  Web requests are growing exponentially since the 90s due to the rapid
development of the Internet. This process was further accelerated by the
introduction of cloud services. It has been observed statistically that memory
or web requests generally follow power-law distribution, Breslau et al.
INFOCOM'99. That is, the $i^{\text{th}}$ most popular web page is requested
with a probability proportional to $1 / i^{\alpha}$ ($\alpha > 0$ is a
constant). Furthermore, this study, which was performed more than 20 years ago,
indicated Zipf-like behavior, i.e., that $\alpha \le 1$. Surprisingly, the
memory access traces coming from petabyte-size modern cloud systems not only
show that $\alpha$ can be bigger than one but also illustrate a shifted
power-law distribution -- called Pareto type II or Lomax. These previously not
reported phenomenon calls for statistical explanation.
  Our first contribution is a new statistical {\it multi-core power-law} model
indicating that double-power law can be attributed to the presence of multiple
cores running many virtual machines in parallel on such systems. We verify
experimentally the applicability of this model using the Kolmogorov-Smirnov
test (K-S test).
  The second contribution of this paper is a theoretical analysis indicating
why LRU and LFU-based algorithms perform well in practice on data satisfying
power-law or multi-core assumptions. We provide an explanation by studying the
online paging problem in the stochastic input model, i.e., the input is a
random sequence with each request independently drawn from a page set according
to a distribution $\pi$. We derive formulas (as a function of the page
probabilities in $\pi$) to upper bound their ratio-of-expectations, which help
in establishing O(1) performance ratio given the random sequence following
power-law and multi-core power-law distributions.
