On weak convergence of long-range-dependent traffic processes

On weak convergence of long-range-dependent traffic processes

A possible model for communication traffic is that the amount of work arriving in successive time intervals is jointly Gaussian. This model seems to fly in the face of certain obvious and characteristic features of real traffic, such as the fact that it arrives in discrete bundles and that there is often a non-zero probability of zero traffic in a time interval of significant length. Also, the Gaussian model allows the possibility of negative traffic, which is clearly unrealistic. As the number of sources of traffic increases and the quantity of traffic in communication networks increases, however, under suitable conditions, the deviation between the distribution of real traffic and the Gaussian model will become less. The appropriate concept of topology/convergence must be used or the result will be meaningless. To identify an appropriate convergence framework, the performance statistics associated with a network, namely cell loss, delay, and, in general, statistics which can be expressed in terms of the network buffers which accumulate in the network may be used as a guide. Weak convergence of probability measures has the property that when the probability measures of traffic processes converge to that of a certain traffic process, the distribution of their performance characteristics, such as buffer occupancy, also converges in the same sense to the performance of the system to which they were converging. Real traffic appears, unambiguously, to be long-range dependent. There is an interesting example where aggregation of traffic does not seem to produce convergence to the queueing behaviour expected of Gaussian traffic, at any rate the tail characteristics do not converge to those of the Gaussian result. However, in Section 4, it is shown that if the variance of one traffic stream is finite and as a proportion of the variance of the whole traffic volume tends to zero, then the traffic in networks can be expected to converge to Gaussian in the sense of weak convergence of probability measures. It is then shown that, as a consequence, the traffic in the paradoxical example does converge in this sense also. The paradox is explained by noticing that asymptotic tail behaviour may become increasingly irrelevant as traffic is aggregated. This fact should sound a warning concerning the cavalier use of tail-behaviour as an indication of performance. Long-range dependence apparently places no inhibition on convergence to Gaussian behaviour. Convergence to a Gaussian distribution of increasing aggregates of traffic is only shown to occur for discrete time models. In fact it appears that continuous time Gaussian models do not share this property and their use for modelling real traffic may be problematic.

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