We study the distribution P(x;α,L) of the relative trend x in long-term correlated records of length L that are characterized by a Hurst exponent α between 0.5 and 1.5. The relative trend x is the ratio between the strength of the trend Δ in the record measured by linear regression and the standard deviation σ around the regression line. We consider L between 400 and 2200, which is the typical length scale of monthly local and annual reconstructed global climate records. Extending previous work by Lennartz and Bunde [S. Lennartz and A. Bunde, Phys. Rev. E 84, 021129 (2011)] we show explicitly that x follows the Student's t distribution P∝[1+(x/a)2/l]-(l+1)/2, where the scaling parameter a depends on both L and α, while the effective length l depends, for α below 1.15, only on the record length L. From P we can derive an analytical expression for the trend significance S(x;α,L)=∫(-x)xP(x';α,L)dx' and the border lines of the 95% significance interval. We show that the results are nearly independent of the distribution of the data in the record, holding for Gaussian data as well as for highly skewed non-Gaussian data. For an application, we use our methodology to estimate the significance of central west Antarctic warming.

We suggest a universal phenomenological description for the collective access patterns in the Internet traffic dynamics both at local and wide area network levels that takes into account erratic fluctuations imposed by cooperative user behaviour. Our description is based on the superstatistical approach and leads to the q-exponential inter-session time and session size distributions that are also in perfect agreement with empirical observations. The validity of the proposed description is confirmed explicitly by the analysis of complete 10-day traffic traces from the WIDE backbone link and from the local campus area network downlink from the Internet Service Provider. Remarkably, the same functional forms have been observed in the historic access patterns from single WWW servers. The suggested approach effectively accounts for the complex interplay of both "calm" and "bursty" user access patterns within a single-model setting. It also provides average sojourn time estimates with reasonable accuracy, as indicated by the queuing system performance simulation, this way largely overcoming the failure of Poisson modelling of the Internet traffic dynamics.

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We conduct the analysis of the traffic measured on the backbone link between the WIDE network that connects a number of leading universities in Japan and the upstream internet service provider. We study its statistical properties such as the packet sizes and time distributions between packets. To reveal the laws governing the end user activity and its impact on the network traffic dynamics, we next extracted inter-packet time and packet sizes for TCP, UDP and ICMP protocols. Additionally, long-term correlation properties of the traffic have been analyzed and its scaling characteristics were estimated. We suggest a statistical model which could be used to describe and simulate the backbone link traffic that is in substantial agreement with the empirical observations both in terms of its statistical and correlation properties.Index Terms-computer networks, protocols, q-exponential distribution, long-term dependence.

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