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)similar to tau(-3/2), when the average rates of task arrival lambda and execution mu satisfy mu <=lambda (as was earlier noted empirically). For mu>lambda, P(tau)similar to tau(-5/2) exp[-(root mu-root lambda)(2)tau].