Performance Analysis with Truncated Heavy-Tailed Distributions

Abstract

This paper deals with queues and insurance risk processes where a generic service time, resp. generic claim, has the form U ∧ 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 ψ(u) or, equivalently, the tail $\mathbb{P}{\left( {W > u} \right)}$ of the M/G/1 steady-state waiting time W. In the first part of the paper, we present numerical values of ψ(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 ψ(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 → ∞ of the asymptotic exponential decay rate γ = γ ^(K) in a more general truncated Lévy process setting, and give a discussion of some of the implications for the approximations.