Question

A cache memory that has a hit rate of 0.8 has an access latency of 10 ns and a miss penalty of 100 ns. An optimization is done on the cache to reduce the miss rate. However, the optimization results in an increase of cache access latency to 15 ns, whereas the miss penalty is not affected. The minimum hit rate (rounded off to two decimal places) needed after the optimization such that it should not increase the average memory access time is \(\underline{\hspace{1cm}}\).

Hint:

To maintain the same or better average memory access time after optimization, adjust the hit rate to compensate for the increase in hit time while keeping the miss penalty constant.

Answer 1

Correct Answer: 0.85

The Average Memory Access Time (AMAT) is given by the standard formula: \[ AMAT = (\text{Hit Rate} \times \text{Hit Time}) + (\text{Miss Rate} \times \text{Miss Penalty}) \]

Before optimization:
Given: - Hit rate \( H = 0.8 \) - Hit time = 10 ns - Miss penalty = 100 ns \[ AMAT_{\text{before}} = (0.8 \times 10) + (0.2 \times 100) \] \[ AMAT_{\text{before}} = 8 + 20 = 28 \text{ ns} \]

After optimization:
Given: - New hit time = 15 ns - Miss penalty = 100 ns - New hit rate = \( H' \) \[ AMAT_{\text{after}} = (H' \times 15) + (1 - H') \times 100 \] Simplifying: \[ AMAT_{\text{after}} = 15H' + 100 - 100H' = 100 - 85H' \]

Condition for no increase in AMAT:
\[ AMAT_{\text{after}} \le AMAT_{\text{before}} \] \[ 100 - 85H' \le 28 \] \[ -85H' \le -72 \] \[ H' \ge \frac{72}{85} \] \[ H' \approx 0.85 \]

Final Answer:
The minimum hit rate required to ensure that the average memory access time does not increase is: \[ \boxed{0.85} \]