Most long-distance telephone service in the United States is provided by three

Chapter 7, Problem 32

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Most long-distance telephone service in the United States is provided by three companies: AT&T, MCI, and Sprint. The three companies are in fierce competition, offering discounts or even cash to those who switch. If the figures advertised by the companies are to be believed, people are switching their long distance provider from one month to the next according to the following diagram:For example, 20% of the people who use AT&T go to Sprint one month later. a. We introduce the state vectora(t) fraction using AT&T x(t) = m(t) fraction using MCI s(t) fraction using Sprint.Find the matrix A such that x(t + 1) = Ax(t), assuming that the customer base remains unchanged. Note that A is a regular transition matrix. b. Which fraction of the customers will be with each company in the long term? Do you have to know the current market shares to answer this question? Use the power method introduced in Exercise 30.