We study the structure and robustness of universality classes for queueing, deriving analytic results for priority-based models with continuous-valued priorities. By mapping one model onto the problem of biased diffusion, we show that its distribution of waiting times, $P(\tau )$, decreases for large times $\tau$ as $P(\tau )\sim {\tau }^{-3/2}$ or as $P(\tau )\sim {\tau }^{-5/2}\exp (-\tau /{\tau }_0)$ in different parameter regimes. In a second model, introducing a cost for switching between different classes of tasks substantially changes the asymptotic behavior of $P(\tau )$.

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