Scaling Limits for Cumulative Input Processes
Scaling Limits for Cumulative Input Processes
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DOI：
10.1287/moor.1070.0267
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发表时间：
2007-11
期刊：
影响因子：
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通讯作者：
T. Mikosch;G. Samorodnitsky
中科院分区：
文献类型：
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作者：
T. Mikosch;G. Samorodnitsky
我们研究了非常一般的电信累积输入过程的不同标度行为。电信系统的活动由标记点过程（（Tn，Zn））n∈Z描述，其中Tn是被带到系统的分组的到达时间或单个源的活动的开始时间，并且标记Zn是在时间Tn被带到系统的工作量。该模型包括流行的ON/OFF过程和无限源泊松模型。除了后面的模型之外，可以灵活地对到达间隔时间Tn-Tn-1的依赖性、由于产生活动流的脉冲的到达而导致的聚类行为、以及到达过程（Tn）与标记（Zn）之间的依赖性进行建模。类似于ON/OFF和无限源泊松模型，我们可以推导出一个源的输入过程或越来越多的这样的源的叠加的众多的标度限制。输入过程中的记忆取决于各种因素，例如到达间隔时间的尾部或在到达Tn时开始的活动分布的尾部，或在Tn时开始的活动的数量。事实证明，在标准的结果在电信中的累积输入过程的标度行为，分数布朗运动或无穷方差列维稳定运动可以发生在标度极限。然而，分数布朗运动是比稳定运动更鲁棒的极限，并且也可能出现许多其他极限。
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 ((Tn, Zn))n∈Z, where Tn 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 Zn is the amount of work brought to the system at time Tn. 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 Tn-Tn-1, clustering behavior due to the arrival of an impulse generating a flow of activities, but also dependence between the arrival process (Tn) and the marks (Zn). 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 Tn, or the number of activities starting at Tn. It turns out that, as in standard results on the scaling behavior of cumulative input processes in telecommunications, fractional Brownian motion or infinite-variance Levy 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.