Two changes that might improve the ranking described above are partitioning memory accesses between the different levels of the memory hierarchy and allowing different rankings based on the processor count. The two possibilities are not entirely independent since running the same size problem on a larger number of processors means a smaller working set on each processor and therefore different cache behavior. However, allowing different rankings for different processor counts takes us away from the original goal of finding a single fixed ranking that can be used as a general guideline.

This leaves us with partitioning memory accesses between the different levels of the memory hierarchy. Unfortunately, as noted previously, this requires us to either choose a representative system or to move towards a more complex model that allows for predictions that are specific to individual machines, as is done in.^{10 19} Therefore, given the level of complexity needed for a ranking method that incorporates so much detail, we simply observe that we achieved a ranking with about 28% more thresholded inversions than the brute-force obtainable optimal ranking on our data set without resorting to anything more complex than partitioning each application’s memory accesses into strided and random accesses. This represents a significant improvement over the ranking based on flops, which was about 75% worse than the optimal ranking.

Now we shift gears and examine how well current performance prediction methodologies rank the relative performance of different machines.

Up to this point our focus has been on generating a single machine ranking that is evaluated across multiple applications run on several different processor counts. In this section we ask how much better we can do by allowing rankings to vary as a function of the application and the processor count. Of course, if we ran all the applications on all the machines, we could then use the actual runtimes to generate an ideal ranking for each application. In practice, for reasons noted previously, such complete runtime data is almost never available. However, we could use any of a number of performance prediction methodologies and then generate machine rankings based on the predicted runtimes. This approach has the advantage of avoiding gaps in the data and also has the potential for allowing designers to rank systems that have yet to be built.

If we could predict performance exactly, then these rankings would be perfect; in practice, performance predictions are never exact. Furthermore, because methodologies can both overpredict and underpredict the real running time, a more accurate performance prediction methodology might not lead to better rankings. In this section we explore the quality of rankings generated through the performance prediction methodologies described in,¹⁰ where the authors evaluate nine increasingly complex and increasingly accurate ways of using combinations of metrics to predict real runtimes of applications on supercomputers.

Table 3 gives the number of thresholded inversions between the predicted and measured runtimes in.¹⁰ Note that their dataset is different from the one used in the previous sections of this paper: we use eight applications to their four, far more processor counts for each application, and 14 machines to their 10. Because of this, the numbers in Table 3 cannot be directly compared with those elsewhere in this paper. However, because the question is how well performance prediction strategies generate application dependent rankings, an exact comparison is neither meaningful nor desirable.

We observe that the first 3 cases in ¹⁰ should produce rankings that are equivalent to using only flops, only the bandwidth of strided accesses to main memory, and only the bandwidth of random accesses to main memory, respectively. We see that the ranking based on the bandwidth of strided access to main memory is the best of the three, which agrees with the data in our Table 1. As expected from the description in,¹⁰ their first and fourth cases produce equivalent rankings.