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On matrix exponential distributions

  • Qi-Ming He (a1) and Hanqin Zhang (a2)


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.

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Corresponding author

Postal address: Department of Industrial Engineering, Dalhousie University, Halifax, NS B3J 2X4, Canada. Email address:
∗∗ Postal address: Institute of Applied Mathematics, Chinese Academy of Sciences, 100080 Beijing, P. R. China. Email address:


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Advances in Applied Probability
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