Conditional limit theorems for queues with Gaussian input, a weak convergence approach

The study relies on the weak convergence in an appropriate space of {Yαt/σ(α):t∈R} to a fractional Brownian motion with Hurst parameter H as α→∞. We prove this weak convergence under a fairly general condition on σ2, sharpening recent results of Kozachenko et al. (Queueing Systems Theory Appl. 42 (2002) 113). The core of the proof consists of a new type of uniform convergence theorem for regularly varying functions with positive index.