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.

翻译：自20世纪90年代以来，随着互联网的快速发展，网络请求呈现指数级增长。云服务的引入进一步加速了这一进程。Breslau等人（INFOCOM'99）的统计研究表明，内存或网络请求通常遵循幂律分布。即，第$i$热门的网页被请求的概率与$1 / i^{\alpha}$成正比（$\alpha > 0$为常数）。此外，这项二十多年前的研究揭示了类Zipf行为，即$\alpha \le 1$。令人惊讶的是，来自PB级现代云系统的内存访问轨迹不仅表明$\alpha$可能大于1，还呈现了偏移幂律分布——称为帕累托II型或洛马克斯分布。这种未被报道的现象亟需统计学解释。我们的第一个贡献是提出新的统计模型——{多核幂律}模型，表明双幂律可归因于此类系统中并行运行的、承载多个虚拟机的多核架构。我们通过科尔莫戈罗夫-斯米尔诺夫检验（K-S检验）实验验证了该模型的适用性。本文的第二个贡献是理论分析，解释为何基于LRU和LFU的算法在实际中能良好处理满足幂律或多核假设的数据。我们通过研究随机输入模型下的在线分页问题提供了解释：输入是根据分布$\pi$从页面集合中独立抽取的随机序列。我们推导了上界期望比值的公式（作为$\pi$中页面概率的函数），从而证明当随机序列遵循幂律和多核幂律分布时，算法可达到O(1)的性能比。