To determine the bandwidth at which the propagation delay equals the transmission delay, we first need to calculate the propagation delay and the transmission delay separately.

Step 1: Calculate the Propagation Delay

The propagation delay is the time it takes for a signal to travel from the source to the destination. It can be calculated using the formula:

$\text{Propagation Delay} = \frac{\text{Distance}}{\text{Propagation Speed}}$

Given:

• Distance = 50 km = 50,000 meters
• Propagation Speed = $2 \times 10^8$ meters/second

$\text{Propagation Delay} = \frac{50,000 \text{ meters}}{2 \times 10^8 \text{ meters/second}} = 0.00025 \text{ seconds} = 250 \text{ microseconds}$

Step 2: Calculate the Transmission Delay

The transmission delay is the time it takes to push all the packet's bits onto the wire. It can be calculated using the formula:

$\text{Transmission Delay} = \frac{\text{Packet Size (in bits)}}{\text{Bandwidth}}$

Let's determine the bandwidth for which the transmission delay equals the propagation delay.

Step 3: Find Bandwidth for 100-byte Packets

First, convert the packet size from bytes to bits:

$\text{Packet Size} = 100 \text{ bytes} = 100 \times 8 = 800 \text{ bits}$

Set the transmission delay equal to the propagation delay:

$\frac{800 \text{ bits}}{\text{Bandwidth}} = 0.00025 \text{ seconds}$

Solve for bandwidth:

$\text{Bandwidth} = \frac{800 \text{ bits}}{0.00025 \text{ seconds}} = 3,200,000 \text{ bits/second} = 3.2 \text{ Mbps}$

Step 4: Find Bandwidth for 512-byte Packets

Similarly, for 512-byte packets:

$\text{Packet Size} = 512 \text{ bytes} = 512 \times 8 = 4096 \text{ bits}$

Set the transmission delay equal to the propagation delay:

$\frac{4096 \text{ bits}}{\text{Bandwidth}} = 0.00025 \text{ seconds}$

Solve for bandwidth:

$\text{Bandwidth} = \frac{4096 \text{ bits}}{0.00025 \text{ seconds}} = 16,384,000 \text{ bits/second} = 16.384 \text{ Mbps}$

Summary of Results:

• For 100-byte packets: Bandwidth = 3.2 Mbps
• For 512-byte packets: Bandwidth = 16.384 Mbps

Would you like more details or have any questions?

Related Questions:

1. How would the results change if the propagation speed was different, say $3 \times 10^8$ m/s?
2. What is the impact of different distances on the required bandwidth for equal propagation and transmission delays?
3. How would these calculations change if the packet size was 1500 bytes, typical for Ethernet?
4. How does the bandwidth affect network latency in point-to-point communication?
5. What are some real-world applications where propagation delay is critical?

Tip:

Remember that while increasing bandwidth reduces transmission delay, it does not affect the propagation delay, which depends solely on the distance and the speed of signal propagation.