Computational forces in the SAGE benchmark

Dimensional analysis applied to a complicated timing formula for the SAGE benchmark yields new insight into the limits to scalability. A single surface, defined by two curvilinear coordinates, describes the parallel efficiency of the benchmark. Each machine, as a function of the number of processors, follows its own path on the surface determined by dimensionless ratios of hardware forces to software forces. Two machines with the same ratios follow the same path and are self-similar, even though the numerical value of each individual force may be different. For this benchmark, latency effects are unimportant relative to bandwidth effects because of the slab decomposition used to distribute the problem across processors. To a good first-order approximation, a single force ratio describes the efficiency as a function of the number of processors. A simpler model, with a single dimensionless exponent, describes the first-order behavior of the computational power as a function of the number of processors.