Methodol Comput Appl Probab (2006) 8: 497–515 DOI 10.1007/s11009-006-0426-9 Passage Times in Fluid Models with Application to Risk Processes V. Ramaswami Received: 23 August 2005 / Revised: 9 February 2006 / Accepted: 21 March 2006 © Springer Science + Business Media, LLC 2006 Abstract 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, ∞) 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. Keywords Insurance risk · Fluid-flow · Transient results · Matrix-geometric method AMS 2000 Subject Classification 60J25 · 60K25 · 60K15 · 60K37 1 Introduction In this paper, we consider the canonical Markov modulated fluid flow model and derive a number of passage time distributions for it exploiting fully the previously obtained results of Ramaswami (1999) and Ahn and Ramaswami (2004, 2005, 2006); our other related papers are Ahn and Ramaswami (2003), Ahn et al. (2005a, 2005b). All passage time distributions considered here are characterized explicitly in terms of the Laplace–Stieltjes transform matrices characterizing the first return time to fluid level 0 in each of two fluid flows that are reflections of each other, given that an upward trajectory of the concerned flow starts at level 0 at the time origin. V. Ramaswami (B ) AT&T Labs-Research, 180 Park Avenue, Florham Park, NJ 07932, USA e-mail: vramaswami@att.com