Gaussian limits for scheduled traffic with super-heavy tailed perturbations

A scheduled arrival model is one in which the jth customer is scheduled to arrive at time jh but the customer actually arrives at time \(jh + \xi _j\), where the \(\xi _j\)’s are independent and identically distributed. It has previously been shown that the arrival counting process for scheduled traffic obeys a functional central limit theorem (FCLT) with fractional Brownian motion (fBM) with Hurst parameter \(H \in (0,1/2)\) when the \(\xi _j\)’s have a Pareto-like tail with tail exponent lying in (0, 1). Such limit processes exhibit less variability than Brownian motion, because the scheduling feature induces negative correlations in the arrival process. In this paper, we show that when the tail of the \(\xi _j\)’s has a super-heavy tail, the FCLT limit process is Brownian motion (i.e., \(H=1/2\)), so that the heaviness of the tails eliminates any remaining negative correlations and generates a limit process with independent increments. We further study the case when the \(\xi _j\)’s have a Cauchy-like tail, and show that the limit process in this setting is a fBM with \(H=0\). So, this paper shows that the entire range of fBMs with \(H \in [0,1/2]\) are possible as limits of scheduled traffic.