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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.

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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