Parallel and distributed computing systems are foundational to the success of cloud computing and big data analytics. These systems process computational workflows in a way that can be mathematically modeled by a fork-and-join queueing network with blocking (FJQN/B). While engineering solutions have long been made to build and scale such systems, it is challenging to rigorously characterize their throughput performance at scale theoretically. What further complicates the study is the presence of heavy-tailed delays that have been widely documented therein. In this article, we utilize an infinite sequence of FJQN/Bs to study the throughput limit and focus on an important class of heavy-tailed service times that are regularly varying with index . The throughput is said to be scalable if the throughput limit infimum of the sequence is strictly positive as the network size grows to infinity. We introduce two novel geometric concepts—scaling dimension and extended metric dimension—and show that an infinite sequence of FJQN/Bs is throughput scalable if the extended metric dimension and only if the scaling dimension . We also show that for the cases where buffer sizes are scaling in an order of , the scalability conditions are relaxed by a factor of . The results provide new insights on the scalability of a rich class of FJQN/Bs with various structures, including tandem, lattice, hexagon, pyramid, tree, and fractals.

中文翻译：

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分叉和加入重尾队列网络：缩放维度和吞吐量限制

并行和分布式计算系统是云计算和大数据分析成功的基础。这些系统处理计算工作流的方式可以通过带有阻塞的分叉连接排队网络 (FJQN/B) 进行数学建模。虽然长期以来一直在制定工程解决方案来构建和扩展此类系统，但在理论上严格描述其大规模吞吐量性能具有挑战性。使该研究进一步复杂化的是其中已广泛记录的重尾延误的存在。在本文中，我们利用无限序列的 FJQN/B 来研究吞吐量限制，并关注一类重要的重尾服务时间，这些服务时间随指数规律变化 . 如果随着网络规模增长到无穷大，序列的吞吐量限制 infimum 严格为正，则称吞吐量是可扩展的。我们引入了两个新的几何概念——缩放维度和扩展度量维度——并表明如果扩展度量维度，无限序列的 FJQN/B 是吞吐量可扩展的 并且仅当缩放维度 . 我们还表明，对于缓冲区大小按以下顺序缩放的情况 , 可扩展性条件放宽了一个因子 . 结果为具有各种结构（包括串联、晶格、六边形、金字塔、树和分形）的丰富类 FJQN/B 的可扩展性提供了新的见解。