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?

To find the long-term distribution of the cars in each location (airport, city, and suburbs), we can model this problem using a **Markov chain**. The problem gives us transition probabilities based on where cars are rented and returned, and we want to find the **steady-state distribution** of the cars after a long period of time.

### Step 1: Define the state space and transition matrix

Let’s define the states:
- \( A \) = airport
- \( C \) = city
- \( S \) = suburbs

The transition matrix \( P \) represents the probabilities of cars being returned to each location based on where they were rented. We can set up the transition matrix as follows, based on the given probabilities:

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

Each row corresponds to where a car was rented, and each column corresponds to where it is returned:
- From the **airport**, 73% are returned to the airport, 2% to the city, and 25% to the suburbs.
- From the **city**, 10% are returned to the airport, 80% to the city, and 10% to the suburbs.
- From the **suburbs**, 25% are returned to the airport, 2% to the city, and 73% to the suburbs.

### Step 2: Set up the steady-state equation

We want to find the steady-state distribution \( \pi = (\pi_A, \pi_C, \pi_S) \), where:
- \( \pi_A \) is the long-term proportion of cars at the airport,
- \( \pi_C \) is the long-term proportion of cars in the city,
- \( \pi_S \) is the long-term proportion of cars in the suburbs.

In a steady state, the distribution \( \pi \) satisfies the equation:

\[
\pi P = \pi
\]

This can be rewritten as the system of equations:

\[
\begin{aligned}
0.73 \pi_A + 0.10 \pi_C + 0.25 \pi_S &= \pi_A \\
0.02 \pi_A + 0.80 \pi_C + 0.10 \pi_S &= \pi_C \\
0.25 \pi_A + 0.10 \pi_C + 0.73 \pi_S &= \pi_S
\end{aligned}
\]

### Step 3: Solve the system of equations

Additionally, the sum of the probabilities must equal 1, since the cars must be somewhere in the system:

\[
\pi_A + \pi_C + \pi_S = 1
\]

We can solve this system of equations to find the steady-state probabilities.

### Step 4: Convert this to real numbers

I will now calculate the steady-state distribution numerically.

Let me do the computation for you.It seems there was an issue solving the system symbolically. I'll adjust the approach by directly solving this system of linear equations numerically. Let me proceed with that method.The steady-state (long-term) distribution of cars is approximately:

- \( \pi_A \approx 0.404 \) (40.4% of cars at the airport),
- \( \pi_C \approx 0.194 \) (19.4% of cars in the city),
- \( \pi_S \approx 0.402 \) (40.2% of cars in the suburbs).

This means that, over time, the proportion of cars in each location stabilizes, with slightly more cars in the airport and suburbs compared to the city.

Would you like further details or have any questions?

Here are 5 related questions for further exploration:
1. How do transition probabilities influence the long-term distribution?
2. What would happen if the rental probabilities were adjusted?
3. How would the model change if cars could be lost or added over time?
4. Can we predict how many cars will be in each location after a fixed number of days?
5. What if there were more locations or more complex rental patterns?

**Tip**: Markov chains are a powerful tool for modeling many real-world systems, especially those involving random, state-based transitions!

Solve a Markov Chain problem involving three car rental locations: airport, city, and suburbs. Learn how to find the steady-state distribution of cars across these locations using a transition matrix and steady-state equations. This problem includes detailed probability analysis and matrix algebra.

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