Applied Probability Trust (15 October 2010) RUIN EXCURSIONS, THE G/G/∞ QUEUE AND TAX PAYMENTS IN RENEWAL RISK MODELS H. ALBRECHER, * University of Lausanne S.C. BORST, ** Eindhoven University of Technology O.J. BOXMA, *** Eindhoven University of Technology J. RESING, **** Eindhoven University of Technology Abstract In this paper we investigate the number and maximum severity of the ruin excursion of the insurance portfolio reserve process in the Cram´ er-Lundberg model with and without tax payments. We also provide a relation of the Cram´ er-Lundberg risk model with the G/G/∞ queue and use it to derive some explicit ruin probability formulas. Finally, the renewal risk model with tax is considered, and an asymptotic identity is derived that in some sense extends the tax identity of the Cram´ er-Lundberg risk model. Keywords: classical risk model; ruin probability; G/G/∞ queue; tax; renewal model 2000 Mathematics Subject Classification: Primary 91B30 Secondary 60K30 1. Introduction Consider the classical Cram´ er-Lundberg model in risk theory to describe the surplus process {R t } at time t of an insurance portfolio. Starting with an initial capital x, premium is collected according to a constant premium intensity (normalized to) 1. * Postal address: Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne, Quartier UNIL-Dorigny, 1015 Lausanne, Switzerland ** Postal address: Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands and Alcatel-Lucent, Bell Labs, 600 Mountain Avenue, P.O. Box 636, Murray Hill, NJ 07974-0636, USA *** Postal address: Eindhoven University of Technology and EURANDOM, P.O. Box 513, 5600 MB Eindhoven, The Netherlands **** Postal address: Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands 1