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Scaling limits for cumulative input processes. (English) Zbl 1279.90033
Summary: We study different scaling behavior of very general telecommunications cumulative input processes. The activities of a telecommunication system are described by a marked-point process \(((T_{n}, Z_{n}))n \in Z\), where \(T_{n}\) is the arrival time of a packet brought to the system or the starting time of the activity of an individual source, and the mark \(Z_{n}\) is the amount of work brought to the system at time \(T_{n}\). This model includes the popular ON/OFF process and the infinite-source Poisson model. In addition to the latter models, one can flexibly model dependence of the interarrival times \(T_{n}-Tn-1\), clustering behavior due to the arrival of an impulse generating a flow of activities, but also dependence between the arrival process \((T_{n})\) and the marks \((Z_{n})\). Similarly to the ON/OFF and infinite-source Poisson model, we can derive a multitude of scaling limits for the input process of one source or for the superposition of an increasing number of such sources. The memory in the input process depends on a variety of factors, such as the tails of the interarrival times or the tails of the distribution of activities initiated at an arrival \(T_{n}\), or the number of activities starting at \(T_{n}\). It turns out that, as in standard results on the scaling behavior of cumulative input processes in telecommunications, fractional Brownian motion or infinite-variance Lévy stable motion can occur in the scaling limit. However, the fractional Brownian motion is a much more robust limit than the stable motion, and many other limits may occur as well.

MSC:
90B18 Communication networks in operations research
60K25 Queueing theory (aspects of probability theory)
60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.)
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