An automobile rental company has three locations, which we designate as P, Q, and R. When an automobile is rented at one of the locations, it may be returned to any of the three locations. Suppose, at some specific time, that there are p cars at location P, q cars at Q, and r cars at R.

Question

An automobile rental company has three locations, which we designate as P, Q, and R. When an automobile is rented at one of the locations, it may be returned to any of the three locations.

Suppose, at some specific time, that there are p cars at location P, q cars at Q, and r cars at R. Experience has shown, in any given week, that the p cars at location P are distributed as follows: 10% are rented and returned to Q, 30% are rented and returned to R, and 60% remain at P (these either are not rented or are rented and returned to P). Similar rental histories are known for locations Q and R, as summarized below.

Weekly Distribution History

[latex] ext { Location } P: \quad 60 \% ext { stay at } P, 10 \% ext { go to } Q, 30 \% ext { go to } R ext {. }[/latex]
[latex] ext { Location } Q: 10 \% ext { go to } P, 80 \% ext { stay at } Q, 10 \% ext { go to } R ext {. }[/latex]
[latex] ext { Location } R: 10 \% ext { go to } P, 20 \% ext { go to } Q, 70 \% ext { stay at } R ext {. }[/latex]

Accepted Answer

Let [latex]x _{k}[/latex] represent the state of the rental fleet at the beginning of week k:

[latex]x _{k}=\left[\begin{array}{l}p(k) \q(k) \r(k)\end{array} ight][/latex].

For the state vector [latex]X _{k}[/latex], p(k) denotes the number of cars at location P, q(k) the number at Q, and r(k) the number at R.
From the weekly distribution history, we see that
p(k + 1) = .6p(k) + .1q(k) + .1r(k)
q(k + 1) = .1p(k) + .8q(k) + .2r(k)
r(k + 1) = .3p(k) + .1q(k) + .7r(k).

(For instance, the number of cars at P when week k + 1 begins is determined by the 60% that remain at P, the 10% that arrive from Q, and the 10% that arrive from R.)
To the extent that the weekly distribution percentages do not change, the rental fleet is rearranged among locations P, Q, and R according to the rule [latex]x _{k+1}=A x _{k}[/latex], k = 0,1,..., where A is the (3 × 3) matrix

[latex]A=\left[\begin{array}{lll}.6 & .1 & .1 \.1 & .8 & .2 \.3 & .1 & .7\end{array} ight][/latex].

Question 4.8.5: An automobile rental company has three locations, which we d...

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