A cloud server processes data batches across three sequential stages (k = 1, 2, 3). At the… — GMAT

A cloud server processes data batches across three sequential stages ( $k = 1, 2, 3$ ). At the start ( $k = 0$ ), the server has a Buffer size of $B_0 = 100\text{ MB}$ and a Queue size of $Q_0 = 50\text{ MB}$ . In each stage $k$ , the sizes are updated according to the following rules:
- $B_k = B_{k-1} - 20\text{ MB}$
- $Q_k = Q_{k-1} + 10\text{ MB}$

Which of the following statements about the state of the server are true?

At the end of Stage 2 ( $k = 2$ ), the Buffer size $B_2$ is $60\text{ MB}$ .Answer
At the end of Stage 3 ( $k = 3$ ), the Queue size $Q_3$ is $80\text{ MB}$ .Answer
C
At the end of Stage 3 ( $k = 3$ ), the Buffer size $B_3$ is $60\text{ MB}$ .
D
At the end of Stage 2 ( $k = 2$ ), the Queue size $Q_2$ is $60\text{ MB}$ .
E
At the end of Stage 1 ( $k = 1$ ), the combined total volume $B_1 + Q_1$ is $150\text{ MB}$ .

Answer

The correct statements are that at the end of Stage 2, the Buffer size is

60\text{ MB}

, and at the end of Stage 3, the Queue size is

80\text{ MB}

The correct statements accurately reflect the iterative evaluation of the state transition formulas: the buffer size decreases by

20\text{ MB}

each stage to reach

60\text{ MB}

at Stage 2, and the queue size increases by

10\text{ MB}

each stage to reach

80\text{ MB}

at Stage 3.

Step-by-Step Solution

Calculate state values at Stage 1 (

k = 1

)

B_1 = 100 - 20 = 80\text{ MB}

and

Q_1 = 50 + 10 = 60\text{ MB}

. Combined total:

140\text{ MB}

Apply the given state transition rules to the initial values at

k = 0

Calculate state values at Stage 2 (

k = 2

)

B_2 = 80 - 20 = 60\text{ MB}

and

Q_2 = 60 + 10 = 70\text{ MB}

Apply the transition rules to the Stage 1 state values.

Calculate state values at Stage 3 (

k = 3

)

B_3 = 60 - 20 = 40\text{ MB}

and

Q_3 = 70 + 10 = 80\text{ MB}

Apply the transition rules to the Stage 2 state values.

Evaluate option statements

The statement specifying

B_2 = 60\text{ MB}

and the statement specifying

Q_3 = 80\text{ MB}

are verified as correct.

Compare calculated stage states directly against the option statements.

Key Concept

Sequential processes require evaluating every state transition iteratively without skipping intermediate stages.

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