You own a trailer rental company in a large city and you have four locations, one in the South East, one in the North East, one in the North West, and one in the South West. Denote these locations by SE, NE, NW, and SW respectively. Suppose that the following table is observed to take place. SE NE NW SW
SE 1/3 1/10 1/10 1/5
NE 1/3 7/10 1/5 1/10
NW 2/9 1/10 3/5 1/5
SW 1/9 1/10 1/10 1/2
In this table, the probability that a trailer starting at NE ends in NW is 1/10, the probability that a trailer starting at SW ends in NW is 1/5, and soforth. Approximately how many will you have in each location after a long time if the total number of trailers is 413?
(a) Do problem 7.3.11 in the text, plus the following related questions
(b) Find an initial vector X(0)メ0 with the property that limn-wAnX(0)-0. Here A is the matrix you should have produced in part (a)
(c) Could the vector from (b) be a valid initial vector in the context of the problem? Could a vector with the property in (b) ever be a valid initial vector in the context of the problem?