Matrix‐Exponential Distributions: Calculus and Interpretations via Flows

Abstract

By considering randomly stopped deterministic flow models, we develop an intuitively appealing way to generate probability distributions with rational Laplace–Stieltjes transforms on [0,∞). 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 allows 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 (MAP) to the setting of matrix-exponential distribution. That process is known as the Rational arrival process (RAP).

Keywords:

• Matrix-exponential distributions
• Phase-type distributions
• Flow models
• Rational Laplace–Stieltjes transforms
• Rational arrival process
• Matrix-analytic methods
• Renewal theory
• Risk theory

Acknowledgments