Traffic equivalence and substitution in a multiplexer with applications to dynamic available capacity estimation

For a multiplexer fed by a large number of sources, we derive conditions under which a given subset of the sources can be substituted for a single source while preserving the buffer overflow probability and the dominant timescales of buffer overflows. This notion of traffic equivalence is stronger than simple effective bandwidth equality and depends on the multiplexing context. We propose several applications of the above traffic substitution conditions. First, we show that fractional Brownian motion as a single source substitute can effectively model a large number of multiplexed sources using information obtained purely from traffic traces; this has direct application to simple but accurate traffic generation. Second, we focus on dynamic (i.e., on-line) estimation of available capacity and buffer overflow probability. This requires the solution of a double optimization problem expressed in terms of functions whose values are obtained from time averages of the traffic traces over a large range of timescales. We show how to solve this problem on-line by reducing it to the calculation of a fixed-point equation that can be solved iteratively by combining traffic substitution using fractional Brownian motion with dynamic measurements of the actual traffic. We have validated this approach by extensive experimentation with large numbers or real traffic sources that are fed to a high bandwidth link, and comparing our on-line estimation of available capacity and the resulting dynamic call admission control with other existing approaches. The superior accuracy of our approach also suggests that taking the buffer size into account, as does our on-line algorithm, may be vital for achieving approximations of practical interest.