Question

1. A car rental company has three locations and keeps statistics as follows: If a car is rented from location 1 on a given day, at the beginning of the next day - 45% of the time the car is returned to location 1. - 15% of the time the car is returned to location 2. - 10% of the time the car is returned to location 3. - 30% of the time the car is still on loan. If a car is rented from location 2 on a given day, at the beginning of the next day - 10% of the time the car is returned to location 1. - 25% of the time the car is returned to location 2. - 20% of the time the car is returned to location 3. - 45% of the time the car is still on loan. If a car is rented from location 3 on a given day, at the beginning of the next day - 15% of the time the car is returned to location 1. - 15% of the time the car is returned to location 2. - 35% of the time the car is returned to location 3. - 35% of the time the car is still on loan. If a car is still on loan on a given day, at the beginning of the next day - 50% of the time the car is returned to location 1. - 15% of the time the car is returned to location 2. - 10% of the time the car is returned to location 3. - 25% of the time the car is still on loan. Model this scenario using a stochastic matrix, and use it to answer the following questions. Justify your answers. (a) If the cars are evenly distributed among the 4 possibilities on a given day, how will they be distributed the next day? (b) If the company has a fleet of 2500 cars, how many parking spaces are needed at each location to adequately accommodate the cars? How many cars are expected to be on loan and not available for rent on a given day?

Answer 1

(a) The next day's car distribution, starting with an even distribution, can be found by multiplying the initial distribution vector by the given stochastic matrix. (b) For a fleet of 2500 cars, the required parking spaces at each location are determined by the next day's distribution, and the expected number of cars on loan is the corresponding value in the "on loan" state of the next day's distribution multiplied by 2500.

Step-by-step explanation:

Let's represent the four possibilities as follows:

Being at location 1

Being at location 2

Being at location 3

Being on loan

The stochastic matrix (transition matrix) for the given scenario is:

P = | 0.45 0.15 0.10 0.30 |

| 0.10 0.25 0.20 0.45 |

| 0.15 0.15 0.35 0.35 |

| 0.50 0.15 0.10 0.25 |

Each row represents the current state, and each column represents the probability of transitioning to the corresponding state.

Now, let's answer the questions:

(a) If the cars are evenly distributed among the 4 possibilities on a given day, how will they be distributed the next day?

If the initial distribution is represented as a row vector [0.25 0.25 0.25 0.25], then the distribution the next day can be obtained by multiplying this vector by the stochastic matrix.

You can perform this matrix multiplication to find the distribution of cars the next day.

(b) If the company has a fleet of 2500 cars, how many parking spaces are needed at each location to adequately accommodate the cars? How many cars are expected to be on loan and not available for rent on a given day?

Let's assume the initial distribution of cars is [0.25 0.25 0.25 0.25] (25% at each location).

Now, if the company has a fleet of 2500 cars, you can multiply the distribution vector by 2500 to get the number of cars in each state.

The number of parking spaces needed at each location would be the number of cars at that location. The number of cars expected to be on loan and not available for rent is the number of cars in the "on loan" state.