Question

1. There are two competing cellphone companies in a city, Company A and Company B. In 2012, 40% of the people in the city use Company A, and 60% of the people use Company B. Each year 40% of the people who use Company A switch to using Company B, and 30% of the people who use Company B switch to using Company A.(Assume that everyone in the city has a cellphone with either Company A or Company B.)

(a) Model this situation with a Markov chain. Find the transition matrix and associated digraph that describes the number of people using each company.

(b) What percent of people will use Company A in 2013?

(c) What percent of people will use Company A in 2014?

Answer 1

(a). Let A_k and B_k represent the market shares of the Companies A and B respectively in the kth year.

Then A_k+1 = 0.60 A_k + 0.30B_k and   B_k+1 = 0.40A_k +0.70 Q_k.

Now, let A =

0.60	0.30
0.40	0.70

Then A is the transition matrix representing the scenario.

(b). The initial vector in 2012 is u = (0.40,0.60)^T.

Now, Au = (21/50,29/50)^T. This means that (21/50)*100 = 42 % of people will use Company A in 2013.

( c). Further, A² u = A.Au = (213/500,287/500)^T. This means that (213/500)*100 = 42.6 % of people will use Company A in 2014.