Question

en+2 (40%) A car rental agency has offices in Phoenix, Denver, Chicago, and Atlanta. The agency allows in-town and one-way rentals, so that cars rented in one location end up in anothet Stai ixiics show that at the end of each week 50% of all rentals are in-town rentals. As for ane wav rentals From , Phoenix, 40% go to Denver, 30% to Chicago, and the rest to Atlanta; . From Denver, 4096 go to Atlanta and 60% to Chicago; . From Chicago, 409% go to Atlanta and the rest to Denver . nom Atlanta, 60% go to Chicago, 30% to Denver, and 10% to Phoenix. (a) (20%) If the company starts the week with 100 cars in each location, what would be the distribution of cars at the end of three weeks? (b) (20%) what is the long-term distribution of cars among the rental locations? Problem 2 (40%) A car rental agency has offices in Phoenix, Denver, Chicago, and Atlanta. The agency allows in-town and one-way rentals, so that cars rented in one location end up in another. Statistics show that at the end of each week 50% of all rentals are in-town rentals. As for one-way rentals, ·From Phoenix, 40% go to Denver, 30% to Chicago, and the rest to Atlanta; From Denver, 40% go to Atlanta and 60% to Chicago; . From Chicago, 40% go to Atlanta and the rest to Denver; From Atlanta, 60% go to Chicago, 3096 to Denver, and 10% to Phoenix. (a) (20%) If the company starts the week with 100 cars in each location, what would be the distribution of cars at the end of three weeks? (b) (20%) what is the long-term distribution of cars among the rental locations?

Answer 1

(a)

We can model the given problem as a Markov chain with four states Phoenix, Denver, Chicago and Atlanta.

50% of all rentals are in-town rentals.

Transition probability from Phoenix to Phoenix, Denver, Chicago and Atlanta are 0.5, 0.5 * 0.4 = 0.2, 0.5 * 0.3 = 0.15 and 0.5(1- (0.4 + 0.3)) = 0.15 respectively.

Transition probability from Denver to Denver, Atlanta and Chicago are 0.5, 0.5* 0.4 = 0.2 and 0.5 * 0.6 = 0.3 respectively.

Transition probability from Chicago to Chicago, Atlanta and Denver are 0.5, 0.5* 0.4 = 0.2 and 0.5 * 0.6 = 0.3 respectively.

Transition probability from Atlanta to Atlanta, Chicago, Denver and Phoenix are 0.5 , 0.5 * 0.6 0.3, 0.5 * 0.3 = 0.15 and 0.5 * 0.1 = 0.05 respectively.

The transition probability matrix is,

States are in the order - Phoenix, Denver, Chicago, Atlanta

$P = \begin{bmatrix} 0.5 & 0.2 & 0.15 & 0.15\\ 0 & 0.5 &0.3 &0.2 \\ 0 & 0.3 & 0.5 &0.2 \\ 0.05 & 0.15 & 0.3 & 0.5 \end{bmatrix}$

$P^3 = \begin{bmatrix} 0.13975 & 0.29475 & 0.312375 & 0.253125\\ 0.018 & 0.334 & 0.3705 &0.2775 \\ 0.018 & 0.326 & 0.3785 & 0.2775 \\ 0.042375 & 0.292875 & 0.363 & 0.30175 \end{bmatrix}$

If the company starts with 100 cars at each location, the initial distribution is [100, 100, 100, 100]

Distribution after three weeks is,

$[100, 100, 100, 100] P^3 = [100, 100, 100, 100] \begin{bmatrix} 0.13975 & 0.29475 & 0.312375 & 0.253125\\ 0.018 & 0.334 & 0.3705 &0.2775 \\ 0.018 & 0.326 & 0.3785 & 0.2775 \\ 0.042375 & 0.292875 & 0.363 & 0.30175 \end{bmatrix}$

= [21.8125 124.7625 142.4375 110.9875]

Thus, there will be 21.8125, 124.7625, 142.4375 and 110.9875 cars at Phoenix, Denver, Chicago, Atlanta respectively.

(b)

Let $\pi$ = (a, b , c, d) be the long-term distribution.

Then $\pi$ P = $\pi$

which gives the below equation

0.5a + 0.05d = a

0.2a + 0.5b + 0.3c + 0.5d = b

0.15a + 0.3b + 0.5c + 0.3d = c

0.15a + 0.2b + 0.2c + 0.5d = d

and a + b + c + d = 1

Solving these equations we get,

a = 0.0283, b =  0.3183, c = 0.3697, d = 0.2837

Thus long term distribution is (0.0283, 0.3183, 0.3697, 0.2837)