For a host machine that uses the token bucket algorithm for congestion control, the token bucket has a capacity of 1 megabyte and the maximum output rate is 20 megabytes per second. Tokens arrive at a rate to sustain output at a rate of 10 megabytes per second. The token bucket is currently full and the machine needs to send 12 megabytes of data. The minimum time required to transmit the data is ____________ seconds.

Let:

• Bucket capacity $C = 1\text{ MB}$

• Maximum burst transmission rate $M = 20\text{ MB/s}$

• Token generation rate $R = 10\text{ MB/s}$

• Total data to transmit $D = 12\text{ MB}$

1. Burst phase (until the bucket becomes empty): During the burst time $t_b$, the output rate is $M = 20\text{ MB/s}$.

   $$M \cdot t_b = C + R \cdot t_b$$

   $$20 \cdot t_b = 1 + 10 \cdot t_b \implies 10 \cdot t_b = 1 \implies t_b = 0.1\text{ seconds}$$

   Data sent during burst phase:

   $$D_{\text{burst}} = M \cdot t_b = 20 \times 0.1 = 2\text{ MB}$$

2. Sustained rate phase: Remaining data to transmit:

   $$D_{\text{rem}} = 12 - 2 = 10\text{ MB}$$

   Since the bucket is empty, data can only be sent at the token arrival rate $R = 10\text{ MB/s}$:

   $$t_{\text{rem}} = \frac{10\text{ MB}}{10\text{ MB/s}} = 1.0\text{ second}$$

3. Total transmission time:

   $$T = t_b + t_{\text{rem}} = 0.1 + 1.0 = 1.1\text{ seconds}$$