The variation of optimal bandwidth and buffer allocation with the number of sources

We consider a single node which multiplexes a large number of traffic sources. We ask a simple question: how do the optimal allocations of bandwidth and buffer vary with the number of sources? We investigate this issue using previous results on the probability of overflow for an aggregate of i.i.d. flows, e.g., overflow resulting from effective bandwidth models. We wish to determine the variation of the minimum cost allocations of bandwidth and buffer with the number of sources, given a cost per unit of each resource. We first consider a class of on/off fluid flows. We find that the optimal bandwidth allocation above the mean rate and the optimal buffer allocation are both proportional to the square root of the number of sources. Correspondingly, we find that the excess cost incurred by a fixed buffer allocation or by linear buffer allocations is proportional to the square of the percentage difference between the assumed number of sources and the actual number of sources and to the square root of the number of sources. We next consider a class of general i.i.d. sources for which the aggregate effective bandwidth is a decreasing convex function of buffer and linearly proportional to the number of sources. We find that the optimal buffer allocation is strictly increasing with the number of sources. Correspondingly, we find that the excess cost incurred by a fixed buffer allocation is an increasing convex function of the difference between the assumed number of sources and the actual number of sources.