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

We consider a buffered queueing system that is fed by a Gaussian source and drained at a constant rate. The fluid offered to the system in a time interval ( 0 , t ] is given by a separable continuous Gaussian process Y with stationary increments. The variance function σ 2 : t ↦ V ar Y t of Y is assumed to be regularly varying with index 2 H , for some 0 < H < 1 . ving conditional limit theorems, we investigate how a high buffer level is typically achieved. The underlying large deviation analysis also enables us to establish the logarithmic asymptotics for the probability that the buffer content exceeds u as u → ∞ . In addition, we study how a busy period longer than T typically occurs as T → ∞ , and we find the logarithmic asymptotics for the probability of such a long busy period. udy relies on the weak convergence in an appropriate space of { Y α t / 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 σ 2 , 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.