When applying numerically the results obtained by the DLAM, we choose , assuming that the cost of these instructions will always be negligible compared to BLACS operations or a Level 3 instruction. We determined as being the achieved peak performance of the BLAS matrix-multiply GEMM. This approximation is incorrect for small block sizes, in which case Level 2 operations are performed and should be set respectively to the achieved peak performance of the BLAS matrix-vector multiply GEMV and zero. Obviously, these coarse approximations could be refined by computing a piece-wise linear approximation of the 's with respect to the problem size. This model smoothes the influence of the physical memory hierarchy and could be adapted to out-of-core BLAS operations.

Modeling the performance of the DLAM network is tightly coupled to the physical network. Experimental values of and can easily be determined for a given machine. If the logical mesh can be embedded into the physical network and the message collisions ignored, is a good approximation of assuming the result has to be left on the processes and neglecting the cost of the local computations; similarly, . When the communications can be pipelined, it is reasonable to estimate by 2. Because this model ignores the probable collision of messages or possible network contention problems, its accuracy depends on the number of physical links. For instance, when comparing the performance obtained on an ideal DLAM with those obtained on an ethernet based network of workstations sharing one physical link, it is important to use appropriate values for . Indeed, an upper bound for is given by . However, for a given value of , it is possible to experimentally determine constants which take into account the cost due to network contention and message collisions. More accurate models taking into account the collisions of messages could be used, but this is beyond the scope of this paper. Finally, the described model could obviously be refined by computing a piece-wise linear approximation of the time for sending a message with respect to the message length.

Fri Mar 31 13:01:26 EST 1995