Question 59

We are given:

• Number of machines to connect: 15
• Each switch has 8 ports and no separate uplink ports.

To connect $k$ switches together into a single loop-free broadcast domain (a connected tree topology), exactly $k - 1$ inter-switch links are required.

Each inter-switch link connects two switch ports (one port on each connected switch). Therefore, $k - 1$ links consume $2(k - 1)$ switch ports in total.

The total number of ports available for connecting end machines across all $k$ switches is:

We require $P_{\text{available}} \ge 15$:

Since $k$ must be an integer:

• For $k = 1$: available ports = $8(1) - 0 = 8 < 15$.
• For $k = 2$: available ports = $6(2) + 2 = 14 < 15$.
• For $k = 3$: available ports = $6(3) + 2 = 20 \ge 15$.

Thus, the minimum number of switches needed is 3.