Positive correlations and buffer occupancy: lower bounds via supermodular ordering

We use recent advances from the theory of multivariate stochastic orderings to formalize the "folk theorem" to the effect that positive correlations lead to increased buffer occupancy and larger buffer levels at a discrete-time infinite capacity multiplexer queue. Input sequences are compared in the supermodular (sm) ordering and buffer contents in the increasing convex (icx) ordering, respectively. Three popular classes of (discrete-time) traffic models are discussed, namely the fractional Gaussian noise traffic model, the on-off source model and the M|G|∞ traffic model. The independent version of an input process in each of these classes of traffic models is a member of the same class. In varying degree of generality, we show that this independent version is smaller than the input sequence itself, and that the corresponding buffer content processes are similarly ordered.