An alternative characterization for matrix exponential distributions

A necessary condition for a rational Laplace–Stieltjes transform to correspond to a matrix exponential distribution is that the pole of maximal real part is real and negative. Given a rational Laplace–Stieltjes transform with such a pole, we present a method to determine whether or not the numerator polynomial admits a transform that corresponds to a matrix exponential distribution. The method relies on the minimization of a continuous function of one variable over the nonnegative real numbers. Using this approach, we give an alternative characterization for all matrix exponential distributions of order three.

[1]  Søren Asmussen,et al.  Computational methods in risk theory: A matrix-algorithmic approach , 1992 .

[2]  Søren Asmussen,et al.  Ruin probabilities , 2001, Advanced series on statistical science and applied probability.

[3]  Mark Fackrell,et al.  Fitting with Matrix-Exponential Distributions , 2005 .

[4]  Marcel F. Neuts,et al.  Matrix-Geometric Solutions in Stochastic Models , 1981 .

[5]  Mogens Bladt,et al.  Point processes with finite-dimensional conditional probabilities , 1999 .

[6]  Colm Art O'Cinneide,et al.  Phase-type distributions and invariant polytopes , 1991, Advances in Applied Probability.

[7]  M. Dehon,et al.  A geometric interpretation of the relations between the exponential and generalized Erlang distributions , 1982, Advances in Applied Probability.

[8]  Colm Art O'Cinneide,et al.  Triangular order of triangular phase-type distributions ∗ , 1993 .

[9]  Marcel F. Neuts,et al.  Matrix‐Exponential Distributions: Calculus and Interpretations via Flows , 2003 .

[10]  C. O'Cinneide.  Characterization of phase-type distributions , 1990 .

[11]  W. G. Marchal,et al.  A note on generalized hyperexponential distributions , 1992 .

[12]  Deep Medhi,et al.  Performance analysis of a digital link with heterogeneous multislot traffic , 1995, IEEE Trans. Commun..

[13]  Lester Lipsky,et al.  Queueing Theory: A Linear Algebraic Approach , 1992 .

[14]  Rembert Reemtsen,et al.  Numerical Methods for Semi-Infinite Programming: A Survey , 1998 .

[15]  E. Page.  The Single Server Queue , 1970 .

[16]  Mogens Bladt,et al.  Renewal Theory and Queueing Algorithms for Matrix-Exponential Distributions , 1996 .

[17]  Nigel G. Bean,et al.  Characterization of Matrix-Exponential Distributions , 2008 .

[18]  Ushio Sumita,et al.  Classes of probability density functions having Laplace transforms with negative zeros and poles , 1987, Advances in Applied Probability.

[19]  D. Cox.  A use of complex probabilities in the theory of stochastic processes , 1955, Mathematical Proceedings of the Cambridge Philosophical Society.

[20]  Walter L. Smith.  On the distribution of queueing times , 1953, Mathematical Proceedings of the Cambridge Philosophical Society.

[21]  Appie van de Liefvoort,et al.  Modeling correlated traffic with a generalized IPP , 2000, Perform. Evaluation.

[22]  A. Zemanian.  On the pole and zero locations of rational Laplace transformations of non-negative functions. II , 1959 .

[23]  D. Cox.  The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables , 1955, Mathematical Proceedings of the Cambridge Philosophical Society.