A New Interpretation of Amdahl's Law and Geometric Scalability

The multiprocessor effect refers to the loss of computing cycles due to processing overhead. Amdahl's law and the Multiprocessing Factor (MPF) are two scaling models used in industry and academia for estimating multiprocessor capacity in the presence of this multiprocessor effect. Both models express different laws of diminishing returns. Amdahl's law identifies diminishing processor capacity with a fixed degree of serialization in the workload, while the MPF model treats it as a constant geometric ratio. The utility of both models for performance evaluation stems from the presence of a single parameter that can be determined easily from a small set of benchmark measurements. This utility, however, is marred by a dilemma. The two models produce different results, especially for large processor configurations that are so important for today's applications. The question naturally arises: Which of these two models is the correct one to use? Ignoring this question merely reduces capacity prediction to arbitrary curve-fitting. Removing the dilemma requires a dynamical interpretation of these scaling models. We present a physical interpretation based on queueing theory and show that Amdahl's law corresponds to synchronous queueing in a bus model while the MPF model belongs to a Coxian server model. The latter exhibits unphysical effects such as sublinear response times hence, we caution against its use for large multiprocessor configurations.