28 : Consider the isolated Swiss town of Andelfingen, inhabited by 1,200 families. Each family takes a weekly shopping trip to the only grocery store in town, run by Mr. and Mrs. Wipf, until the day when a new, fancier (and cheaper) chain store, Migros, opens its doors. It is not expected that everybody will immediately run to the new store, but we do anticipate that 20% of those shopping at Wipf’s each week switch to Migros the following week. Some people who do switch miss the personal service (and the gossip) and switch back: We expect that 10% of those shopping at Migros each week go to Wipf’s the following week. The state of this town (as far as grocery shopping is concerned) can be represented by the vector
where w(t) and m(t) are the numbers of families shopping at Wipf’s and at Migros, respectively, t weeks after Migros opens. Suppose w(0) = 1,200 and m(0) = 0.
a. Find a 2 × 2 matrix A such that role="math" localid="1659586084144" . Verify that A is a positive transition matrix. See Exercise 25.
b. How many families will shop at each store after t weeks? Give closed formulas. c. The Wipfs expect that they must close down when they have less than 250 customers a week. When does that happen?
Transition matrix may refer to: The matrix associated with a change of basis for a vector space. Stochastic matrix, a square matrix used to describe the transitions of a Markov chain. State-transition matrix, a matrix whose product with the state vector. at an initial time.
(A) Clearly we see that,
(B) The matrix A is positive transition matrix, we know its eigenvectors are
Hence, the final answer is : (a)
(c) The wipfs will never have to close.
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