Network Design and Control Using On-Off and Multi-level Source Traffic Models with Long-Tailed Distributions

N. G. Duffield  and W. Whitt
AT&T Laboratories,
Room {A175, A117}, 180 Park Avenue, Florham Park, NJ 07932-0971, USA
E-mail {duffield,wow}@research.att.com

Abstract:

A major challenge for designing and controlling emerging high-speed integrated-services communication networks is to develop methods for analyzing more realistic source traffic models that are consistent with recent traffic measurements. We consider the familiar on-off source traffic model, but we allow the on and off times to have long-tailed distributions such as the Pareto and Weibull distributions. We also consider a more general traffic model in which the required bandwidth (arrival rate) as a function of time for each source is represented as the sum of two stochastic processes: (1) a macroscopic (longer-time-scale) level process and (2) a microscopic (shorter-time-scale) within-level variation process. We let the level process be a finite-state semi-Markov process (SMP), allowing general (possibly long-tailed) level holding-time distributions, and we let the within-level variation process be a zero-mean piecewise-stationary process. However, the fine structure of the within-level variation process turns out not to matter in our analysis. We make design and control decisions based on the likelihood that aggregate demand (the input rate from a set of sources) will exceed capacity (the maximum possible output rate), using a specification of the sources and their source traffic models to predict demand. This approach to model analysis avoids the customary queueing detail (focus on buffer content and overflow).

We propose using transient analysis, exploiting asymptotics associated with multiplexing a large number of sources. A conditional law of large numbers supports approximating the future aggregate demand conditional on current state information by its conditional mean value, conditional on the levels and elapsed times in levels of the sources. The conditional aggregate mean can be expressed compactly in terms of its Laplace transform and efficiently calculated by numerical transform inversion. We supply further approximations which enable the rapid calculation of the conditional mean without using numerical transform inversion.

As an application in control, we formulate an integer program in which to evaluate the outcomes of decision within a given cost structure. As an application in design, we describe a simple approximate scheme in which a network link can be dimensioned to achieve multiplexing gain while keeping overdemand sufficiently rare and short.

Keywords: source traffic model, admission control, congestion control, overload control, transient analysis, deterministic fluid approximation, long-tailed distributions, Laplace transforms, numerical transform inversion, statistical multiplexing, value of information