Question

How long does it take to transmit \(x\) KB over a \(y\)-Mbps link? Give your answer as a ratio of \(x\) and \(y\).

Short answer

\(\frac{8x}{y \times 1000}\) seconds

Step-by-step solution

Convert KB to Kbits

First, convert the data size from kilobytes (KB) to kilobits (Kbits). Since 1 byte = 8 bits, multiply the given data size by 8. So, if the data size is given as \(x\) KB, it will be \(8x\) Kbits.

Calculate the Transmission Time

Now, use the link speed in Mbps to calculate the transmission time. The link speed given is \(y\) Mbps, which means \(y\) megabits per second. To find the transmission time in seconds, divide the total data size in Kbits by the link speed in Mbps.The formula for the transmission time \(T\) is: \[ T = \frac{8x \text{ Kbits}}{y \text{ Mbps}} \] Since 1 Mbps = 1,000 Kbits, you need to divide the numerator by y*1000 to convert the denominator appropriately.

Simplify the Expression

Simplify the formula to find the final time taken: \[ T = \frac{8x}{y \times 1000} \text{ seconds} \] So, the transmission time is given as the ratio \( \frac{8x}{y \times 1000}\) seconds.

Key concepts

Understanding Data Size Conversion

Before diving into the calculation of transmission time, it's important to understand how to convert data sizes. Data sizes in computer systems are typically measured in bytes (B) or bits (b). Here's a quick guide to converting kilobytes (KB) to kilobits (Kbits):

1. A byte consists of 8 bits. When you have data in kilobytes (KB), multiply by 8 to convert it to kilobits (Kbits).
2. For example, if you have a file size of 2 KB:
- 2 KB x 8 = 16 Kbits

This step is crucial because transmission speeds are often given in megabits per second (Mbps) and not in megabytes per second (MBps). Converting data size to kilobits ensures that we use the correct units in our formula for transmission time.

Understanding Transmission Speed

Transmission speed refers to the rate at which data is transmitted from one point to another in a given amount of time. This speed is often given in megabits per second (Mbps), which stands for millions of bits per second. Here are some key points to consider:

- 1 Mbps equals 1,000 Kbits per second.
- A higher transmission speed means faster data transfer.

For example, a transmission speed of 10 Mbps means that 10 million bits can be transmitted every second. Knowing the transmission speed helps us determine how quickly data can be sent over a network.

Formula for Transmission Time

Once you've converted the data size and understood the transmission speed, you can calculate the transmission time using a simple formula. The key to solving this problem is the formula:

\[ T = \frac{8x \text{ Kbits}}{y \text{ Mbps}} \tag{1} \]

Where:
- \( T \) is the transmission time in seconds
- \( x \) is the data size in kilobytes (KB)
- \( y \) is the transmission speed in megabits per second (Mbps)

To simplify it further, since 1 Mbps = 1,000 Kbits, we can rewrite the formula as:

\[ T = \frac{8x}{y \times 1000} \text{ seconds} \tag{2} \]

This means the time it takes to transmit the data is given by the ratio of \( \frac{8x}{y \times 1000} \) seconds. This formula helps you quickly determine how long it will take to send a certain amount of data over a network with a given speed.