Scaling and memory in recurrence intervals of Internet traffic

By studying the statistics of recurrence intervals, $\tau$, between volatilities of Internet traffic rate changes exceeding a certain threshold $q$, we find that the probability distribution functions, $P_{q}(\tau)$, for both byte and packet flows, show scaling property as $P_{q}(\tau)=\frac{1}{\overline{\tau}}f(\frac{\tau}{\overline{\tau}})$. The scaling functions for both byte and packet flows obeys the same stretching exponential form, $f(x)=A\texttt{exp}(-Bx^{\beta})$, with $\beta \approx 0.45$. In addition, we detect a strong memory effect that a short (or long) recurrence interval tends to be followed by another short (or long) one. The detrended fluctuation analysis further demonstrates the presence of long-term correlation in recurrence intervals.