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(tau) of the waiting times tau of executed tasks for this model is shown to behave asymptotically as a power law, P(tau) approximately tau(-3/2), when the average rates of task arrival lambda and execution micro satisfy micro<or=lambda (as was earlier noted empirically). For micro>lambda, P(tau) approximately tau(-5/2)exp[-(sqrt[micro]-sqrt[lambda])2tau].