Today we continue to review the WRL system for trace generation. We read up on the effect of direct-mapped versus associative second level cache. When associativity is increased, miss ratio can decrease but the overall performance may not improve or even degrade. If the cache is very large, it is likely to have a low missed ratio already. Increasing the associativity might provide little or no benefits. Associativity helps with the miss ratio but it adds up to the cost of every reference to a cache.
If we ignore this cost, we can bound the benefits of associativity with a comparison between the direct mapped and the fully associative caches. The results showed that the cumulative CPI for a fully associative second level cache with four different sizes were minimally effective for large caches but more effective for small caches.
 A more useful measure for the benefits of associativity may be to see the effect on the total CPI as a function of the cost per reference of the associativity.
We know that the CPI for the associative case will be a delta more than the CPI for the direct mapped case. This delta is the combination of the first level miss ratio and the difference in the average cost per reference to the second level cache. This difference can be elaborated in terms of the cost of a hit in the associative cache (ha), the cost of a hit in a direct mapped cache (hd), the cost of a second level cache miss (same direct-mapped and associative) (m), the miss ration for a second level associative cache and the miss ratio for a second level direct-mapped.
In this case, the delta in the average cost per reference can be worked out as the difference in the average cost per reference in the associative case and in the direct mapped case. The first term is sum of the costs of access to  the associative second level and the miss from this level. The associative second level cost of access is a combination of the cost of a hit and the miss ratio for that cache. One minus this miss ratio times the cost of a second level cache miss which is the same for a direct mapped and the associative case is the cost of a miss from this level. This kind of elaboration is true for the second term as well in the case of the direct mapped second-level cache. Writing the costs of a hit in terms of a ratio, we get the delta in the average cost per reference to be weighted based on the difference in the miss ratios for a second level associative cache and the second level direct mapped cache
We can now plot the graph of the cumulative CPI in both cases, and see that there is a cross over.  The distance between the points at which the crossover occurs is the maximum gain to be expected from the associativity.

#codingexercise
Decimal GetEvenNumberRangeMin(Decimal [] A)
{
if (A == null) return 0;
Return A.EvenNumberRangeMin ();
}