Question

Suppose that a certain communications protocol involves a per-packet overhead of 100 bytes for headers and framing. We send 1 million bytes of data using this protocol; however, one data byte is corrupted and the entire packet containing it is thus lost. Give the total number of overhead + loss bytes for packet data sizes of \(1000,5000,10,000\), and 20,000 bytes. Which size is optimal?

Short answer

The optimal packet size is 10000 bytes with a total overhead and loss of 20100 bytes.

Step-by-step solution

- Understanding the Overhead

Each packet includes an overhead of 100 bytes. This overhead is always added to the data packet size.

- Calculation for 1000 Bytes Packet Size

To calculate total overhead for sending 1 million bytes:\[ \text{Number of packets} = \frac{1000000}{1000} = 1000 \] Each packet has an overhead of 100 bytes: \[ \text{Total overhead} = 1000 \times 100 = 100000 \text{ bytes} \] If one packet is lost, the total lost data and overhead is: \[ 1000 + 100 = 1100 \text{ bytes} \] Thus, the total cost: \[ 100000 + 1100 = 101100 \text{ bytes} \]

- Calculation for 5000 Bytes Packet Size

To calculate total overhead for sending 1 million bytes: \[ \text{Number of packets} = \frac{1000000}{5000} = 200 \] Each packet has an overhead of 100 bytes: \[ \text{Total overhead} = 200 \times 100 = 20000 \text{ bytes} \] If one packet is lost, the total lost data and overhead is: \[ 5000 + 100 = 5100 \text{ bytes} \] Thus, the total cost: \[ 20000 + 5100 = 25100 \text{ bytes} \]

- Calculation for 10000 Bytes Packet Size

To calculate total overhead for sending 1 million bytes: \[ \text{Number of packets} = \frac{1000000}{10000} = 100 \] Each packet has an overhead of 100 bytes: \[ \text{Total overhead} = 100 \times 100 = 10000 \text{ bytes} \] If one packet is lost, the total lost data and overhead is: \[ 10000 + 100 = 10100 \text{ bytes} \] Thus, the total cost: \[ 10000 + 10100 = 20100 \text{ bytes} \]

- Calculation for 20000 Bytes Packet Size

To calculate total overhead for sending 1 million bytes: \[ \text{Number of packets} = \frac{1000000}{20000} = 50 \] Each packet has an overhead of 100 bytes: \[ \text{Total overhead} = 50 \times 100 = 5000 \text{ bytes} \] If one packet is lost, the total lost data and overhead is: \[ 20000 + 100 = 20100 \text{ bytes} \] Thus, the total cost: \[ 5000 + 20100 = 25100 \text{ bytes} \]

- Determining the Optimal Packet Size

Compare the total cost for each packet size: \[ \begin{align*} 1000 \text{ bytes packet: } & 101100 \text{ bytes} \ 5000 \text{ bytes packet: } & 25100 \text{ bytes} \ 10000 \text{ bytes packet: } & 20100 \text{ bytes} \ 20000 \text{ bytes packet: } & 25100 \text{ bytes} \end{align*} \] The optimal packet size is the one with the least total cost, which is 10000 bytes.

Key concepts

communications protocol

A communications protocol is a set of rules and standards designed to allow electronic devices to communicate with each other. This includes computers, phones, and routers, among others. Protocols define how data is formatted, transmitted, and received, ensuring that everything works smoothly.
In our exercise, the protocol specifies a **per-packet overhead** of 100 bytes. This overhead acts as extra information added to each packet of data to handle various tasks like addressing, error checking, and more. Understanding overhead is crucial because it affects the total amount of data that needs to be transmitted. For instance, even if you're only sending 1000 bytes of data, with an overhead of 100 bytes, you're actually sending 1100 bytes. Knowing how protocols work helps us optimize data transmission.

packet size

Packet size refers to the amount of data (in bytes) contained within a single packet. The size of the packet can greatly affect both efficiency and reliability when transmitting data over a network.
In this exercise, different packet sizes are considered: 1000, 5000, 10000, and 20000 bytes. Each packet includes 100 bytes of overhead.
Here's how it works:

• For a 1000-byte packet, each packet also contains 100 bytes of overhead, resulting in a total of 1100 bytes per packet.
• For a 5000-byte packet, it’s 5100 bytes per packet.
• For a 10000-byte packet, the total becomes 10100 bytes.
• For a 20000-byte packet, it's 20100 bytes per packet.

Choosing the right packet size is a balance between network efficiency and the likelihood of data loss. Larger packets reduce the overhead percentage but have higher data loss costs if a packet is corrupted. Smaller packets increase the overhead percentage but limit loss impact.

data loss calculation

Data loss calculation involves figuring out the total amount of data lost when packets are corrupted during transmission. In our problem, packet corruption means losing the entire packet, including its overhead.
To calculate data loss for a given packet size:

• Find the total overhead and data for all packets needed to send 1 million bytes.
• Add the data lost plus the overhead of one corrupted packet.

For example:

• 1000-byte packets: Total overhead for 1 million bytes is 100000 bytes. Adding one corrupted packet (1100 bytes) results in 101100 bytes total loss.
• 5000-byte packets: Total overhead is 20000 bytes. Adding one corrupted packet (5100 bytes) results in 25100 bytes total loss.
• 10000-byte packets: Total overhead is 10000 bytes. Adding one corrupted packet (10100 bytes) results in 20100 bytes total loss.
• 20000-byte packets: Total overhead is 5000 bytes. Adding one corrupted packet (20100 bytes) results in 25100 bytes total loss.

These calculations help identify which packet size is optimal by minimizing total data loss.

optimal packet size

An optimal packet size minimizes the total bytes sent when considering both the overhead and the potential data loss. In our scenario, the total bytes include the overhead for all packets and the lost bytes when one packet is corrupted.
To determine the optimal packet size:

• Calculate the total cost (overhead + loss) for each packet size.
• Compare these costs to see which one is the lowest.

From our calculations:

• 1000-byte packets: Total cost = 101100 bytes
• 5000-byte packets: Total cost = 25100 bytes
• 10000-byte packets: Total cost = 20100 bytes
• 20000-byte packets: Total cost = 25100 bytes

The packet size with the smallest total cost is 10000 bytes, making it the optimal size for this specific scenario. This size provides a good balance, minimizing both the overhead and the loss due to corruption, leading to more efficient data transmission.