×

Matrix-exponential distributions: Calculus and interpretations via flows. (English) Zbl 1020.60005

Summary: By considering randomly stopped deterministic flow models, we develop an intuitively appealing way to generate probability distributions with rational Laplace-Stieltjes transforms on [0,\(\infty\)). That approach includes and generalizes the formalism of PH-distributions. That generalization results in the class of matrix-exponential probability distributions. To illustrate the novel way of thinking that is required to use these in stochastic models, we retrace the derivations of some results from matrix-exponential renewal theory and prove a new extension of a result from risk theory. Essentially the flow models allow for keeping track of the dynamics of a mechanism that generates matrix-exponential distributions in a similar way to the probabilistic arguments used for phase-type distributions involving transition rates. We also sketch a generalization of the Markovian arrival process to the setting of matrix-exponential distribution. That process is known as the rational arrival process.

MSC:

60E05 Probability distributions: general theory
60K05 Renewal theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1142/9789812779311 · doi:10.1142/9789812779311
[2] Asmussen S., Matrix-Analytic Methods in Stochastic Models pp 313– (1996) · doi:10.1201/b17050-17
[3] DOI: 10.1016/S0304-4149(99)00006-X · Zbl 0997.60077 · doi:10.1016/S0304-4149(99)00006-X
[4] Asmussen S., Scand. Actuarial J. 1 pp 19– (1996) · Zbl 0876.62089 · doi:10.1080/03461238.1996.10413960
[5] Knight, F.B. 1981. ”Essays on Prediction Processes”. Vol. 1, Hayward, CA: The Institute of Mathematical Statistics. · Zbl 1356.60005
[6] Knight F.B., Foundation of Prediction Processes (1992)
[7] Neuts M.F., Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach (1981) · Zbl 0469.60002
[8] DOI: 10.1080/15326349908807134 · Zbl 0701.62021 · doi:10.1080/15326349908807134
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.