OK i think I got it and my initial guess is correct. Rereading the original paper, I see that the parameters L and \epsilon match up with U and \alpha in the Simpson paper. So then according to Equation (3.2),
P(\alpha > L) = \epsilon \quad \implies \quad \exp(-\lambda L) = \epsilon \quad \implies \quad \lambda = -\frac{\ln(\epsilon)}{L}.