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

0.92 0.04 0.02 0.05 0.895 0.015 0.03 0.065 0.965

After a period of 30 days we have

30 0.92 0.04 0.02 0.27714 0.24431 0.22097 0.05 0.895 0.015 0.21937 0.21138 0.17854 0.50348 0.54431 0.60049 0.03 0.065 0.965

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

0.59 11800 1 0.31 6200 20000 2000 Po 0.1

$\blacksquare$

Hope this was helpful. Please do leave a positive rating if you liked this answer. Thanks and have a good day!