Question

Markov Chains

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) Find the steady state vector q of the Markov chain using the fact that P has an eigenvalue of 1 with

corresponding eigenvector q. What are the probable long-term market shares of each company?

Answer 1

1) The matrix of the Markov Chain is

For steady state vector, we must have

Along with (as these are steady state probabilities)

Which has the solution

And so

So the steady state vector is

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