Question

Imagine that you have trained your St. Bernard, Bernie, to carry a box of three 8-mm tapes instead of a flask of brandy. (When your disk fills up, you consider that an emergency.) These tapes each contain 10 gigabytes. The dog can travel to your side, wherever you may be, at \(18 \mathrm{~km} /\) hour. For what range of distances does Bernie have a higher data rate than a transmission line whose data rate (excluding overhead) is 150 Mbps? How does your answer change if (i) Bernie's speed is doubled; (ii) each tape capacity is doubled; (iii) the data rate of the transmission line is doubled.

Short answer

Bernie is faster for distances less than 29,496 km. If his speed or tape capacity are doubled, Bernie's efficient range increases, while doubling the line's rate decreases it.

Step-by-step solution

Understand the Problem

We need to compare the data transfer capability of Bernie the dog carrying three tapes against a transmission line. The tapes each have a 10GB capacity, Bernie has a speed of 18 km/h, and the transmission line has a data rate of 150 Mbps.

Calculate Data Capacity Carried by Bernie

Bernie carries three tapes, each holding 10 gigabytes, so the total capacity carried by Bernie is:\[3 \times 10 \text{ GB} = 30 \text{ GB}\]Convert this to megabits:\[30 \text{ GB} \times 1024 \text{ MB/GB} \times 8 \text{ b/MB} = 245,760 \text{ Mb}\]

Determine Bernie's Data Rate

To find Bernie's data rate, consider how long it takes Bernie to travel a certain distance and deliver the data. If Bernie travels at 18 km/h, it takes \( t \) hours to cover distance \( d \) (in km):\[t = \frac{d}{18}\]The data rate is the total data capacity divided by the time:\[\text{Data rate}_{Bernie} = \frac{245,760 \text{ Mb}}{t} = \frac{245,760 \text{ Mb}}{\frac{d}{18}} = \frac{245,760 \times 18}{d} \text{ Mbps}\]That simplifies to:\[\frac{4,424,480}{d} \text{ Mbps}\]

Set Up Inequality for Bernie's Advantage

We need Bernie's data rate to be higher than the transmission line's data rate:\[\frac{4,424,480}{d} > 150\]Solving for \( d \):\[d < \frac{4,424,480}{150}\]This gives:\[d < 29,496.53 \text{ km}\]Thus, Bernie has a higher data rate for any distance less than approximately 29,496 km.

Consider the Change in Variables

(i) If Bernie's speed is doubled to 36 km/h: - Re-calculate the data rate and inequality: \[ \frac{8,848,960}{d} > 150 \Rightarrow d < 58,993.07 \text{ km} \]Bernie is more efficient for larger distances.(ii) If each tape capacity is doubled to 20 GB: - Re-calculate the total data: 61,440 MB - Inequality: \( \frac{8,448,960}{d} > 150 \Rightarrow d < 58,993.07 \text{ km} \)(iii) If the line data rate doubles to 300 Mbps: - Inequality: \( \frac{4,424,480}{d} > 300 \Rightarrow d < 14,748.27 \text{ km} \) - Bernie is only more efficient for shorter distances.

Key concepts

Network Performance

When considering network performance, it's important to understand how data transfer rates affect how quickly and efficiently data can be moved from one place to another. In our scenario, Bernie the St. Bernard plays the unconventional role of a data carrier, challenging a digital transmission line. We measure network performance based on the data rate, which is the speed at which data is sent. Here, Bernie's network performance depends on the physical speed at which he can deliver the tapes and the total data capacity he can carry.

Key to understanding Bernie's network performance is his speed and the timescale over which he delivers data. When Bernie travels at 18 km/h with three tapes, the expressions for performance rely on how quickly he covers the distance relative to the data size. To evaluate performance, we calculate Bernie's comparative advantage over a given digital line, defined by its data rate in megabits per second (Mbps). Analyzing this balance helps us understand how traditional transportation compares with a digital network in specific scenarios. This direct comparison offers insights into how various factors influence overall network throughput and data delivery efficiency.

Bandwidth Comparison

Bandwidth is a crucial factor in comparing data transfer methods, as it indicates the maximum data that can be transferred over a network in a given time. Bernie’s carrying capacity involves three tapes, each with a substantial 10GB data payload. Adding the tapes' data capacity gives a total of 30GB, which translates to 245,760 megabits.

When considering bandwidth in terms of distance, time, and capacity, Bernie's bandwidth evaluation comes from how the physical transportation of data competes with electronic means. This means comparing our unconventional dog courier to a standard 150 Mbps transmission line. As we see in the calculations, Bernie manages a higher bandwidth up to a distance of approximately 29,496 km. If Bernie's speed or tape capacity increases, his effective bandwidth expands, surpassing electronic methods for longer distances. Understanding these calculations aids in discerning benchmarks for different data transfer methods, showcasing how effective certain strategies may be under diverse conditions.

Information Theory

Information Theory gives us tools to quantify and compare different data transfer mechanisms, focusing on the efficiency and effectiveness of information passing from sender to receiver. In comparing Bernie to a communication line, we leverage principles of this theory to analyze relative effectiveness. Concepts like entropy and channel capacity highlight limits on data transfer rates.

In this problem, we look at the transmission line's stated data rate of 150 Mbps. Information theory helps us lay down strategies like maximizing tape capacity or improving Bernie's speed to enhance the data rate. Effective use of information theory allows us to directly calculate how adjustments in variables like Bernie's speed, tape size, or the line data rate affect outcomes and optimal data strategy choices.

• Doubling Bernie's speed significantly increases his effective data transfer range.
• Increasing tape capacity directly enhances total transferable data.
• Changes in the line's data rate allow us to see how improvements affect efficiency and capability.

These adjustments are all fundamental to analyzing straightforward problems using advanced principles from information theory, helping to precisely evaluate the efficiency of complex data transfer settings.