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We present the first algorithm that samples$\max_{n\geq0}\{S_{n}-n^{\alpha}\},$ where $S_n$ is a mean zero random walk,and $n^{\alpha}$ with $\alpha\in(1/2,1)$ defines a nonliner boundary. We showthat our algorithm has finite expected running time. We also apply thealgorithm to construct the first exact simulation method for the steady-statedeparture process of a $GI/GI/\infty$ queue where the service time distributionhas infinite mean.

Exact Sampling of the Infinite Horizon Maximum of a Random Walk Over a Non-linear Boundary