# Markov Chains Help

• Apr 21st 2012, 09:13 PM
paolo
Markov Chains Help
hello i would like u to help me with this markov chains problem thanks for ur help

The best-selling college statistics text, The Thrill of Statistics, sells 5 million copies every fall. Some users keep the book, and some sell it back to the bookstore. Suppose that 90% of all students who buy a new book sell it back, 80% of all students who buy a once-used book sell it back, and 60% of all students who buy a twice-used book sell it back. If a book has been used four or more times, the cover falls off, and it cannot be sold back.
a In the steady state, how many new copies of the book will the publisher be able to sell each year?
b Suppose that a bookstore’s proﬁt on each type of book is as follows:
New book: $6 Once-used book:$3
Twice-used book: $2 Thrice-used book:$1
If the steady-state census is representative of the bookstore’s sales, what will be its average proﬁt per book?
• Apr 25th 2012, 06:46 AM
tom@ballooncalculus
Re: Markov Chains Help
So if they sell it back they do so before next fall?

So, let an, bn, cn and dn be the number of new, second-hand, third-hand and fourth-hand books sold during fall in year n.

a0 = 5 million; b0 = c0 = d0 = 0.

a1 = 5 million - b1; b1 = 0.9 a0; c1 = d1 = 0.

a2 = 5 million - b2 - c2; b2 = 0.9 a1; c2 = 0.8 b1; d2 = 0.

a3 = 5 million - b3 - c3 - d3; b3 = 0.9 a2; c3 = 0.8 b2; d3 = 0.6 c2.

Or eliminate b, c and d to get an in terms of an-1, an-2 and an-3.

Make a spreadsheet from either to see it stabilise at a60.

Homogenise the second as directed here Recurrence relation - Wikipedia, the free encyclopedia, and you can make a transition matrix,

$\left( \begin{array}{cccc}0 & 1 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \\0.432 & 0.238 & 0.18 & 0.1 \end{array} \right)$