Continuous models for communication density constraints on multiprocessor performance

Fundamental limits on the communication capabilities of massively parallel multiprocessors are investigated. It is shown that in the limit of machines of infinite extent in which the number of processors per unit volume is constant and in which the communication bandwidth from each processor to its neighbors depends only on their separation distance, interprocessor communication must fall off faster than the fourth power of distance. For machines of finite size, communication energy density is used as a metric to compare various machine sizes and packaging densities. For instance, for machines with spherical symmetry and uniform communication requirements, the peak density depends on the number of processors to the 4/3 power and the number of processors per unit volume to the 2/3 power