A car rental service in a certain town has a fleet of about 600 cars, at three l

ID: 3185555 • Letter: A

Question

A car rental service in a certain town has a fleet of about 600 cars, at three locations. A car rented at one location may be returned to any of the three locations. The various fractions of cars returned to the three locations are shown in the matrix to the right. Suppose that on Monday there are 355 cars at the airport (or rented from there), 75 cars at the east side office, and 170 cars at the west side office. What will be the approximate distribution of cars on Wednesday? Cars Rented From: East West Returned To: Airport 0.92 0.04 0.10 Airport 0.00 0.08 0.04 0.85 West 0.920.05East On Wednesday, cars will be at the airport, cars will be at the East location, and cars will be at the West location Round to the nearest integer as needed.)

Explanation / Answer

Let the given matrix be denoted by A. It may be observed that A is a stochastic matrix.

Let X = (355,75,170)T which is the distribution of cars amongst the airport, the east side office and the west side office respectively. The the distribution of cars on Tuesday and Wednesday will be given by AX and A2X respectively. Now, A2 = A.A =

534/625

97/1250

179/1000

1/250

2121/2500

177/2000

177/1250

37/500

293/400

Then A2X = (340,80,180)T ( on rounding off to the nearest whole numbers).

Thus, the approximate distribution of cars on Wednesday will be 340 cars at the airport,80 cars at the the east side office and 180 cars at the the west side office respectively.