Matrix-Exponential Distributions: Calculus and Interpretations via Flows

• Published in: Stochastic models 2003-01, Vol.19 (1), p.113-124
• Main Authors: Bladt, Mogens, Neuts, Marcel F.
• Format: Article
• Language: English
• Subjects: Distribution theory; Exact sciences and technology; Flow models; Inference from stochastic processes; time series analysis; Mathematics; Matrix-analytic methods; Matrix-exponential distributions; Phase-type distributions; Probability and statistics; Probability theory and stochastic processes; Rational arrival process; Rational Laplace-Stieltjes transforms; Renewal theory; Risk theory; Sciences and techniques of general use; Statistics
• ISSN: 1532-6349; 1532-4214
• DOI: 10.1081/STM-120018141

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).