GATE CS 2017 Set 2 Q.29 (CS Q.29): Memory Hierarchy: Cache, Main Memory and Secondary Storage

In a two-level cache system, the access times of $L_1$ and $L_2$ caches are 1 and 8 clock cycles, respectively. The miss penalty from the $L_2$ cache to main memory is 18 clock cycles. The miss rate of $L_1$ cache is twice that of $L_2$ . The average memory access time (AMAT) of this cache system is 2 cycles. The miss rates of $L_1$ and $L_2$ respectively are:

0.111 and 0.056

0.056 and 0.111

0.0892 and 0.1784

0.1784 and 0.0892

Official answer and solution

Official answer: A
From: Official GATE 2017 answer key

Solution

Set up the Average Memory Access Time (AMAT) formula for a two-level cache hierarchy and solve the resulting quadratic equation for the miss rates.

Let $m_1$ and $m_2$ denote the local miss rates of the $L_1$ and $L_2$ caches, respectively.
Given:

$L_1$ hit access time $T_1 = 1$ clock cycle
$L_2$ hit access time $T_2 = 8$ clock cycles
Miss penalty from $L_2$ to main memory $T_{\text{MM}} = 18$ clock cycles
$m_1 = 2m_2 \implies m_2 = \frac{m_1}{2}$
$\text{AMAT} = 2$ clock cycles

The AMAT for a two-level cache system is:

\text{AMAT} = T_1 + m_1 \left( T_2 + m_2 \cdot T_{\text{MM}} \right)

Substitute the given values:

2 = 1 + m_1 \left( 8 + 18 m_2 \right)

1 = m_1 \left( 8 + 18 \cdot \frac{m_1}{2} \right) = m_1 \left( 8 + 9m_1 \right)

9m_1^2 + 8m_1 - 1 = 0

Factor the quadratic equation:

(9m_1 - 1)(m_1 + 1) = 0

Since miss rate must be positive, $m_1 = \frac{1}{9} \approx 0.111$ .
Then $m_2 = \frac{m_1}{2} = \frac{1}{18} \approx 0.0556 \approx 0.056$ .

Therefore, the miss rates of $L_1$ and $L_2$ are 0.111 and 0.056 respectively, which matches option A.

Final answer

Common trap

Using global miss rate instead of local miss rate for $L_2$ , or inverting the relation between $m_1$ and $m_2$ .

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2017 Computer Science and Information Technology (CS) Session 2
Question: 29 of 65, printed as CS Q.29
Section: Computer Science and Information Technology

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