Hitting and ruin probabilities for compound Poisson processes and the cycle maximum of the M/G/1 queue

Abstract

We derive a new formula for the probability that a compound Poisson process with positive jumps hits a lower straight line before it crosses a parallel upper line. This yields a new approach to determine the distribution of the cycle maximum of the M/G/1 queue. Moreover, we express the hitting probabilities in terms of the corresponding ruin probabilities.