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.

In this paper we present results which suggest that Internet packet delay, when viewed as a time series, is self-similar in nature. A self-similar phenomenon displays the same or similar statistical properties across a wide range of time scales. Self-similarity has previously been observed in the magnitude of traffic transmitted on local-area and wide-area networks. Our research is the first to explore the self-similarity of traffic delay rather than traffic magnitude. Our results imply that the delay experienced by application layer protocols is extremely bursty and that traditional Poisson models cannot accurately predict the degree of burstiness. Furthermore, we present evidence that the degree of self-similarity for a round-trip path in the Internet may be correlated with the packet loss observed on that path.

We are concerned with characterizing the variation of multiplexing gains with source type and burstiness in integrated service systems such as ATM. We model a fixed capacity high speed bit pipe that multiplexes a moderate number of bit streams with minimal buffer under a low loss constraint. Each service type is defined by its instantaneous bitrate distribution, but the bitrate distribution of multiplexed streams is approximated as Gaussian. The Gaussian approximation is not as accurate as Chernoff bounds, but it allows for stronger characterization of multiplexing gains. We consider three schemes for allocating bandwidth to services: by individual user, by path and service type, and by path only. We find explicit formulae for sensitivities of required capacity to source rate mean and variance and to loss rate. We characterize multiplexing gains and costs to identify the benefits of each allocation policy. We find that the capacity savings resulting from sharing resources is proportional to the square root of the ratio of source rate variance to source rate mean. This suggests that although bursty sources require more bandwidth, multiplexing gains are increasing with burstiness. We also find that the extra capacity required to multiplex dissimilar source types is increasing with the difference between their burstinesses. This suggests that when bit streams are partially grouped, it is most important first, to group similar source types.