Advances in Applied Probability

On matrix exponential distributions

Qi-Ming He and Hanqin Zhang

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

Adv. in Appl. Probab. Volume 39, Number 1 (2007), 271-292.

First available in Project Euclid: 30 March 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60A99: None of the above, but in this section
Secondary: 15A18: Eigenvalues, singular values, and eigenvectors

Hankel matrix matrix exponential distribution phase-type distribution matrix-analytic methods


He, Qi-Ming; Zhang, Hanqin. On matrix exponential distributions. Adv. in Appl. Probab. 39 (2007), no. 1, 271--292. doi:10.1239/aap/1175266478.

Export citation


  • Aldous, D. and Shepp, L. (1987). The least variable phase type distribution is Erlang. Stoch. Models 3, 467--473.
  • Asmussen, S. (1987). Applied Probability and Queues. John Wiley, New York.
  • 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 A. S. Alfa and S. Chakravarthy, Marcel Dekker, New York, pp. 313--341.
  • Benvenuti, L. and Farina, L. (2004). A tutorial on the positive realization problem. IEEE Trans. Automatic Control 49, 651--664.
  • Bladt, M. and Neuts, M. F. (2003). Matrix-exponential distributions: calculus and interpretations via flows. Stoch. Models 19, 113--124.
  • Commault, C. and Chelma, J. P. (1993). On dual and minimal phase-type representations. Stoch. Models 9, 421--434.
  • 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.
  • 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.
  • Fackrell, M. (2003). Characterization of matrix-exponential distributions. Doctoral Thesis, University of Adelaide.
  • Fackrell, M. (2005). Fitting with matrix-exponential distributions. Stoch. Models 21, 377--400.
  • Gilbert, E. J. (1959). On the identifiability problem for functions of finite Markov chains. Ann. Math. Statist. 30, 688--697.
  • He, Q. and Zhang, H. (2005). An algorithm for computing minimal Coxian representations. To appear in INFORMS J. Computing.
  • 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.
  • 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.
  • 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.
  • Johnson, M. A. and Taaffe, M. R. (1993). Tchebycheff systems for probabilistic analysis. Amer. J. Math. Manag. Sci. 13, 83--111.
  • Karlin, S. (1968). Total Positivity, Vol. I. Stanford University Press.
  • Karlin, S. and Studden, W. J. (1966). Tchebycheff Systems: With Applications in Analysis and Statistics. John Wiley, New York.
  • Lancaster, P. and Tismenetsky, M. (1985). The Theory of Matrices. Academic Press, New York.
  • Latouche, G. and Ramaswami, V. (1999). Introduction to Matrix Analytic Methods in Stochastic Modelling. Society for Industrial and Applied Mathematics, Philadelphia, PA.
  • Lipsky, L. (1992). Queueing Theory: A Linear Algebraic Approach. McMillan, New York.
  • Luenberger, D. G. (1979). Positive linear systems. In Introduction to Dynamic Systems: Theory, Models, and Applications, John Wiley, New York.
  • Neuts, M. F. (1975). Probability distributions of phase type. In Liber Amicorum Prof. Emeritus H. Florin, University of Louvain, pp. 173--206.
  • Neuts, M. F. (1981). Matrix-Geometric Solutions in Stochastic Models. An Algorithmic Approach. The Johns Hopkins University Press, Baltimore, MD.
  • O'Cinneide, C. A. (1989). On non-uniqueness of representations of phase-type distributions. Stoch. Models 5, 247--259.
  • O'Cinneide, C. A. (1990). Characterization of phase-type distributions. Stoch. Models 6, 1--57.
  • O'Cinneide, C. A. (1991). Phase-type distributions and invariant polytopes. Adv. Appl. Prob. 23, 515--535.
  • O'Cinneide, C. A. (1993). Triangular order of triangular phase-type distributions. Stoch. Models 9, 507--529.
  • Rockafellar, R. T. (1970). Convex Analysis. Princeton University Press.
  • 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.
  • Van de Liefvoort, A. and Heindl, A. (2005). Approximating matrix-exponential distributions by global randomization. Stoch. Models 21, 669--693.