Question

Markov Chains

(a) Three companies provide internet service in a city of 20,000 people. At the beginning of the year, the

market shares of each company are as follows: 11,800 people use company A, 6200 people use company

B, and only 2000 people use company C. Each month, 5% of company A’s customers switch to company

B, and 3% of company A’s customers switch to company C. During the same time, 4% of company B’s

customers switch to A, and 6.5% switch to C. Also during the same time, 2% of company C’s customers

switch to company A, and 1.5% switch to company B. Let A_k represent the probability that a randomly

chosen consumer on day k will be using company A’s service, B_k be the probability they will be using

company B’s service, and C_k be the probability that they go with company C.

1) What is the probability that a person who started with company A stayed with company A after the

first month? What is the probability that they switched to a different company?

2)Define a state vector p_k whose entries are the probabilities that a randomly selected person in the city will be using each companies service during month k. What is the initial state vector?

Answer 1

1) The daily transition matrix is

After a period of 30 days we have

So the required probability that a person with company A remains with A is 0.27714 and the probability that they left is 1-0.27714=0.7229

2) The initial state vector depends on the initial values

This means is the required initial vector

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