MaPhySto
The Danish National Research Foundation:
Network in Mathematical Physics and Stochastics

MPS-RR 2004-24
November 2004

# Performance analysis with truncated heavy-tailed distributions

by: Søren Asmussen, Mats Pihlsgård

## Abstract

This paper deals with queues and insurance risk processes where a generic service time, resp. generic claim, has the form $U\wedge K$ for some r.v. $U$ with distribution $B$ which is heavy-tailed, say Pareto or Weibull, and a typically large $K$, say much larger than $\mathbb{E} U$. We study the compound Poisson ruin probability $\psi(u)$ or, equivalently, the tail $\mathbb{P}(W>u)$ of the $M/G/1$ steady-state waiting time $W$. In the first part of the paper, we present numerical values of $\psi(u)$ for different values of $K$ by using the classical Siegmund algorithm as well as a more recent algorithm designed for heavy-tailed claims/service times, and compare the results to different approximations of $\psi(u)$ in order to figure out the threshold between the light-tailed regime and the heavy-tailed regime. In the second part, we investigate the asymptotics as $K\to\infty$ of the asymptotic exponential decay rate ${\gamma}={\gamma}^{(K)}$ in a more general truncated Lévy process setting, and give a discussion of some of the implications for the approximations.

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