*Tid:* **4 december 2006 kl 15.15-17.00 **

*Plats :* **Seminarierummet 3733**, Institutionen för
matematik, KTH, Lindstedts väg 25, plan 7. Karta!

*Föredragshållare:*
**
Professor Thomas Mikosch, Laboratory of Actuarial Mathematics,
University of Copenhagen.
**

**Titel:**
Scaling limits for workload processes.

**Sammanfattning:**
We study different scaling behaviour of a very general
telecommunications workload process. The activities of a
telecommunication system are described by a marked point process
((T_{n},Z_{n}))_{n∈ 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 inter-arrival times T_{n}-T_{n-1}, clustering
behaviour 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 workload
process of one source or for the superposition of an increasing
number of such sources. The memory in the workload depends on a
variety of factors such as the tails of the inter-arrival 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 behaviour of workload
processes in telecommunications, fractional Brownian motion or
infinite variance stable Lévy motion can occur in the scaling
limit. However, fractional Brownian motion is a much more robust
limit than the stable motion, and many other limits may occur as
well.

The talk is based on joint work with Gennady Samorodnitsky, Cornell
University.

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