Asymptotics of the maximum workload for a class of gaussian queueing systems

The asymptotics of the maximum workload in a fluid queueing system fed by a process containing a random component are described by a centered Gaussian process. It is assumed that the variance of the process is a regularly varying at infinity function with index belonging to interval $(0,2)$. Such class of processes includes, in particular, a sum of independent fractional Brownian motions. It is shown that, under an appropriate scaling, the maximum workload over interval $[0,t]$ converges in probability to an explicitly given constant as $t$ increases.