On a single server queue fed by scheduled traffic with Pareto perturbations

Abstract

A“scheduled” arrival process is one in which the nth arrival is scheduled for time n, but instead occurs at $n+\xi _n$ , where the $\xi _j$’s are i.i.d. We describe here the behavior of a single server queue fed by such traffic in which the processing times are deterministic. A particular focus is on perturbations with Pareto-like tails but with finite mean. We obtain tail approximations for the steady-state workload in both cases where the queue is critically loaded and under a heavy-traffic regime. A key to our approach is our analysis of the tail behavior of a sum of independent Bernoulli random variables with parameters of the form $p_n\sim c \,n^{-\alpha }$ as $n\rightarrow \infty$, for $c>0$ and $\alpha >1$.