Fork and Join Queueing Networks with Heavy Tails: Scaling Dimension and Throughput Limit

Abstract

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.