A file system uses an in-memory cache to cache disk blocks. The miss rate of the cache is shown in the figure. The latency to read a block from the cache is 1 ms and to read a block from the disk is 10 ms. Assume that the cost of checking whether a block exists in the cache is negligible. Available cache sizes are in multiples of 10 MB. The smallest cache size required to ensure an average read latency of less than 6 ms is ________ MB.

Let $m$ be the miss rate of the cache, and $h = 1 - m$ be the hit rate.

The average read latency $T_{\text{avg}}$ is given by:

Given:

• $T_{\text{cache}} = 1\text{ ms}$

• $T_{\text{disk}} = 10\text{ ms}$

• Desired $T_{\text{avg}} < 6\text{ ms}$

Substitute the values into the inequality:

From the given miss rate graph:

• For cache size $= 10\text{ MB}$, miss rate $= 80\%$

• For cache size $= 20\text{ MB}$, miss rate $= 60\%$

• For cache size $= 30\text{ MB}$, miss rate $= 40\%$

• For cache size $= 40\text{ MB}$, miss rate $= 35\%$

For $T_{\text{avg}} < 6\text{ ms}$, we require $m < 55.56\%$.

• At $20\text{ MB}$, $m = 60\% > 55.56\%$ ($T_{\text{avg}} = 0.4(1) + 0.6(10) = 6.4\text{ ms}$).

• At $30\text{ MB}$, $m = 40\% < 55.56\%$ ($T_{\text{avg}} = 0.6(1) + 0.4(10) = 4.6\text{ ms}$).

Thus, the smallest available cache size meeting the requirement is 30 MB.