Problem 28 - A First Course in Probability

Question

The \(k\) -of-r-out-of- \(n\) circular reliability system, \(k \leq\) \(r \leqq n,\) consists of \(n\) components that are arranged in a circular fashion. Each component is either functional or failed, and the system functions if there is no block of \(r\) consecutive components of which at least \(k\) are failed. Show that there is no way to arrange 47 components, 8 of which are failed, to make a functional 3 -of- 12 -out-of- 47 circular system.

Step-by-Step Solution

Answer

In order to create a functional 3-out-of-12-out-of-47 circular system, there must not be any block of 12 consecutive components with at least 3 failed components. Distributing the 8 failed components as evenly as possible in the circle, we find the arrangement: F - OOOOO - F - OOOOO - F - OOOOO - F - OOOOO - F - OOOOO - F - OOOOO - F - OOOOOF - OOOOO - F Even in this "best" arrangement, there will always be at least one block of 12 consecutive components containing 3 failed components, making the system non-functional. Therefore, it is impossible to arrange 47 components, 8 of which are failed, to form a functional 3-out-of-12-out-of-47 circular system.

Step 1: Understand the requirements for a functional system

To have a functional 3-out-of-12-out-of-47 circular system, there must not be any block of 12 consecutive components in which there are at least 3 failed components. Otherwise, the system would fail.

Step 2: Find the best possible arrangement

To minimize the chances of having a block of 12 consecutive components with at least 3 failed components, we should try to distribute the failed components as evenly as possible. In this case, since we have 47 components and 8 failed components, we can try to place each failed component after every 5 or 6 functional components.

Step 3: Count failed components in consecutive blocks

Following the best possible arrangement from step 2, let's place a failed component after each 5 or 6 functional components: F - OOOOO - F - OOOOO - F - OOOOO - F - OOOOO - F - OOOOO - F - OOOOO - F - OOOOOF - OOOOO - F In this arrangement, we placed a failed component (F) after every 5 functional components (O), with the exception of one failed component placed after 6 functional components.

Step 4: Check for failed blocks

Now, we'll check for blocks of 12 consecutive components. A block will be considered failed if it contains at least 3 failed components. In the arrangement from step 3, we can see that there will be an overlap between consecutive blocks. However, no matter where you position the starting point of the 12-component block, there will always be at least one block with 3 failed components (e.g., F - OOOO - F - OOOOO - F).

Step 5: Prove that the system is not functional

Since we were trying to find the best possible arrangement of failed components to create a functional 3-out-of-12-out-of-47 circular system, the result we reached in step 4 shows that it's impossible to construct a functional system in this case. Therefore, we can conclude that there is no way to arrange 47 components, 8 of which are failed, to make a functional 3-out-of-12-out-of-47 circular system.

Key Concepts

Circular Systems

A circular system is a unique arrangement where components are placed in a loop, or like beads in a circle.
This configuration is commonly used in reliability analysis because it simulates scenarios like network loops, ring circuits, or even certain biological processes.
In a circular system, each component interacts with its neighbors in a continuous line. This cyclic nature means it wraps around, making the system more complex than a simple linear setup:

• Every component is equally important since they directly influence both left and right neighbors.
• The circular structure ensures there's no true 'start' or 'end' point, affecting how systems are analyzed for reliability.

When evaluating or creating circular systems, the goal is often to maximize functionality and minimize failure risks. Understanding this setup is crucial when studying how failures can impact system performance, especially in a resourceful arrangement like a 3-out-of-12-out-of-47 system in this exercise.

Component Arrangement

In reliability systems, the arrangement of components, especially when dealing with failed or potentially failing components, can significantly impact overall functionality.
This involves deciding how to distribute these components to avoid redundant paths of failure.
To optimize reliability, a strategic arrangement can balance the number of failed and functional components:

• The aim is to ensure that no continuous series of components exceeds allowed limits of failure.
• In the context of the 3-out-of-12-out-of-47 system, the goal was to distribute failed components in a way that any block of 12 components would not contain 3 or more failed components.

Smart placement reduces risk; however, the wrong arrangement can lead to inevitable overlap of failed segments, as seen when attempting to achieve functionality in the given problem.

Failed Components

Failed components in any system hinder its overall performance and can lead to cascading failures unless appropriately managed.
In reliability systems, failed components pose a twofold challenge: they must be recognized, and, if possible, isolated to prevent broader dysfunction.
In the configuration challenge posed by the exercise:

• There are 47 components, out of which 8 have failed.
• The task was to distribute these failures in a way that the system remains functional (i.e., blocks of 12 components containing less than 3 failed ones).

Despite efforts, certain density or groupings of these failed parts within the system cause unavoidable overlapping, where multiple layers or areas within the circular setup end up with too many failures. This highlights the importance of strategic distribution efforts and their limitations in real-world applicability, as witnessed by the exercise result.

Probability

Probability plays a crucial role in the analysis and design of reliability systems.
It helps in forecasting how likely a system will continue performing without failure over a given period.
This exercise involves understanding the likelihood of components failing and how their arrangement impacts system performance:

• The probability that any block of 12 components contains 3 or more failed ones determines the system's success or failure.
• With 8 failed out of 47 components, calculating how consistently failures align within blocks ties directly to notions of probability.

The arrangement attempted in the exercise shows that probability isn't just about numbers, but also patterns:

Random distribution seldom achieves the desired functionality without strategic planning. It's a game of chance versus choice, often tilting towards failure when not thoughtfully engineered.

By understanding probability within such arrangements, engineers can better predict outcomes and adjust systems to mitigate unforeseen failures.