III. The public library has three locations in Raleigh: downtown, Hilldale, and
ID: 3147553 • Letter: I
Question
III. The public library has three locations in Raleigh: downtown, Hilldale, and College Station. The policy is that you may return your books to any of the three branches (not necesarily the one you rented it from·The lead librarian has discovered that of the books borrowed from the downtown branch, 25% are returned downtown. Of the books borrowed from Hilldale. 40 are returned to Hilldale. And of the books borrowed from College Station. 20% are returned there. Furthermore, of the books borrowed downtown. 67.5% are returned to Hilldale and 7.5% are returned to College Station. Of the books borrowed frorn Hilldale, 50% are returned downtown and 10 % returned to College. And of the books borrowed from College Station, 50% are returned downtown and 30% are returned to Hilldale 12. Give linear equations describing the above information, similar to the ones given in the middle of this page (e.g. 125ik + .26mk = yk +1. Be sure to give definitions for all of your variables. To simplify matters, let's assume all books are borrowed weekly, so each is borrowed and returned 7 days later, and let's assume it's a busy period of moving with al boos begin borrowed again immediately.) 13. Does this system have a steady-state vector? Show all work, indicating any computations done on MATLAB. 14. Suppose there are 60 books total distributed amongst the three branches. In the long term, how many books can we expect to find in each location at a given time? Explain briefly how you know.Explanation / Answer
11) The basic idea here is to form Linear Equations out of the Information given.
So Lets denote General Returnable Books for Hildale as H, For College Station as C, Downtown as D.
So by given Information 0.25 D + 0.5 C + +0.5H -> Downtown
0. 4 H + 0. 675 D + 0.3 C -> Hildale
0.2 C + 0.075 D + 0.10H -> College,
Nown Lets assume this is the First Iteration so after a week when all the books are returned
The next Iteration will take place so Lets write Equations for the K Iteration
0.25 Dk + 0.5 Ck + +0.5Hk -> Dk+1
0. 4 Hk + 0. 675 Dk + 0.3 Ck -> Hk+1
0.2 Ck + 0.075 Dk + 0.10Hk -> Ck+1
12 ) Thus Matrix for this process can be
[ 0.25 0.5 0.5 [ D
0.675 0.3 0.4 C = [D C H]
0.075 0.2 0.1] H]
To Acheive a Steady state for the Above Matrix M ,
M .P =P
i.e MP - P =0 . i.e P(M - 1) = 0
So We have to Subtract Identity Matrix I i,e [1 0 0
0 1 0 from Matrix M
Till we get Reducable Rows in the Matrix. Then Again equate the Reducable matrix.
We have to continue this M can be possibly Reduced to I
So that above conitions are saitisfied. If this is achieved Steady state vector exists and if not id doesnt exist.
This can be Programmed in Matlab to get the Result Practically.
14. Here the Question tells H = 60 , D = 60 and C =60 for a Cetrain K.
and it is asking what will be there values as the iteration K becomes very large
i.e K -> Large number.
Again we have to program this in matlab
Pogram a K to K+1 iteration for the changing values of H, C and D using the above Discussed Matrix M.
[ 0.25 0.5 0.5 [ D
0.675 0.3 0.4 C = [D C H]
0.075 0.2 0.1] H]
Then for very large values of K we will get the desired Result. ( If it is a steady state Matrix , Result will be a an
Exact state where no more modification are possible)