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 \(\alpha >1\).