In this article we study Gaussian queues (that is, queues fed by Gaussian processes, such as fractional Brownian motion (fBm) and the integrated Ornstein-Uhlenbeck (iOU) process), with a focus on the dependence structure of the workload process. The main question is to what extent does the workload process inherit the dependence properties of the input process? We first present a specific notion of dependence that allows (in asymptotic regimes) explicit analysis. For the special cases of fBm and iOU, we analyze the behavior of this metric under a many sources scaling. Relying on (the generalized version of) Schilder's theorem, we are able to characterize its decay. We observe that the dependence structure of the input process essentially carries over to the workload process (in the asymptotic regime that we have chosen, in terms of our specific notion of dependence).