MaPhySto
The Danish National Research Foundation:
Network in Mathematical Physics and Stochastics

MPS-RR 2004-20
October 2004

# Activity Rates with Very Heavy Tails

by: Thomas Mikosch, Sidney Resnick

## Abstract

Consider a data network model in which sources begin to transmit at renewal time points ${S_n}$. Transmissions proceed for random durations of time ${T_n}$ and transmissions are assumed to proceed at fixed rate unity. We study $M(t)$, the number of active sources at time $t$, a process we term the activity rate process, since $M(t)$ gives the overall input rate into the network at time $t$. Under a variety of heavy-tailed assumptions on the inter-renewal times and the duration times, we can give results on asymptotic behavior of $M(t)$ and the cumulative input process $A(t)= \int_0^t M(s)\, ds$.

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