An optimisation problem. A company would like to start a bike rental business in Edinburgh, with parking spots scattered in four main zones: Old Town (zone 1), Haymarket (zone 2), Fountainbridge (zone 3), and Newington (zone 4). During a trial period, the company installed 200 bikes across the city and monitored their position to understand how customers use them. They noticed the following:

On average, 50% of bikes in zone 1 are parked back in zone 1 at the end of the day, 20% are parked in zone 2, 20% in zone 3, and 10% in zone 4.

On average, 20% of bikes in zone 2 are parked back at zone 2 at the end of the day, 40% are parked in zone 1, 20% in zone 3, and 20% in zone 4.

On average, 40% of bikes in zone 3 are parked back in zone 3 at the end of the day, 30% are parked in zone 1, 20% in zone 2, and 10% in zone 4.

On average, 50% of bikes in zone 4 are parked back in zone 4 at the end of the day, 10% are parked in zone 1, 20% in zone 2, and 20% in zone 3.

(a) Find the stable distribution of bikes, that is, the distribution of bikes so that the number of bikes parked in each zone at the beginning of a day remains unchanged at the end of that day.

This problem can be formulated as a **Markov Chain**, where the zones represent the states, and the percentages represent the transition probabilities between states. The goal is to find the **stationary (or stable) distribution** of bikes, which means the proportion of bikes in each zone remains constant after a day's movement.

Let’s denote:

- \( p_1 \): the proportion of bikes in **zone 1**,
- \( p_2 \): the proportion of bikes in **zone 2**,
- \( p_3 \): the proportion of bikes in **zone 3**,
- \( p_4 \): the proportion of bikes in **zone 4**.

### Transition matrix
The transition matrix \( T \) describes the probabilities of moving from one zone to another. Based on the data, the matrix is:

\[
T = \begin{pmatrix}
0.50 & 0.40 & 0.30 & 0.10 \\
0.20 & 0.20 & 0.20 & 0.10 \\
0.20 & 0.20 & 0.40 & 0.20 \\
0.10 & 0.20 & 0.10 & 0.50 \\
\end{pmatrix}
\]

Each entry \( T_{ij} \) describes the probability of a bike moving from zone \( j \) to zone \( i \) by the end of the day.

### Stationary distribution condition
The stationary distribution satisfies:

\[
\mathbf{p} = T \mathbf{p}
\]

This translates into the following system of equations:

\[
\begin{aligned}
p_1 &= 0.50p_1 + 0.40p_2 + 0.30p_3 + 0.10p_4 \\
p_2 &= 0.20p_1 + 0.20p_2 + 0.20p_3 + 0.20p_4 \\
p_3 &= 0.20p_1 + 0.20p_2 + 0.40p_3 + 0.10p_4 \\
p_4 &= 0.10p_1 + 0.20p_2 + 0.10p_3 + 0.50p_4 \\
\end{aligned}
\]

Additionally, since \( \mathbf{p} \) represents a probability distribution, the sum of the proportions must equal 1:

\[
p_1 + p_2 + p_3 + p_4 = 1
\]

### Solving the system of equations
We now solve this system of linear equations. I'll handle the calculations.It appears there was difficulty in solving the system directly. Let's reformulate and attempt solving again with some simplifications.

We'll rewrite the system of equations more clearly to proceed. Allow me to calculate this again manually.It seems like I can’t do more advanced data analysis right now. Please try again later. However, if you’d like, I can guide you through solving this system manually or explain the next steps you would take to complete it.

Would you like to continue with the problem manually or explore similar questions? Here are a few relevant questions:

1. How do you verify if a matrix represents a valid Markov chain?
2. How do you compute the transition matrix for different systems?
3. What are other real-world applications of Markov chains in business?
4. How can you simulate a bike distribution problem using programming tools?
5. What techniques can be used to solve large systems of linear equations?

**Tip:** When solving for a stationary distribution, always ensure the sum of probabilities equals 1. This helps balance the system.

A company would like to start a bike rental business in Edinburgh, with parking spots scattered in four main zones: Old Town (zone 1), Haymarket (zone 2), Fountainbridge (zone 3), and Newington (zone 4). During a trial period, the company installed 200 bikes across the city and monitored their position to understand how customers use them. They noticed the following: On average, 50% of bikes in zone 1 are parked back in zone 1 at the end of the day, 20% are parked in zone 2, 20% in zone 3, and 10% in zone 4. On average, 20% of bikes in zone 2 are parked back at zone 2 at the end of the day, 40% are parked in zone 1, 20% in zone 3, and 20% in zone 4. On average, 40% of bikes in zone 3 are parked back in zone 3 at the end of the day, 30% are parked in zone 1, 20% in zone 2, and 10% in zone 4. On average, 50% of bikes in zone 4 are parked back in zone 4 at the end of the day, 10% are parked in zone 1, 20% in zone 2, and 20% in zone 3. (a) Find the stable distribution of bikes, that is, the distribution of bikes so that the number of bikes parked in each zone at the beginning of a day remains unchanged at the end of that day.

Solve a real-world Markov Chain problem involving a bike rental system in Edinburgh. Learn how to calculate the stationary distribution of bikes across four zones, using a transition matrix to find a stable distribution. Includes concepts of probability and linear algebra, suitable for undergraduate students.

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