Comparing the optimal performance of different MIMD multiprocessor architectures

We compare the performance of systems consisting of one large cluster containing q processors with systems where processors are grouped into k clusters containing u processors each. A parallel program, consisting of n processes, is executed on this system. Processes may be relocated between the processors in a cluster. They may, however not be relocated from one cluster to another. The performance criterion is the completion time of the parallel program. We present two functions: g(n,k,u,q) and G(k,u,q). Provided that we can find optimal or near optimal schedules, these functions put optimal upper bounds on the gain of using one cluster containing q processors compared to using k clusters containing u processors each. The function g(n,k,u,q) is valid for programs with n processes, whereas G(k,u,q) only depends on the two multiprocessor architectures. By evaluating g(n,k,u,q) and G(k,u,q) we show that the gain of increasing the cluster size from 1 to 2 and from 2 to 4 is relatively large. However, the gain of using clusters larger than 4 is very limited.