In a large city, a car rental company has three locations, the airport, the city, and the suburbs. You have data on which location the cars are returned daily. 1. rented at airport, 2% are returned to the city and 25% to the suburbs. The rest are returned to the airport. 2. rented in city, 10% returned to airport, 10% returned to suburbs. The rest are returned to the city. 3. Rented in suburbs, 25% are returned to the airport and 2% to the city. The rest are returned to the suburbs. If initially there are 35 cars at the airport, 150 in the city and 35 in the suburbs, what is the longest term distribution of the cars?

This problem can be solved using **Markov Chains** since we have transitions between different states (airport, city, suburbs) with fixed probabilities. We aim to find the long-term distribution of the cars across the three locations.

### Step 1: Define the system

Let the three locations be represented as states:
- \( A \) for the Airport,
- \( C \) for the City,
- \( S \) for the Suburbs.

### Step 2: Set up the transition matrix

We create a **transition matrix** \( P \) where each entry \( P_{ij} \) represents the probability of transitioning from location \( i \) to location \( j \).

From the problem statement:

1. **Cars rented at the Airport**:
   - 2% returned to the City.
   - 25% returned to the Suburbs.
   - The rest (73%) returned to the Airport.

2. **Cars rented in the City**:
   - 10% returned to the Airport.
   - 10% returned to the Suburbs.
   - The rest (80%) returned to the City.

3. **Cars rented in the Suburbs**:
   - 25% returned to the Airport.
   - 2% returned to the City.
   - The rest (73%) returned to the Suburbs.

Thus, the transition matrix \( P \) is:

\[
P = \begin{pmatrix}
0.73 & 0.02 & 0.25 \\
0.10 & 0.80 & 0.10 \\
0.25 & 0.02 & 0.73
\end{pmatrix}
\]

### Step 3: Find the long-term distribution

The **long-term distribution** is a vector \( \mathbf{x} = \begin{pmatrix} x_A & x_C & x_S \end{pmatrix} \), where \( x_A \), \( x_C \), and \( x_S \) represent the proportions of cars in the Airport, City, and Suburbs, respectively. 

In the long term, the system reaches a steady state, which means \( \mathbf{x} \) satisfies the following equation:
\[
\mathbf{x}P = \mathbf{x}
\]

This can be written as:
\[
\begin{pmatrix} x_A & x_C & x_S \end{pmatrix} \begin{pmatrix}
0.73 & 0.02 & 0.25 \\
0.10 & 0.80 & 0.10 \\
0.25 & 0.02 & 0.73
\end{pmatrix} = \begin{pmatrix} x_A & x_C & x_S \end{pmatrix}
\]

This results in the following system of linear equations:

1. \( 0.73x_A + 0.10x_C + 0.25x_S = x_A \)
2. \( 0.02x_A + 0.80x_C + 0.02x_S = x_C \)
3. \( 0.25x_A + 0.10x_C + 0.73x_S = x_S \)

Additionally, the sum of the probabilities must equal 1:
\[
x_A + x_C + x_S = 1
\]

### Step 4: Solve the system

We can solve this system of equations to find the values of \( x_A \), \( x_C \), and \( x_S \).

Let me calculate the solution.It seems that the system of equations didn't yield a solution in its current form, which might be due to rounding or approximation errors in the coefficients. Let me adjust the setup and solve it again using more precise values.The long-term distribution of the cars across the three locations is:

- \( x_A = 0.4545 \) (about 45.45% of the cars will be at the airport),
- \( x_C = 0.0909 \) (about 9.09% of the cars will be in the city),
- \( x_S = 0.4545 \) (about 45.45% of the cars will be in the suburbs).

This means that in the long run, the cars will be almost equally distributed between the airport and the suburbs, with a smaller portion remaining in the city.

Would you like further details or have any questions?

Here are 5 related questions you might find helpful:
1. How do we compute the steady state for a Markov chain with more states?
2. How can we use eigenvectors to find the steady state distribution?
3. What are the real-world applications of Markov chains in business?
4. How can we model car rental behavior using a different set of conditions?
5. How would adding another location affect the distribution?

**Tip:** Markov chains can be applied to a variety of processes involving transitions between states, including economics, genetics, and game theory!

In a large city, a car rental company has three locations, the airport, the city, and the suburbs. You have data on which location the cars are returned daily. 1. rented at airport, 2% are returned to the city and 25% to the suburbs. The rest are returned to the airport. 2. rented in city, 10% returned to airport, 10% returned to suburbs. The rest are returned to the city. 3. rented in suburbs, 25% are returned to the airport and 2% to the city. The rest are returned to the suburbs. If initially there are 35 cars at the airport, 150 in the city and 35 in the suburbs, what is the longest term distribution of the cars?

Explore the long-term distribution of rental cars across three locations using Markov Chains. This problem involves calculating steady-state distributions using a transition matrix and solving a system of equations. Perfect for advanced high school or college-level students.

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