Power-law and exponential tails in a stochastic priority-based model queue

We derive exact asymptotic results for a stochastic queueing model in which tasks are executed according to a continuous-valued priority. The distribution $P\left(\tau \right)$ of the waiting times $\tau$ of executed tasks for this model is shown to behave asymptotically as a power law, $P\left(\tau \right)\sim {\tau }^{-3∕2}$, when the average rates of task arrival $\lambda$ and execution $\mu$ satisfy $\mu \le \lambda$ (as was earlier noted empirically). For $\mu >\lambda$, $P\left(\tau \right)\sim {\tau }^{-5∕2}\exp [-(\sqrt{\mu }-\sqrt{\lambda }{)}^2\tau ]$.