The Effect of Positive Correlations on Buffer Occupancy: Comparison and 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 leads to increased buffer occupancy and larger buffer levels at a discrete time multiplexer queue of infinite capacity. We do so by comparing input sequences in the supermodular (sm) ordering and the corresponding buffer contents in the increasing convex (icx) ordering, respectively. Three popular classes of (discrete-time) traffic models are considered here, namely, the fractional Gaussian noise (FGN) traffic model, the on-off source model and the M|G|infinity traffic model. The independent version of an input process in each of these classes of traffic models is a member of the same class. We show that this independent version is smaller than the input sequence itself and that the corresponding buffer content processes are ordered in the same direction. For each traffic model, we show by simulations that the first and second moments of buffer levels are ordered in agreement with the comparison results. The more general version of the folk theorem, namely "the larger the positive correlations of input traffic, the higher the buffer occupancy levels" is established in some cases. For the FGN traffic models, we show that the process with higher Hurst parameter is larger than the process with smaller Hurst parameter. In the case of the M|G|infinity model, the effect of session-duration variability is discussed and the comparison result is obtained in the bivariate case