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

Victor F. Araman, Hong Chen, Peter W. Glynn, Li Xia

Abstract: A“scheduled” arrival process is one in which the nth arrival is scheduled for time n, but instead occurs at $$n+\xi _n$$ n + ξ n , where the $$\xi _j$$ ξ 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 $$p_n\sim c \,n^{-\alpha }$$ p n ∼ c n - α as $$n\rightarrow \infty $$ n → ∞ , for $$c>0$$ c > 0 and $$\alpha >1$$ α > 1 .