matrix-exponential distributions: calculus and interpretations via flows. Stoch. Models 19, 113–124.Google Scholar
Commault, C. and Chelma, J. P. (1993). On dual and minimal phase-type representations. Stoch. Models 9, 421–434.Google Scholar
Commault, C. and Mocanu, S. (2003). Phase-type distributions and representations: some results and open problems for system theory. Internat. J. Control 76, 566–580.CrossRefGoogle Scholar
Dehon, M. and Latouche, G. (1982). A geometric interpretation of the relations between the exponential and the generalized Erlang distributions. Adv. Appl. Prob. 14, 885–897.CrossRefGoogle Scholar
Fackrell, M. (2003). Characterization of matrix-exponential distributions. , University of Adelaide.Google Scholar
Fackrell, M. (2005). Fitting with matrix-exponential distributions. Stoch. Models 21, 377–400.CrossRefGoogle Scholar
Gilbert, E. J. (1959). On the identifiability problem for functions of finite Markov chains. Ann. Math. Statist. 30, 688–697.Google Scholar
He, Q. and Zhang, H. (2005). An algorithm for computing minimal Coxian representations. To appear in INFORMS J. Computing.Google Scholar
He, Q. and Zhang, H. (2005). Spectral polynomial algorithms for computing bi-diagonal representations for phase type distributions and matrix exponential distributions. Stoch. Models 22, 289–317.CrossRefGoogle Scholar
Ito, H., Amari, S.-I. and Kobayashi, K. (1992). Identifiability of hidden Markov information sources and their minimum degrees of freedom. IEEE Trans. Inf. Theory 38, 324–333.Google Scholar
Johnson, M. A. and Taaffe, M. R. (1989). Matching moments to phase distributions: mixtures of Erlang distributions of common order. Stoch. Models 5, 711–743.Google Scholar
Johnson, M. A. and Taaffe, M. R. (1993). Tchebycheff systems for probabilistic analysis. Amer. J. Math. Manag. Sci. 13, 83–111.Google Scholar
Karlin, S. (1968). Total Positivity, Vol. I. Stanford University Press.Google Scholar
Karlin, S. and Studden, W. J. (1966). Tchebycheff Systems: With Applications in Analysis and Statistics. John Wiley, New York.Google Scholar
Lancaster, P. and Tismenetsky, M. (1985). The Theory of Matrices. Academic Press, New York.Google Scholar
Latouche, G. and Ramaswami, V. (1999). Introduction to Matrix Analytic Methods in Stochastic Modelling. Society for Industrial and Applied Mathematics, Philadelphia, PA.Google Scholar
Lipsky, L. (1992). Queueing Theory: A Linear Algebraic Approach. McMillan, New York.Google Scholar
Luenberger, D. G. (1979). Positive linear systems. In Introduction to Dynamic Systems: Theory, Models, and Applications, John Wiley, New York.Google Scholar
Neuts, M. F. (1975). Probability distributions of phase type. In Liber Amicorum Prof. Emeritus H. Florin, University of Louvain, pp. 173–206.Google Scholar
Neuts, M. F. (1981). Matrix-Geometric Solutions in Stochastic Models. An Algorithmic Approach. The Johns Hopkins University Press, Baltimore, MD.Google Scholar
O'Cinneide, C. A. (1989). On non-uniqueness of representations of phase-type distributions. Stoch. Models 5, 247–259.Google Scholar
O'Cinneide, C. A. (1990). Characterization of phase-type distributions. Stoch. Models 6, 1–57.Google Scholar
O'Cinneide, C. A. (1991). Phase-type distributions and invariant polytopes. Adv. Appl. Prob. 23, 515–535.Google Scholar
O'Cinneide, C. A. (1993). Triangular order of triangular phase-type distributions. Stoch. Models 9, 507–529.CrossRefGoogle Scholar
Rockafellar, R. T. (1970). Convex Analysis. Princeton University Press.Google Scholar
Ryden, T. (1996). On identifiability and order of continuous-time aggregated Markov chains, Markov-modulated Poisson processes and phase-type distributions. J. Appl. Prob. 33, 640–653.Google Scholar
Van de Liefvoort, A. and Heindl, A. (2005). Approximating matrix-exponential distributions by global randomization. Stoch. Models 21, 669–693.CrossRefGoogle Scholar