GATE CS 1998 Q2.18 — Memory Management and Virtual Memory

To find the effective instruction execution time, we can analyze the average time spent over a sequence of $k$ instructions.

According to the given information:

• Each instruction takes a base execution time of $i$ microseconds.
• A page fault occurs once every $k$ instructions on average.
• Each page fault incurs an additional overhead of $j$ microseconds.

Let us calculate the total time taken to execute $k$ instructions:

1. The base execution time for $k$ instructions is: $$T_{\text{base}} = k \times i \text{ microseconds}$$
2. Since one page fault occurs every $k$ instructions, the additional overhead for these $k$ instructions is: $$T_{\text{overhead}} = 1 \times j = j \text{ microseconds}$$

Thus, the total time to execute $k$ instructions is: $$T_{\text{total}} = k \cdot i + j$$

The effective instruction time ($T_{\text{eff}}$) is the average time per instruction: $$T_{\text{eff}} = \frac{T_{\text{total}}}{k} = \frac{k \cdot i + j}{k} = i + \frac{j}{k}$$

Therefore, the correct option is A.