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A geometric interpretation of the relations between the exponential and generalized Erlang distributions

Published online by Cambridge University Press:  01 July 2016

Michel Dehon  and
Guy Latouche
Michel Dehon*
Affiliation:
Université Libre de Bruxelles
Guy Latouche*
Affiliation:
Université Libre de Bruxelles
*
∗Postal address: Université Libre de Bruxelles, Faculté des Sciences-CP 212, Laboratoire d'Informatique Theorique, Boulevard du Triomphe, B-1050 Bruxelles, Belgium.
∗Postal address: Université Libre de Bruxelles, Faculté des Sciences-CP 212, Laboratoire d'Informatique Theorique, Boulevard du Triomphe, B-1050 Bruxelles, Belgium.
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Abstract

Linear combinations of exponential distribution functions are considered, and the class of distribution functions so obtainable is investigated. Convex combinations correspond to hyperexponential distributions, while non-convex combinations yield, among other, generalized Erlang distributions obtainable as sums of independent exponential random variables with different parameters.

For a given number n of different exponential distributions, the class investigated is an (n – 1)-dimensional convex subset of the n-dimensional real vector space generated by the n distribution functions. The geometric aspect of this subset is revealed, together with the location of hyperexponential and generalized Erlang distributions.

Keywords

EXPONENTIAL DISTRIBUTIONSERLANG DISTRIBUTIONSLINEAR COMBINATION OF DISTRIBUTION FUNCTIONSVECTOR SPACESCONVEX SETS
Type
Research Article
Information
Advances in Applied Probability , Volume 14 , Issue 4 , December 1982 , pp. 885 - 897
Copyright
Copyright © Applied Probability Trust 1982 

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References

[1] Latouche, G. (1982) A phase-type semi-Markov point process. SIAM J. Algebraic and Discrete Methods 3, 77–90.CrossRefGoogle Scholar
[2] Neuts, M. F. (1975) Probability distributions of phase type. In Liber Amicorum Prof. Emeritus H. Florin, Université de Louvain, Belgium, 173–206.Google Scholar
[3] Neuts, M. F. (1979) A versatile Markovian point process. J. Appl. Prob. 16, 764–779.CrossRefGoogle Scholar
[4] Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models. An Algorithmic Approach. The Johns Hopkins University Press, Baltimore, Md.Google Scholar