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Fig. 15.16 (a) Arrival sequence of 34,152 arrivals from a CMU dataset. (b) Empirical and tted cumulative distributions of the interarrival times of the simulations.

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By design the nonstationary Poisson process follows the original arrival sequence very accurately; this is not the case for the Weibull time series. This would indicate that the stationary Weibull model is not appropriate. Yet, given that the simulation results2 indicate otherwise, consider the implications of Fig. 15.16(b). The plot indicates that the cumulative distribution of the interarrival times of the total nonstationary Poisson process is much closer to the tted exponential distribution than to the one of the original data (that it is trying to model). This implies that the nonstationary Poisson process is not modeling the short interarrivals very accurately. Therefore, since exactly these interarrival times are crucial for the burstiness of the arrival process, the stationary Weibull model can outperform the nonstationary Poisson process for the connection-admission problem. In addition, it explains why the stationary Weibull model loses prediction accuracy as the average bandwidth requirements of the connections are decreased. With decreased bandwidth requirements large time scale load variations can dominate load variations on smaller time scales and the nonstationary Poisson process may match the relevant time scale of interest. Therefore, a decrease of the segment sizes leads to only a small improvement in predicting the blocking probability. In summary, the Weibull distribution matches the interarrival time distribution over more time scales than the nonstationary Poisson model. If short time scale interactions within the arrival process are relevant to the network management problem, then a Weibull arrival process may well be an appropriate model to capture the burstiness of the arrival process.

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Traditional models such as exponential models are insuf cient to explain the observed distributions of interarrival times of TCP connections. Instead, we nd that most parameters are statistically better tted using distributions with heavy tails, such as Weibull, Pareto, and lognormal distributions. The Weibull distributions, especially, yield superior ts for all interarrival times, all application-speci c interarrival times that are not timer driven, and time-dependent interarrival times, implying that the number of long and short interarrival times and therefore the number and intensity of bursts are higher than traditional traf c models suggest. Indeed, we observed that the increased burstiness of the arrival process for some applications is consistent with a self-similar process. This means it is scale-invariant behavior for application types such as Web and FTP. Simulation results for a connection-admission problem let us suggest the Weibull model as an alternative model for analysis and simulations of this and other network resource allocation problems.

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2 While the plots show only one simulation result, we evaluated several simulations, each using a different seed for the random number generator. Each simulation run resulted in quantitatively similar results.

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