Passage times in fluid models with application to risk processes

An efficient quadratically convergent algorithm has been derived earlier by Ahn and Ramaswami for computing the busy period distribution of the canonical fluid flow model. In this paper, we derive formulae for a variety of passage time distributions in the canonical fluid flow model in terms of its busy period distribution and that of its reflection about the time axis. These include several passage time distributions with taboo not only of the fluid level 0 but also of a set [a, infinity) of levels. These are fundamental to the analysis of a large set of complex applied probability models, and their use is illustrated in the context of a general insurance risk model with Markovian arrival of claims and phase type distributed claim sizes, a context in which we have also introduced some new ideas that make the analysis very transparent.