In high performance computing environments, we observe an ongoing increase in the available numbers of cores. This development calls for re-emphasizing performance (scalability) analysis and speedup laws as suggested in the literature (e.g., Amdahl's law and Gustafson's law), with a focus on asymptotic performance. Understanding speedup and efficiency issues of algorithmic parallelism is useful for several purposes, including the optimization of system operations, temporal predictions on the execution of a program, and the analysis of asymptotic properties and the determination of speedup bounds. However, the literature is fragmented and shows a large diversity and heterogeneity of speedup models and laws. These phenomena make it challenging to obtain an overview of the models and their relationships, to identify the determinants of performance in a given algorithmic and computational context, and, finally, to determine the applicability of performance models and laws to a particular parallel computing setting. In this work, we provide a generic speedup (and thus also efficiency) model for homogeneous computing environments. Our approach generalizes many prominent models suggested in the literature and allows showing that they can be considered special cases of a unifying approach. The genericity of the unifying speedup model is achieved through parameterization. Considering combinations of parameter ranges, we identify six different asymptotic speedup cases and eight different asymptotic efficiency cases. Jointly applying these speedup and efficiency cases, we derive eleven scalability cases, from which we build a scalability typology. Researchers can draw upon our typology to classify their speedup model and to determine the asymptotic behavior when the number of parallel processing units increases. In addition, our results may be used to address various extensions of our setting.

翻译：在高性能计算环境中，可用的核心数量持续增长。这一发展要求我们重新聚焦于文献中提出的性能（可扩展性）分析与加速比定律（如Amdahl定律和Gustafson定律），特别是其渐近性能。理解算法并行化的加速比与效率问题，有助于优化系统运行、预测程序执行时间、分析渐近特性以及确定加速比上界。然而，现有文献较为分散，加速比模型与定律呈现出显著的多样性和异质性。这一现象使得人们难以全面把握各类模型及其相互关系，难以在特定算法与计算上下文中识别性能决定因素，更难以确定性能模型与定律在特定并行计算场景中的适用性。本文针对同构计算环境提出了一个通用加速比（进而也是效率）模型。我们的方法推广了文献中提出的众多主流模型，并证明这些模型均可视为该统一方法的特例。通过参数化实现了统一加速比模型的通用性。通过考量参数范围的组合，我们识别出六种不同的渐近加速比情形和八种不同的渐近效率情形。将加速比与效率情形联合应用，推导出十一种可扩展性情形，并以此构建了可扩展性分类体系。研究者可借助该分类体系对其加速比模型进行归类，并在并行处理单元数量增加时确定渐近行为。此外，我们的结果还可用于解决本设置的各种扩展问题。